research article cascading failures in weighted...

Research ArticleCascading Failures in Weighted Complex Networks ofTransit Systems Based on Coupled Map Lattices

Ailing Huang12 H Michael Zhang2 Wei Guan1 Yang Yang1 and Gaoqin Zong1

1MOE Key Laboratory for Urban Transportation Complex Systems Theory and Technology Beijing Jiaotong UniversityBeijing 100044 China2Department of Civil and Environment Engineering University of California Davis CA 95616 USA

Correspondence should be addressed to H Michael Zhang hmzhangucdavisedu

Received 24 April 2014 Revised 4 July 2014 Accepted 4 July 2014

Academic Editor Jose R C Piqueira

Copyright copy 2015 Ailing Huang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Study on the vulnerability and robustness of urban public transit networks (PTNs) has great implications for PTNs planning andemergency management particularly considering passengersrsquo dynamic behaviors We made a complex weighted network analysisbased on passenger flow for Beijingrsquos bus stop network and multimodal transit network coupled with bus and urban rail systemsThe analysis shows that there are small-world or scale-free properties in these two networks which make them display differentrobustness under link or node failuresWith consideration of the dynamic flow redistribution we propose amodel based on coupledmap lattices to analyze the cascading failures of these two weighted networks We find that the dynamic flow redistribution cansignificantly improve the tolerance of small-world or scale-free PTN against random faults Because of the coupling of bus and railsystems the multimodal network with scale-free topology and flow distribution structures displays an increasing tolerance evenagainst intentional attack however its cascade is also much more intense once the failure is triggered We find some thresholds oftopological and flow coupling strength in the spreading process which can be exploited to develop strategies to control cascadefailures

1 Introduction

In the last few years the error tolerance and attack vul-nerability of real networks have attracted the attention ofmany researchers since the discovery of the ubiquitoussmall-world [1] and scale-free properties [2] A scale-freenetwork displays a property of heterogeneity with a power-law degree distribution 119901(119896) sim 119896

minus120574 [2] where 119896 is thedegree of a randomly chosen node and 120574 is the scale-freeexponent while a small-world network compared to theformer is more homogeneous with an exponential degreedistribution a greater clustering coefficient and a shorteraverage distance [1] Some empirical studies report that manyreal networksrsquo topological structures including public transitnetworks (PTNs) are found to be scale-free or small-worldfor example the Boston subway network [3 4] the Indianrailway network [5] the Chinese railway network [6] andthe public transport systems in Poland [7] in Germany andin France [8] The scale-free networks become ldquorobust yet

fragilerdquo [9 10] that is these networks are very sensitive tothe intentional removal of central nodes but become robustto the random removal of some other nodesThere have beena few recent empirical studies on the structural robustnessof many real complex networks for example the proteinnetwork [11] the food webs [12] the email network [13]the internet network [14 15] the computer network [16]the cyber-physical network [17] and the uncertain system[18 19]

In addition there exists the physical flow (also definedas load) in real-world networks such as the data transmit-ted in communication networks the electrical currents inpower grids the vehicles in transportation networks and thepassenger flow in PTNs As the physical flow is dynamicthe failure of one or a few nodesedges could lead to thefailures of other nodesedges by the coupling relationshipbetween nodesedges and the redistribution of flow over thewhole networks triggering chain reactions that lead to thecollapse of a few or all nodesedges in the network This

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 940795 16 pageshttpdxdoiorg1011552015940795

2 Mathematical Problems in Engineering

phenomenon is called cascading failure [20 21] The largestblackout event in the northeastern US power grid in August2003 for example is due to cascading failure

Urban PTN composed of routes stops and the passengerflow traveling in network is a complicated giant system Ina PTN if one or some key stations or routes are destroyedby natural disasters or their carrying capacity is in shortsupply due to sudden increases of passenger flow (such asa large number of people coming from stadiums or theatersafter matches or concerts) one part of or even the wholenetwork may be overloaded and out of operation A caselike this occurred on July 21 2012 when the heaviest rainin the last 60 years in Beijing shut down a metro lineand caused 100 bus routes to detour dump stop or stopoperation completely This resulted in more than 200 busroutes extending their service hours to transport passengersAnother example occurred on January 3 2010 when heavysnow caused more than 60 bus routes to detour or be takenout of operation These failures greatly impacted passengertravel and severely disrupted the socialeconomic activities ofthe city These events motivated us to study the dependenceof avalanches and dynamic robustness in PTNs on the initialperturbation network structure and flow behavior

Some models and empirical research on cascading fail-ures in different networks have been reported in the recentliterature and many of them showed that a networkrsquos struc-ture has great impact on its function They include theempirical studies on theUS power grid [22] and ecoindustrialsymbiosis network considering the load redistribution thecascading dynamics in small-world network [23 24] scale-free network [25] weighted complex network [26] andindependent network [27] For PTNs however the majorityof the research focuses on the (static) robustness of networktopology based on the idea of Albert et al [9] by removingone or some nodes or edges in the network A typical exampleis the analysis of attack vulnerability of 14 PTNs in the L-and P-spaces (explained in Figure 1) [28 29] respectivelyTopological vulnerability of other PTN networks such as therailway network of Switzerland [30] the subway network ofShanghai [31] the air flight network of China [32] is alsostudied

The research on the dynamic robustness of PTNs how-ever is still limited Although Wu et al [33 34] and Zhenget al [35] have studied the cascading behavior in scale-freenetworks based on the traffic dynamics theory and Yang etal [36] have simulated the epidemic dynamic behaviors inPTNs based on the SIS spreadmodel the cascading dynamicsin weighted PTNs considering the dynamical passenger flowhas not yet been reported Also the empirical studies onthe dynamical characteristics of weighted PTNs based onpassengerrsquos travel behavior are not given adequate consid-eration As far as we know only the studies of the Seoulsubway system [37] Singapore public transportation system[38] and Beijing bus routes system [39] considered passengerflow in their analysisThe former two studies constructed thecomplex network according to passengersrsquo travel routes butthe underlying physical structure is not considered

In light of the review several questions remain open in thestudy of PTNs considering flow dynamics For example what

is the structure of a real PTN measured in the P-space [4]or L-space [5] How robust are these PTNs against randomfault or intentional attack when considering passenger flowredistribution in the network What are the thresholds forcascading failures and spreading dynamics in different real-world PTN structures In seeking answers to these questionswe were able to identify the threshold to avoid cascadingfailures in PTNs and find the strategies to avoid such failureswhich provides an important theoretical basis for protectingreal PTNs and improving their robustness

In this paper we propose a model to explore thedynamic robustness in actual PTNs of Beijing consideringpassenger flow behavior Here we study not only the busstop network (BSN) but also the multimodal public transitnetwork (MPTN) coupled with bus and urban rail transitsystems which will help us compare the PTNsrsquo robustnesswith different weighted structures The paper is organizedas follows In the next section we will make an analysis ofthe statistical properties of the weighted complex networkof Beijingrsquos BSN and MPTN in L-space In Section 3 acascading failure analysis model for weighted network basedon coupled map lattices is proposed and in Sections 4 and5 the dynamic robustness of BSN and MPTN in Beijing isstudied respectively Conclusions and outlook are given inSection 6

2 Statistical Properties of WeightedComplex Network Derived from BeijingPublic Transit System in L-Space

21 Data Processing and Networks Generation In this workthe weighted complex network for public transit stops inBeijingrsquos bus and rail transit networks is built to explorethe impact of connectivity between stops on passenger flowdistribution Here we use the L-space method [5] to describethe network that is nodes represent bus trolley bus orrail stops and a link between two nodes exists if they areconsecutive stops on a route The weight 119908

119894119895

between node119894 and node 119895 is represented by the sectional passenger flow 119875

119894119895

that represents the number of passengers through the sectionbetween stop 119894 and adjacent stop 119895 in a weekday Althoughthe passenger flow is directed in general the daily flow isbidirectionally balanced for a whole day For simplicity wemake the assumption that passenger flow is undirected (119908

119894119895

=

119908119895119894

) by averaging the in and out edge weightsIn Beijingrsquos bus transit network a flat fare system is

applied in most of bus routes which means commuters onlyswipe the smart card one time when they board the busesso it is difficult to capture the exact data about sectionaloccupational passengers through the intelligent card (IC)system A good way to acquire this information is by theride check We obtained the ride-check data of 100 routes ina weekday from Beijing Public Transportation CorporationThese routes are so typical that they were selected to investi-gate and on each route the occupational passengers of busin each section along the route for more than 200 bus tripsper day were counted According to data of 100 routes stopsand sectional passengers in Beijing we develop a weighted

Mathematical Problems in Engineering 3

67

891

2

3

4

5

Line ALine B

(a) L-space

Line ALine B

5

1

2

3

67

4

98

(b) P-space

Figure 1 Explanation of L-space (a) and P-space (b) Refer to [7]

bus stops network in the L-space with 1388 nodes and 1555edges

In theMPTN both for bus and rail stops based on follow-ing methodology the sectional passenger flow between twoadjacent rail stops will be calculated in terms of passengers inand out of rail stops that are recorded by the IC system

Firstly we use the gravitymodel to estimate the origin anddestination (OD) flow 119902

119894119895

from stop 119894 to 119895

119902119894119895

= 119886119874119894

119863119895

119891 (119909119894119895

) (1)

where 119886 is the trip generation coefficient119874119894

is the passengersgoing into stop 119894 119863

119895

is the passengers out from stop 119895 119891(119909119894119895

)

denotes an impedance function from stop 119894 to stop 119895 119909119894119895

is an impedance parameter As a flat fare is also appliedin the whole urban rail network without any transfer feehere 119909

119894119895

is represented by the travel times between stops 119894and 119895 composed of the train running time dwell time instops and transfer time in transfer stationsThe train runningtime is estimated based on the distance between stops andaverage train running velocity the dwell time and transfertime are obtained by historical data and field investigationsrespectively

Secondly considering the travel times to be theimpedance we assign the OD flow between stops intothe rail network by all-or-nothing assignment through theTransCAD (TransCAD is a commercial software packagethat performs various types of traffic assignment) softwareand then the sectional passenger flow between adjacent stopsis calculated Thus based on the topology and flow data of 13rail lines and 100 bus routes the weighted complex MPTN inthe L-space is established with 1504 nodes and 1728 edges

22 Degree Distribution In the L-space the degree 119896 of anode represents the number of its consecutive neighbor stopsit is linked to It is calculated that the degree interval ofMPTNin Beijing is 119896

119861+119877

isin [1 10] which is wider than that ofBSN 119896

119861

isin [1 8] although their average degrees are quitesimilar ⟨119896

119861

⟩ = 224 and ⟨119896119861+119877

⟩ = 229 indicating that theconnectivity of MPTN has been improved in part because ofthe addition of the rail system As shown in Figure 2(a) thedegree distribution of BSN appears exponentially distributed119901(119896) sim 04636119890

minus03286119896 and thus is small-world which is

consistent with Suirsquos PTN analysis [40] and similar to thedegree distribution exponent of the North American powergrid that is 120574 asymp 05 [4] However for the MPTN (shown inFigure 2(b)) its degree distribution plot is nearly a straightline on a log-log scale (red dots) and follows a power-lawdistribution 119901(119896) sim 1067119896

minus4042 So the MPTN is scale-free after the bus stops network is coupled with the railnetwork It shows that the topology of Beijingrsquos MPTN ismuch more heterogeneous than BSN and some nodes arehighly concentrated due to the limit of two-dimensionalgeographic space This can be explained by the fact that theldquorich get richerrdquo phenomenon exists in PTNs just because busroutes tend to connect with rail stops

23 Strength Distribution The strength 119904119894

of node 119894 is definedas 119904119894

= sum119895

119908119894119895

[41] thus a stoprsquos strength in a PTN representsthe average daily passenger flow passing the sections betweenthis stop and its connected neighboring stops Likewisebecause of the rail system the MPTN possesses wider nodersquosstrength interval 119904

119861+119877

isin [80 1056343e + 06] and higheraverage strength ⟨119904

119861+119877

⟩ = 39500 than that of BSN 119904119861

isin

[64 60918] and ⟨119904119861

⟩ = 9320 As shown in Figure 3 there arelong tails for strength distribution (blue dots) in both BSNandMPTN especially in the latter indicating that some stopstransport quite large volume of passengers every day whichis muchmore than that of most stops It is noted that the stopwith large traffic may differ from the one with large degreeNot considering some stops with very small strengths in theBSN both the strength distributions of BSN and MPTN onlog-log scales are nearly linear and appear to follow power-law distributions 119901(119904) sim 119888119904minus120574 with 120574 = 142 and 120574 = 1209respectivelyTheMPTNrsquos power-law exponent is smaller thanthat of BSN which reveals that the heterogeneity of MPTNhas been enhanced when the bus network is coupled withthe rail networkThe strong heterogeneity ofMPTN indicatesthat there are a few nodes called hubs in the network thathave higher strength than other nodes and play a dominantrole in preserving connections and serving traffic of theoverall network Such heterogeneity makes the scale-freenetwork more robust to random failures but more vulnerableto deliberate attacks [9]

In summary we find that both the topology and flowdistribution structures of MPTN show the scale-free features


k

P(k)

k

1 2 3 4 5 6 7 8

07

06

05

04

03

02

01

0

(a) BSN degree distribution

10minus1

10minus2

10minus3

10minus4

P(k)

k

k

lg(k)

lg(P(k

))

07

06

05

04

03

02

01

01 2 3 4 5 6 7 8 9 10

100

100101

(b) MPTN degree distribution and its log-log scale plot

Figure 2 Node degree distribution for Beijingrsquos BSN and MPTN

s

s

014

012

01

008

006

004

002

00 1 2 3 4 5 6 7

100102 103 104 105

10minus1

10minus2

10minus3

10minus4

lg(s)

P(s)

lg(P(s

))

times104

(a) BSN strength distribution

s

s

103 104 105 106 107

100

10minus1

10minus2

10minus3

10minus4

lg(s)

P(s)

35

3

25

2

15

1

05

00 2 4 6 8 10 12

lg(P(s

))

times105

(b) MPTN strength distribution

Figure 3 Nodersquos strength distribution of Beijingrsquos weekday weighted BSN and MPTN and their log-log scale plot

while the BSN is coupled with small-world topology andscale-free flow distribution which could make them displaydifferent tolerance and robustness under random fault orintentional attack so they will be further studied in thefollowing sections

3 A Cascading Failure Analysis Based onCoupled Map Lattices

31 ACMLsModel forWeightedComplexNetwork Acoupledmap lattice (CML) is a dynamical system that models thebehavior of nonlinear systems (especially partial differentialequations) [42]They are predominantly used to qualitativelystudy the chaotic dynamics of spatially extended systems

Studied systems include populations convection fluid flowchemical reactions and biological networks More recentlyCMLs have been applied to computational networks iden-tifying detrimental attack methods and cascading failuresHowever the current literature of cascading failure in CMLsis almost based on topological network [43ndash48] and thereis few research on static weighted network [49 50] not tomention ones in view of dynamic weight In fact in thereal world not only topology but also the flow distributioninfluences the cascading failure of the network It is oftento see that when the flow is beyond the carrying capacitycascading failures occur for example in the urban trans-portation network the Internet and the power grid networkThere are also similar cascading failures in a PTN namely


the redistribution of passenger flow caused by failures inone or a few stops or sections could lead to overload oroutage of other nodes and then result in the whole networkrsquosfailure In a PTN failures may be triggered by the change ofnetworkrsquos topology or the redistribution of passenger flowwhereas when failures occur the topology and passengerflow are affected at the same time so the occurrence andspread of cascading failures are based on the network coupledwith ldquotopological structurerdquo and ldquoflow distribution structurerdquoTherefore when the passenger flow is loaded on a topologicalPTN as the weight it is significant to study the cascadingfailures and their dynamical behaviors in the weighted PTNwhich is represented by the coupling function betweenldquotopological structurerdquo and ldquoflow distribution structurerdquo indifferent strength

Here we propose a model to describe the cascading fail-ures in weighted networks based on a CML model presentedby Wang and Xu [43 44] The model is demonstrated asfollows

119909119894

(119905 + 1) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1205761

minus 1205762

) 119891 (119909119894

(119905)) + 1205761

119873

sum

119895=1119895 =119894

119886119894119895

119891 (119909119895

(119905))

119896119894

+1205762

119873

sum

119895=1119895 =119894

119908119894119895

119891 (119909119895

(119905))

119904119894

10038161003816100381610038161003816100381610038161003816100381610038161003816

119894 = 1 2 119873

(2)

where 119909119894

(119905) is a state variable of the 119894th node at the 119905th timestepThe connection information among119873 nodes is given bythe adjacencymatrix119860 = (119886

119894119895

)119873times119873

If there is an edge betweennode 119894 and node 119895 119886

119894119895

= 119886119895119894

= 1 otherwise 119886119894119895

= 119886119895119894

= 0119896119894

denotes the degree of node 119894 119908119894119895

represents the weight ofedge between node 119894 and node 119895 119904

119894

is the strength of node 1198941205761

means the coupling strength of topological structure and1205762

represents that of weighted network which is loaded bypassenger flow respectively where 120576

1

1205762

isin (0 1) 1205761

+ 1205762

lt 1The function 119891 demonstrates the local dynamic behaviorswhich is chosen in this work as the chaotic logistic map119891(119909) = 4119909(1 minus 119909) When 0 le 119909 le 1 and 0 le 119891(119909) le 1 weuse absolute value notation in (2) to guarantee that the stateof each node is always nonnegative

Here the nonlinear function 119891 can be defined as theevolution rule of node 119894rsquos capacity According to the mappingmethod of L-space node 119894 is said to be in a normal stateand its flow is smaller than its capacity at the 119897th time stepif 0 lt 119909

119894

(119905) lt 1 119905 le 119897 In contrast if 119909119894

(119897) ge 1 it meansthat the flow is overloaded on node 119894 at the 119897th time step andthen node 119894 is said to be failed and we assume in this casethat 119909

119894

(119905) equiv 0 119905 gt 119897 The state of a node evolves in accordancewith the function if the initial states of all 119873 nodes in thenetwork are in the interval (0 1) and there is not any externalperturbation or the flow surges then all nodes in the networkwill be in a normal state forever and the PTN will operatenormally

32 Methods for Analysis of Dynamic Robustness Based onCML Model In order to show how an initial shock at a

i

k

jwij

wjk rarr wjk + Δwjk

Figure 4 The adjustment rule of 119908119894119895

after node 119894 is failed

single node triggers cascading failures we add an externalperturbation 119877 ge 1 to node 119894 at the119898th time step as follows

119909119894

(119898) = 119877 +

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1205761

minus 1205762

) 119891 (119909119894

(119898 minus 1)) +1205761

+1205761

119873

sum

119895=1119895 =119894

119886119894119895

119891 (119909119895

(119898 minus 1))

119896119894

+1205762

119873

sum

119895=1119895 =119894

119908119894119895

119891 (119909119895

(119898 minus 1))

119904119894

10038161003816100381610038161003816100381610038161003816100381610038161003816

(3)

In this case node 119894 will be failed at the 119898th timestep namely stop 119894 of a PTN is out of operation due tonatural events or other emergencies and then the topologicalstructure will change accordingly and the passenger flow willbe redistributed Therefore we have 119909

119894

(119905) equiv 0 for all 119905 gt 119898At the (119898 + 1)th time step the states of those nodes directlyconnected with node 119894 will be affected by 119909

119894

(119898) according to(2) and their states may also be larger than 1 and thus maylead to a new round of nodesrsquo failures Nodesrsquo failures will leadto a redistribution of the flow and spread out with a continuedprocess

The rule of flow redistribution is like this when node 119894failed node 119894 and the edges connected with it will be excludedfrom the network as shown in Figure 4 and119908

119894119895

the weight ofedge linkedwith node 119895 and the failed node 119894 will be shared byother edges connected with node 119895 so their original weightswill be changed119908

119895119896

rarr 119908119895119896

+Δ119908119895119896

these edges share the flowaccording to theirweightsΔ119908

119895119896

= 119908119894119895

lowast119908119895119896

119904119895

Generally thisrule exists in real PTNs because passengers would be muchmore likely to choose the nearby transit services to escapefrom the confusion and accident as soon as possible when astop or route suffers from disaster or attack

During the spreading process of cascading failures theproblem needed to be studied is as follows what is therelationship between the scale of cascading failures and thestrength of external perturbation in real weighted PTNs Isthere any threshold to avoid the cascading failures What isthe relationship between the spread of cascading failures anddifferent coupling strength of topology and flow distributionstructures under different attacks And what are the similar-ities and dissimilarities of the cascading failures in differentPTN structures

Before following simulations the initial states of nodesare all chosen randomly from the interval (0 1) A pertur-bation with 119877 ge 1 is added to a node 119888 119868(119905) is defined asthe total number of failed nodes in the network before the119905th time step and failure proportion is the ratio of 119868(119905) to


the networkrsquos scale 119873 The failed nodes and failed edges areexcluded from the network and flow is redistributed in termsof rule of Figure 4 then the spreading process of failures canbe observed in the network When there are no failures 119868is defined as the scale of the failures in the network 119868 equivlim119905rarrinfin

119868(119905)

4 Analysis of Cascading Failures inBeijingrsquos Weighted BSN

In Sections 4 and 5 we will take Beijingrsquos BSN and MPTNsystems as cases with consideration ofmodes of random faultand intentional attack to analyze their tolerance and robust-ness with the above CMLs model for weighted network Inthe followingmode of random fault all simulation results areaverages over 50 random realizations

41 Threshold for Cascading Failure

411 When a Randomly Chosen Node Fails Here underrandom fault the thresholds to trigger global cascadingfailures in Beijingrsquos BSN are studied with and without flowredistribution respectively based on ourCMLsmodel Given1205761

= 1205762

= 025 a perturbation is added to a randomlyselected node As shown in Figure 5(a) considering flowredistribution based on the rule shown in Figure 4 when119877 gt 119877

119877

1198611

= 3 failures start to be triggered in thenetwork however when not considering flow redistributionthe threshold 119877119877

1198612

= 2 becomes smaller (Figure 5(b)) Itreflects that the passengersrsquo travel behaviors with autonomyand flexibility can reduce the risk of cascade failure in theBSN At the same time we find that since the topology ofBeijingrsquos BSN is small-world and the weighted one is scale-free the thresholds 119877119877

1198611

1198771198771198612

are both relatively small whichis in accordance with the results of [43 44] that cascadingfailures are much easier to occur in small-world and scale-free topological CMLs than in global CMLs

As shown in Figure 6 we find that there is a normaldistribution relation between the scale and time step ofcascading failures When considering flow redistribution(Figure 6(a)) no matter how much 119877 changes the failure istriggered earlier peak times are concentrated in a smallerrange 15 lt 119905

1199011

lt 25 and nodes are all failed in anearlier time step 119905119877

1

(119890) = 80 comparing with that withoutflow redistribution (Figure 6(b)) It implies that the flowredistribution has the potential to enhance the ability ofsynchronization in bus transit system

412 When a Specified Node Is Intentionally Attacked Herean intentional perturbation is added to nodes with the largestdegree or strength respectively namely the most importantstations are intentionally attacked in the BSN which areGuangrsquoanmennei station with 119896 = 8 and Military Museumstation with 119904 = 60918 Similarly we set 120576

1

= 1205762

= 025 Whenthe stop with the largest degree is attacked (Figure 7(a))considering flow redistribution the threshold of failure is119877119870

119861

= 2 which is smaller than that of the stop with the largest

strength 119877119878119861

= 3 (Figure 8(a)) indicating that the topologyis more vulnerable than the flow distribution structure Thethreshold is less than or equal to that of random fault namely119877119870

119861

lt 119877119877

1198611

but 119877119878119861

= 119877119877

1198611

revealing that topological structureis more vulnerable against intentional attack but that ofthe flow distribution structure surprisingly keeps the samerobustness against random fault or intentional attack Wealso find that the thresholds of degree-based and strength-based attacks are the same with those of the case withoutconsidering flow redistribution respectively (Figures 7(b)and 8(b)) which reflects that when the most central nodes inthe network are intentionally attacked though passengers canchange their travel routes the flow changes in a small scopeare not sufficient to reduce the impact of failures because thedamage is very huge and failures spread much faster

The further analysis of Figure 7(a) presents that when thedegree-based attack occurs and 119877 gt 119877119870

119861

there is a diversityfor the failure proportion distribution and peak time withthe change of 119877 When the perturbation is small that is119877 = 3 4 the failures do not spread very quickly at first until119905 = 20 and there is a turning point that the failures increasesharply before which there could be a good opportunity tocontrol failure to spread further but when 119877 ge 5 it seemsthat the spreading processes are synchronized with different119877 and the peak proportions are all 119901

119896

asymp 007 at a similarpeak time 119905

119901119896

= 12 However when the strength-basedattack triggers failures (Figure 8(a)) the spreading speeds aresimilar for different 119877 values and the peak proportions are119901119904

isin (006 007) in a quite small time interval 119905119901

119904

isin [12 18]indicating that once failure is triggered the perturbationstrength has little effect on the failure spreading process forthe weighted BSN

42 Effect of Flow Coupling Coefficient 1205762

The following willstudy the impacts of different flow coupling coefficients 120576

2

on the robustness of Beijingrsquos BSN Firstly a perturbation119877 = 5 gt 119877

119877

1198611

is added to a randomly selected node with1205761

= 025 and 1205762

taking the values of 015 025 035 045and 055 respectively Both Figures 9(a) and 9(b) show thaton the whole with 120576

2

increasing the failures occur earlier andspread faster with and without flow redistribution Howeverwhen considering passengers change their travel routes 120576

2

has less influence on spreading processes of failures (shown inFigure 9(a)) comparing with that without flow redistribution(shown in Figure 9(b)) and there is less diversity between theinitial occurrence times or spreading speeds of failures withdifferent 120576

2

In addition as shown in Figure 9(a) the failuresspread faster These characteristics imply that because of theflow redistribution in network it is equivalent to increasingthe coupling strength further between other stops and thenpromoting the spread of failure It is important to note thatwhen 120576

2

le 1205761

the peak proportion with flow redistributionis smaller than that of without it indicating that flexibilityof passengersrsquo travel behaviors will be likely to alleviate thedamage of perturbation under a small flow coupling strength

For the case of intentional attack with the same param-eter set as the above a perturbation 119877 = 5 gt 119877

119870

119861

and119877119878

119861

is added to the node with the largest degree or strength


0 20 40 60 80 1000

02

04

06

08

1Cu

mul

ativ

e fai

lure

pro

port

ion

1205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

(a) Considering flow redistribution

0 10 20 30 40 50 60 70 80 90 1000

010203040506070809

1

R = 2

R = 3

R = 4

R = 5

R = 6

Cum

ulat

ive f

ailu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step

(b) Without considering flow redistribution

Figure 5 When 1205761

= 1205762

a randomly chosen node fails the spreading process of cascading failure with different perturbations 119877 in Beijingrsquosweighted BSN

0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

01

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762

a randomly chosen node fails the relationship between failure scale and time step with different perturbations 119877 inBeijingrsquos weighted BSN

respectively The simulations (shown in Figure 10) show thatwhether the stop with the largest degree or strength isintentionally attacked the spreading processes of failures arealmost synchronized for different 120576

2

that is the peak timesall occur in a small range 119905

119901

119896

asymp 119905119901

119904

isin [12 16] and theirfailure proportions are 119901

119896

isin [007 008] and 119901119904

isin [006 007]respectively which reflects that although the topologicalnetwork is small-world and the flow distribution is scale-free the flow dynamic behaviors may nearly eliminate thecascading difference between these two structures The time

step that all of nodes are failed is 119905119896

(119890) = 74 when the largestdegree stop is attacked which is later than that of the largeststrength one that is 119905

119904

(119890) = 69 but they are both earlier thanthat in the mode of random fault 119905(119890) = 82 revealing thatthe spreading process is the most intense when the largeststrength node is intentionally attacked

43 Effect of Topological Coupling Coefficient 1205761

Here westudy the impacts of different topological coupling coeffi-cients 120576

1

on the network Assume that 119877 = 5 1205762

= 025 and 1205761


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n1205761 = 1205762 = 025

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762

the node with the largest degree is intentionally attacked the relationship between failure scale and time step withdifferent perturbations 119877 in Beijingrsquos weighted BSN

0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762

the node with the largest strength is intentionally attacked the relationship between failure scale and time step withdifferent perturbations 119877 in Beijingrsquos weighted BSN

takes the values of 015 025 035 045 and 055 respectivelyWhen a perturbation is added to a randomly selected nodeaccording to Figure 11 we find that when considering flowredistribution the spreading process of failures is muchconsistent and less diverse compared with that without flowredistribution and both the failure peak times 119905

119901

119903

isin [18 20]

and proportions 119901119903

isin [007 009] occur in a small range Itshows that in random fault mode because of the couplingfunction by flow redistribution the impact of topological

coupling strength on the spreading process of failure has beenmitigated

When the node with the largest degree or strength isintentionally attacked as shown in Figure 12 if 120576

1

lt 1205762

thefailures spread more slowly correspondingly and there aresome fluctuations for failure proportion at certain time stepsunlike other cases which failures always increase before thepeak and the time 119905

119901

119896

= 33 that reaches the failure peakunder the degree-based attack (Figure 12(a)) is later than that


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

011205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step


Figure 9 When a randomly chosen node fails the relationship between failure scale and time step with different flow coupling coefficients1205762

in Beijingrsquos weighted BSN

0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step

(a) Node with the largest degree is intentionally attacked

0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

0071205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step

(b) Node with the largest strength is intentionally attacked

Figure 10 When a specified node is intentionally attacked considering flow redistribution the relationship between failure scale and timestep with different flow coupling coefficients 120576

2


of strength-based attack 119905119901

119904

= 22 (Figure 12(b)) Howeverwhen 120576

1

gt 1205762

and 1205761

+ 1205762

gt 05 the spreading processesare nearly consistent for these two attacks that is the peaktimes are 119905

119901

119896

asymp 119905119901

119904

asymp 15 and the peak proportion for degree-based attack is 119901

119896

isin [007 008] and for strength-based is119901119904

isin [006 007] the times when all nodes fail are 119905119896

(119890) = 74

and 119905119904

(119890) = 69 respectively which are also earlier than thatof random fault mode To sum up when 120576

1

lt 1205762

intentional

attack to the most central node in weighted BSN (strength-based attack) will trigger a more intense spreading process offailure than that in the topology (degree-based attack) butwhen 120576

1

gt 1205762

hardly does 1205761

influence the failure occurrencetime spreading speed and scale implying that in this caseto optimize the flow distribution structure for example bymaking it more homogenous or evacuating the passengersin the most central stop maybe is much more significant


0 10 20 30 40 50 60 70 80 90 1000

001002003004005006007008009

011205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

0091205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n

Time step


Figure 11 When a randomly chosen node fails the relationship between failure scale and time step with different topological couplingcoefficients 120576

1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Failu

re p

ropo

rtio

n

Time step

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205762 = 025 R = 5


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Failu

re p

ropo

rtio

n

Time step

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205762 = 025 R = 5


Figure 12 When a specified node is intentionally attacked considering flow redistribution the relationship between failure scale and timestep with different topological coupling coefficients 120576

1


since the topological coupling strength has little impact onthe cascade while the flow redistribution can influence thespread of failure

5 Analysis of Cascading Failures inBeijingrsquos Weighted MPTN

51 Threshold for Cascading Failure Here the thresholds forcascading failures in MPTN are studied with consideration

of flow redistribution and the parameters are set the sameas those in Section 41 As shown in Figure 13 in the randomfault mode when 119877 gt 119877119877

119861+119877

= 2 failures start to be triggeredin the network representing the fact that the threshold ofMPTN 119877119877

119861+119877

= 2 is smaller than that of BSN 1198771198771198611

= 3It reveals that the MPTN is more vulnerable than that ofBSN under random fault because both the topology andflow distribution of MPTN display much more heterogeneityaccording to our analysis in Sections 22 and 23 which isconsistent with earlier works on cascading failures [21 44]


0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 1205762 = 025

R = 2

R = 3

R = 4

R = 5

R = 6

(a) Spreading process of failures

0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

1205761 = 1205762 = 025

Failu

re p

ropo

rtio

n(b) Relationship between failure scale and time step


= 1205762

and a randomly chosen node fails considering flow redistribution the threshold for cascading failure with differentperturbations 119877 in Beijingrsquos weighted MPTN

0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

0091205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n



= 1205762

a specified node is intentionally attacked considering flow redistribution the relationship between failure scaleand time step with different perturbations 119877 in Beijingrsquos weighted MPTN

where it has been shown that large-scale cascading failuresare much less likely to happen in a homogeneous networkthan in a heterogeneous network In addition as shown inFigure 13(b) when 119877 gt 119877

119877

119861+119877

the initial occurrence timeof failure becomes earlier while peak proportion could besmaller with the increase of 119877 for example 119877 = 4 or 5indicating that the perturbation strength is not linear with thepeak failure

When the node with the largest degree or strength isattacked respectively (Figure 14) that is Sanyuanqiao stationwith 119896 = 10 and Jianguomen station with 119904 = 1056343the thresholds of failures in MPTN are the same as 119877119870

119861+119877

=

119877119878

119861+119877

= 3 Also they are the same with BSN under strength-based attack 119877119878

119861

= 3 larger than BSN under degree-basedattack 119877119870

119861

= 2 and even larger than itself under randomfault 119877119877

119861+119877

= 2 which implies that because of the flow


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055


0 20 40 60 80 1000

0005

001

0015

002

0025

003

0035

004

0045

005

Time step

Failu

re p

ropo

rtio

n

1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

(b) Relationship between failure scale and time step

Figure 15When a randomly chosen node fails considering flow redistribution the spreading process of cascading failure with different flowcoupling coefficients 120576

2

in Beijingrsquos weighted MPTN

redistribution and the coupling function with two transitmodes of bus and rail though the most important station isattacked the ability of MPTN to resist perturbation can besurprisingly improved At the same time we find that unlikeBSN since both topology and flow structures of MPTNare more heterogeneous whether the node with the largestdegree or strength is attacked the initial failure is muchmoreeasily triggered with the increase of 119877 and failure spreadsfaster than that of random fault mode It also shows that thespreading process under strength-based attack is much fasterand more intense than that under degree-based attack as theflow distribution ofMPTN ismuchmore heterogeneous thanthe topology


As shown inFigure 15 a perturbation119877 = 5 is added to a randomly chosennode When 120576

2

= 035 failure spreads more slowly thanthose with other values of 120576

2

and the peak comes later itsproportion 119901119877

119861+119877

(1205762

= 035) = 0034 is smaller or equal tothosewith other values of 120576

2

(119901119877119861+119877

isin [0034 005]) indicatingthere could be an opportunity to control the cascading failureWe also find that unlike BSN there is no trend showing that1205762

is linearly correlated with time step and failure proportionin the MPTN and when 120576

2

increases the dynamic flowredistribution in the network even can delay the failure initialtime and reduce the peak scale because of the heterogeneityof MPTN

When the node with the largest degree is attacked (shownin Figure 16(a)) as 120576

1

is being kept at the same value there isa threshold 120576119870

lowast

2

= 04 which means that when 1205762

isin (0 04] or1205762

isin (04 1) no matter how much 1205762

changes the spreadingprocesses of failures present two synchronous curves and

the spreading processes with 1205762

isin (0 04] and those with1205762

isin (04 1) are spectacularly consistent at the same timeFigure 16(a) indicates that the failure initial times are earlierand the average peaks of failures are larger with 120576

2

isin (04 1)

than those with 1205762

isin (0 04]This weak sensitivity of effect on cascading failure from 120576

2

becomes more prominent when the largest strength node isattacked As shown in Figure 16(b) the spreading processesare always the same and totally not affected by 120576

2

valuesAccording to the relationship between failure scale and timestep under strength-based attack the cascading failure ismuch more intense than that under degree-based attack andthe failure peak occurs much earlier that is 119905

119901

119904

= 12 lt 119905119901

119896

=

29 (when 1205762

isin (0 04]) or 21 (when 1205762

isin (04 1)) with higherpeak value 119901

119904

= 0086 gt 119901119896

= 006 (when 1205762

isin (0 04]) or0072 (when 120576

2

isin (04 1)) the time of network failed globallyis also earlier 119905

119904

(119890) = 63 lt 119905119896

(119890) = 78 which is also earlierthan the time 119905(119890) = 92 in random fault


In randomfault mode as shown in Figure 17 similar to BSN withconsideration of flow redistribution 120576

1

has less effect onthe spreading processes of failures in MPTN with differentvalues and the peak times and peak proportions of failuresappear in a small range that is 119905119877

119861+119877

isin [14 24] and 119901119877119861+119877

isin

[0062 0078] Notably as shown in Figure 17(b) when a largevalue of 120576

1

is set as 1205761

= 055 the failure peak time is relativelylate and its proportion is the smallest indicating that the flowredistribution in a topology with high coupling strength mayplay a role to relieve the impact of cascading failure

Figure 18 plots the spreading process of cascading failurewhen the node with the largest degree is intentionallyattacked with different 120576

1

and 1205762

= 015 or 1205762

= 025 We


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035 1205762 = 065

1205762 = 04

1205762 = 045

1205762 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205761 = 025 R = 5

Failu

re p

ropo

rtio

n

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055


Figure 16 When a specified node is intentionally attacked considering flow redistribution the spreading process of cascading failures withdifferent flow coupling coefficients 120576

2

in the Beijingrsquos weighted MPTN

0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

Failu

re p

ropo

rtio

n

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055


Figure 17 When a randomly chosen node fails considering flow redistribution the spreading process of cascading failure with differenttopological coupling coefficients 120576

1


find that there is a threshold of 120576119870lowast

1

= 045 namely when 1205761

ge

045 whatever 1205761

increases the spreading processes reach asynchronizationmeanwhile when 120576

1

ge 045 the initial timesof failures are earlier than those when 120576

1

lt 045 and theaverage peak proportion is also relatively larger Likewiseunder strength-based attack (shown in Figure 19) there isalso a synchronization for spreading process with a threshold

120576119878

lowast

1

= 025 and when 1205761

ge 025 the initial and peak times offailures are earlier and the peak value is larger

Similarly we find that no matter how topological cou-pling strength 120576

1

changes the spreading process of cascadingfailure is the most intense when the node with the largeststrength is attacked the threshold 120576119878

lowast

1

= 025 in the casethat the synchronization of cascade occurs is smaller than


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205762 = 015 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205761 = 065

(a) When 1205762

= 015

0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n(b) When 120576

2

= 025 relationship between failure scale and time step

Figure 18When the nodewith the largest degree is intentionally attacked considering flow redistribution the spreading process of cascadingfailure with different topological coupling coefficients 120576

1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n

Figure 19 When the node with the largest strength is intentionallyattacked considering flow redistribution the spreading process ofcascading failures with different topological coupling coefficients 120576

1


that of 120576119870lowast

1

= 045 when the largest degree one is attackedafter occurrence of synchronization the peak time 119905

119901

119904

= 12

under strength-based attack is earlier than that of 119905119901

119896

= 17

under degree-based attack and the peak proportion is higher119901119904

= 0085 gt 119901119896

= 0072 It reveals that when the mostimportant station in a weighted network is attacked eventhough the topology is not coupled so much the MPTN will

suffer much heavier damage as both of its topology and flowdistribution structures are scale-free which is consistent withthe conclusions of [2 43 44]

6 Conclusions

In summary we find that both weighted BSN and MPTNin Beijing are scale-free networks while the topology of theformer is small-world and that of the latter is scale-free whichmake them show different robustness against different kindof faults or attacks According to the thresholds for cascadingfailures we find that when considering the dynamic flowbehavior the network displays a higher tolerance against ran-dom fault than that without flow redistribution Particularlyin the weighted PTN based on passenger flow as a scale-free network even the station with the largest strength isintentionally attacked because of the dynamic flow redis-tribution the network displays a surprisingly increasingdegree of robustness comparedwith topological networkwithsmall-world structure against intentional attack or even withscale-free one against random fault This should encouragethe development of multimodal transit networks in largemetropolitan areas It is the passengersrsquo choice behaviors thatplay an important role to improve the robustness of PTN Itcan explain why real PTNs rarely suffer complete breakdownsdespite frequent disruptions from natural and man-madedisasters

However once cascading failures are triggered in thenetwork we find that when the station with the largeststrength is intentionally attacked the damage to the networkis much heavier than that of the largest degree-based attackalso heavier than that of a randomly chosen node failedThisreminds us that we should pay more attention to the most


central node (station) that is the onewith the largest strengthin a weighted network to control the cascading failure sincethe control window is short when cascading failure occursAs the weighted PTN in Beijing is a scale-free networkthese characteristics of vulnerability are consistent with thediscoveries of Motter [21]

In essence the flow redistribution has the ability topromote the synchronization during the spread of failureUnder intentional attack although there is a certain degree ofdiversity for the spreading process with different topologicalcoupling strength 120576

1

some synchronizations tend to occurHowever the flow coupling strength 120576

2

has little impact on thefailure spread especially for strength-based attack and thesynchronizations are much more likely to appear indicatingthat when the most important station in the weighted PTNis attacked the dynamic changes and redistribution of pas-senger flow can intensify the spreading process of cascadingfailure no matter how much the initial flow is coupledFortunately we can find some key thresholds of 120576

1

and 1205762

that lead to synchronizations ormake the cascading failure betriggered later or spread more slowly which will help us findstrategies to contain the spreading and avoid further damage

The flow redistribution behavior adopted in this workis still primitive More realistic behavior adaptations ofmajor failures in transportation systems such as the globalredistribution of OD flows should be explored In additionwe only considered failures in nodes while in the real worldfailures in links also existWhen a route or route link that goesthrough a node fails the node (and all connecting edges) doesnot necessarily have to be excluded from the network but onlythe failed links Link failures in transportation system will beexplored in our future study Last but not least further studiesto better understand the complicated intrinsic propertiesof cascading failures and the essence of synchronization inweighted complex public transit networks are also worthpursuing

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported in part by the Fundamental ResearchFunds for the Central Universities of China (Approval no2014JBM071) the Major State Basic Research DevelopmentProgram of China (973 Program) (Grant no 2012CB725403-5) the Key Program of National Natural Science Foundationof China (Grant no 71131001-2) and China ScholarshipCouncil (no 201207095022)

References

[1] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[2] A Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquo Science vol 286 no 5439 pp 509ndash512 1999

[3] V Latora and M Marchiori ldquoIs the Boston subway a small-world networkrdquo Physica A vol 314 no 1 pp 109ndash113 2002

[4] K A Seaton and L M Hackett ldquoStations trains and small-world networksrdquo Physica A vol 339 no 3-4 pp 635ndash644 2004

[5] P Sen S Dasgupta A Chatterjee P A Sreeram G Mukherjeeand S S Manna ldquoSmall-world properties of the Indian railwaynetworkrdquo Physical Review E vol 67 no 3 Article ID 036106 5pages 2003

[6] YWang T Zhou J Shi JWang and D He ldquoEmpirical analysisof dependence between stations in Chinese railway networkrdquoPhysica A vol 388 no 14 pp 2949ndash2955 2009

[7] J Sienkiewicz and J A Holyst ldquoStatistical analysis of 22 publictransport networks in Polandrdquo Physical Review E vol 72 no 4Article ID 046127 11 pages 2005

[8] C Von Ferber T Holovatch Y Holovatch and V PalchykovldquoPublic transport networks empirical analysis and modelingrdquoThe European Physical Journal B vol 68 no 2 pp 261ndash2752009

[9] R Albert H Jeong and A-L Barabasi ldquoError and attacktolerance of complex networksrdquo Nature vol 406 no 6794 pp378ndash382 2000

[10] P Holme B J Kim C N Yoon and S K Han ldquoAttackvulnerability of complex networksrdquo Physical Review E vol 65no 5 Article ID 056109 14 pages 2002

[11] H Jeong S P Mason A-L Barabasi and Z N OltvaildquoLethality and centrality in protein networksrdquo Nature vol 411no 6833 pp 41ndash42 2001

[12] J A Dunne R JWilliams andNDMartinez ldquoNetwork struc-ture and biodiversity loss in food webs robustness increaseswith connectancerdquo Ecology Letters vol 5 no 4 pp 558ndash5672002

[13] M E J Newman S Forrest and J Balthrop ldquoEmail networksand the spread of computer virusesrdquo Physical Review E vol 66no 3 Article ID 035101 4 pages 2002

[14] R Cohen K Erez D Ben-Avraham and S Havlin ldquoResilienceof the Internet to random breakdownsrdquo Physical Review Lettersvol 85 no 21 pp 4626ndash4628 2000

[15] R Cohen K Erez D Ben-Avraham and S Havlin ldquoBreakdownof the internet under intentional attackrdquoPhysical Review Lettersvol 86 no 16 pp 3682ndash3685 2001

[16] B K Mishra and A K Singh ldquoTwo quarantine models on theattack ofmalicious objects in computer networkrdquoMathematicalProblems in Engineering vol 2012 Article ID 407064 13 pages2012

[17] M Li and W Zhao ldquoVisiting power laws in cyber-physicalnetworking systemsrdquo Mathematical Problems in Engineeringvol 2012 Article ID 302786 13 pages 2012

[18] M Hua P Cheng J Fei J Zhang and J Chen ldquoNetwork-based robust 119867

infin

filtering for the uncertain systems withsensor failures and noise disturbancerdquo Mathematical Problemsin Engineering vol 2012 Article ID 945271 19 pages 2012

[19] M Li and W Zhao ldquoOn 1119891 noiserdquo Mathematical Problems inEngineering vol 2012 Article ID 673648 23 pages 2012

[20] V Latora and M Marchiori ldquoEfficient behavior of small-worldnetworksrdquo Physical Review Letters vol 87 no 19 Article ID198701 4 pages 2001

[21] A E Motter and Y Lai ldquoCascade-based attacks on complexnetworksrdquo Physical Review E vol 66 no 6 Article ID 0651024 pages 2002

[22] J Wang and L Rong ldquoCascade-based attack vulnerability onthe US power gridrdquo Safety Science vol 47 no 10 pp 1332ndash13362009


[23] Y Xia J Fan and D Hill ldquoCascading failure in Watts-Strogatzsmall-world networksrdquo Physica A vol 389 no 6 pp 1281ndash12852010

[24] M Babaei H Ghassemieh and M Jalili ldquoCascading failuretolerance of modular small-world networksrdquo IEEE Transactionson Circuits and Systems II vol 58 no 8 pp 527ndash531 2011

[25] J Wang and L Rong ldquoEdge-based-attack induced cascadingfailures on scale-free networksrdquo Physica A vol 388 no 8 pp1731ndash1737 2009

[26] BMirzasoleimanM BabaeiM Jalili andM Safari ldquoCascadedfailures in weighted networksrdquo Physical Review E vol 84 no 4Article ID 046114 8 pages 2011

[27] S V Buldyrev R Parshani G Paul H E Stanley and S HavlinldquoCatastrophic cascade of failures in interdependent networksrdquoNature vol 464 no 7291 pp 1025ndash1028 2010

[28] C von Ferber T Holovatch and Y Holovatch ldquoAttack vulnera-bility of public transport networksrdquo inTraffic andGranular Flowrsquo07 pp 721ndash731 Springer Berlin Germany 2007

[29] B Berche C Von Ferber T Holovatch and Y HolovatchldquoResilience of public transport networks against attacksrdquo TheEuropean Physical Journal B vol 71 no 1 pp 125ndash137 2009

[30] R Dorbritz and U Weidmann ldquoStability of public transporta-tion systems in case of random failures and intended attacks-a case study from Switzerlandrdquo in Proceedings of the 4th IETInternational Conference on Systems Safety Incorporating theSaRS Annual Conference pp 1ndash6 2009

[31] J Zhang X Xu L Hong S Wang and Q Fei ldquoNetworkedanalysis of the Shanghai subway network in Chinardquo Physica Avol 390 no 23-24 pp 4562ndash4570 2011

[32] Y Dang F Ding and F Gao ldquoEmpirical analysis on flightflow network survivability of Chinardquo Journal of TransportationSystems Engineering and Information Technology vol 12 no 6pp 177ndash185 2012

[33] J J Wu Z Y Gao and H J Sun ldquoModel for dynamic trafficcongestion in scale-free networksrdquo Europhysics Letters vol 76no 5 pp 787ndash793 2006

[34] J JWuH J Sun andZ YGao ldquoCascading failures onweightedurban traffic equilibrium networksrdquo Physica A vol 386 no 1pp 407ndash413 2007

[35] J Zheng Z Gao and X Zhao ldquoModeling cascading failuresin congested complex networksrdquo Physica A vol 385 no 2 pp700ndash706 2007

[36] X Yang B Wang S Chen and W Wang ldquoEpidemic dynamicsbehavior in some bus transport networksrdquo Physica A vol 391no 3 pp 917ndash924 2012

[37] K LeeW Jung J S Park andM Y Choi ldquoStatistical analysis ofthe Metropolitan Seoul Subway System network structure andpassenger flowsrdquoPhysicaA vol 387 no 24 pp 6231ndash6234 2008

[38] H Soh S Lim T Zhang et al ldquoWeighted complex networkanalysis of travel routes on the Singapore public transportationsystemrdquo Physica A vol 389 no 24 pp 5852ndash5863 2010

[39] A L Huang W Guan B H Mao and G Z Zang ldquoStatisticalanalysis of weighted complex network in Beijing Public TransitRoutes System based on passenger flowrdquo Journal of Transporta-tion Systems Engineering and Information vol 13 no 6 pp 198ndash204 2013

[40] Y Sui F Shao R Sun and S Li ldquoSpace evolution modeland empirical analysis of an urban public transport networkrdquoPhysica A vol 391 no 14 pp 3708ndash3717 2012

[41] A BarratM Barthelemy R Pastor-Satorras andAVespignanildquoThe architecture of complex weighted networksrdquo Proceedings

of the National Academy of Sciences of the United States ofAmerica vol 101 no 11 pp 3747ndash3752 2004

[42] K Kaneko ldquoOverview of coupled map latticesrdquo Chaos vol 2no 3 pp 279ndash282 1992

[43] X F Wang and J Xu ldquoCascading failures in coupled maplatticesrdquo Physical Review E vol 70 no 5 Article ID 0561132004

[44] J Xu and X F Wang ldquoCascading failures in scale-free coupledmap latticesrdquo Physica A vol 349 no 3-4 pp 685ndash692 2005

[45] D Cui Z Gao and X Zhao ldquoCascades in small-world modularnetworks with CMLrsquoS methodrdquo Modern Physics Letters B vol21 no 30 pp 2055ndash2062 2007

[46] D Cui Z Gao and X Zhao ldquoCascades with coupled map lat-tices in preferential attachment community networksrdquo ChinesePhysics B vol 17 no 5 pp 1703ndash1708 2008

[47] Z J Bao Y J Cao L J Ding G Z Wang and Z X HanldquoSynergetic behavior in the cascading failure propagation ofscale-free coupled map latticesrdquo Physica A vol 387 no 23 pp5922ndash5929 2008

[48] C Di G Zi-You and Z Jian-Feng ldquoTolerance of edge cascadeswith coupled map lattices methodsrdquo Chinese Physics B vol 18no 3 pp 992ndash996 2009

[49] X G Chen J Zhou and Z T Zhu ldquoCascading failures study ofurban traffic system based on CMLrdquo Mathematics in Practicesand Theory vol 39 no 7 pp 79ndash84 2009

[50] X B Lu and B Z Qin ldquoCascading failures in weightedheterogeneous coupled map latticesrdquo in Proceedings of the 1stInternational Conference on Sustainable Power Generation andSupply (SUPERGEN rsquo09) pp 1ndash4 April 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


phenomenon is called cascading failure [20 21] The largestblackout event in the northeastern US power grid in August2003 for example is due to cascading failure

Urban PTN composed of routes stops and the passengerflow traveling in network is a complicated giant system Ina PTN if one or some key stations or routes are destroyedby natural disasters or their carrying capacity is in shortsupply due to sudden increases of passenger flow (such asa large number of people coming from stadiums or theatersafter matches or concerts) one part of or even the wholenetwork may be overloaded and out of operation A caselike this occurred on July 21 2012 when the heaviest rainin the last 60 years in Beijing shut down a metro lineand caused 100 bus routes to detour dump stop or stopoperation completely This resulted in more than 200 busroutes extending their service hours to transport passengersAnother example occurred on January 3 2010 when heavysnow caused more than 60 bus routes to detour or be takenout of operation These failures greatly impacted passengertravel and severely disrupted the socialeconomic activities ofthe city These events motivated us to study the dependenceof avalanches and dynamic robustness in PTNs on the initialperturbation network structure and flow behavior

Some models and empirical research on cascading fail-ures in different networks have been reported in the recentliterature and many of them showed that a networkrsquos struc-ture has great impact on its function They include theempirical studies on theUS power grid [22] and ecoindustrialsymbiosis network considering the load redistribution thecascading dynamics in small-world network [23 24] scale-free network [25] weighted complex network [26] andindependent network [27] For PTNs however the majorityof the research focuses on the (static) robustness of networktopology based on the idea of Albert et al [9] by removingone or some nodes or edges in the network A typical exampleis the analysis of attack vulnerability of 14 PTNs in the L-and P-spaces (explained in Figure 1) [28 29] respectivelyTopological vulnerability of other PTN networks such as therailway network of Switzerland [30] the subway network ofShanghai [31] the air flight network of China [32] is alsostudied

The research on the dynamic robustness of PTNs how-ever is still limited Although Wu et al [33 34] and Zhenget al [35] have studied the cascading behavior in scale-freenetworks based on the traffic dynamics theory and Yang etal [36] have simulated the epidemic dynamic behaviors inPTNs based on the SIS spreadmodel the cascading dynamicsin weighted PTNs considering the dynamical passenger flowhas not yet been reported Also the empirical studies onthe dynamical characteristics of weighted PTNs based onpassengerrsquos travel behavior are not given adequate consid-eration As far as we know only the studies of the Seoulsubway system [37] Singapore public transportation system[38] and Beijing bus routes system [39] considered passengerflow in their analysisThe former two studies constructed thecomplex network according to passengersrsquo travel routes butthe underlying physical structure is not considered

In light of the review several questions remain open in thestudy of PTNs considering flow dynamics For example what

is the structure of a real PTN measured in the P-space [4]or L-space [5] How robust are these PTNs against randomfault or intentional attack when considering passenger flowredistribution in the network What are the thresholds forcascading failures and spreading dynamics in different real-world PTN structures In seeking answers to these questionswe were able to identify the threshold to avoid cascadingfailures in PTNs and find the strategies to avoid such failureswhich provides an important theoretical basis for protectingreal PTNs and improving their robustness

In this paper we propose a model to explore thedynamic robustness in actual PTNs of Beijing consideringpassenger flow behavior Here we study not only the busstop network (BSN) but also the multimodal public transitnetwork (MPTN) coupled with bus and urban rail transitsystems which will help us compare the PTNsrsquo robustnesswith different weighted structures The paper is organizedas follows In the next section we will make an analysis ofthe statistical properties of the weighted complex networkof Beijingrsquos BSN and MPTN in L-space In Section 3 acascading failure analysis model for weighted network basedon coupled map lattices is proposed and in Sections 4 and5 the dynamic robustness of BSN and MPTN in Beijing isstudied respectively Conclusions and outlook are given inSection 6

2 Statistical Properties of WeightedComplex Network Derived from BeijingPublic Transit System in L-Space

21 Data Processing and Networks Generation In this workthe weighted complex network for public transit stops inBeijingrsquos bus and rail transit networks is built to explorethe impact of connectivity between stops on passenger flowdistribution Here we use the L-space method [5] to describethe network that is nodes represent bus trolley bus orrail stops and a link between two nodes exists if they areconsecutive stops on a route The weight 119908

119894119895

between node119894 and node 119895 is represented by the sectional passenger flow 119875

119894119895

that represents the number of passengers through the sectionbetween stop 119894 and adjacent stop 119895 in a weekday Althoughthe passenger flow is directed in general the daily flow isbidirectionally balanced for a whole day For simplicity wemake the assumption that passenger flow is undirected (119908

119894119895

=

119908119895119894

) by averaging the in and out edge weightsIn Beijingrsquos bus transit network a flat fare system is

applied in most of bus routes which means commuters onlyswipe the smart card one time when they board the busesso it is difficult to capture the exact data about sectionaloccupational passengers through the intelligent card (IC)system A good way to acquire this information is by theride check We obtained the ride-check data of 100 routes ina weekday from Beijing Public Transportation CorporationThese routes are so typical that they were selected to investi-gate and on each route the occupational passengers of busin each section along the route for more than 200 bus tripsper day were counted According to data of 100 routes stopsand sectional passengers in Beijing we develop a weighted


67

891

2

3

4

5

Line ALine B

(a) L-space

Line ALine B

5

1

2

3

67

4

98

(b) P-space





119894119895

from stop 119894 to 119895

119902119894119895

= 119886119874119894

119863119895

119891 (119909119894119895

) (1)



119895


)



119894119895




119861+119877


119861


119861

⟩ = 224 and ⟨119896119861+119877







= sum119895

119908119894119895


119861+119877


119861+119877

⟩ = 39500 than that of BSN 119904119861

isin

[64 60918] and ⟨119904119861




k

P(k)

k

1 2 3 4 5 6 7 8

07

06

05

04

03

02

01

0


10minus1

10minus2

10minus3

10minus4

P(k)

k

k

lg(k)

lg(P(k

))

07

06

05

04

03

02

01

01 2 3 4 5 6 7 8 9 10

100

100101



s

s

014

012

01

008

006

004

002

00 1 2 3 4 5 6 7

100102 103 104 105

10minus1

10minus2

10minus3

10minus4

lg(s)

P(s)

lg(P(s

))

times104


s

s

103 104 105 106 107

100

10minus1

10minus2

10minus3

10minus4

lg(s)

P(s)

35

3

25

2

15

1

05

00 2 4 6 8 10 12

lg(P(s

))

times105










119909119894

(119905 + 1) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1205761

minus 1205762

) 119891 (119909119894

(119905)) + 1205761

119873

sum

119895=1119895 =119894

119886119894119895

119891 (119909119895

(119905))

119896119894

+1205762

119873

sum

119895=1119895 =119894

119908119894119895

119891 (119909119895

(119905))

119904119894

10038161003816100381610038161003816100381610038161003816100381610038161003816

119894 = 1 2 119873

(2)

where 119909119894


119894119895

)119873times119873


119894119895

= 119886119895119894

= 1 otherwise 119886119894119895

= 119886119895119894

= 0119896119894



119894




1

1205762

isin (0 1) 1205761

+ 1205762



119894



119894



i

k

jwij





119909119894

(119898) = 119877 +

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1205761

minus 1205762

) 119891 (119909119894

(119898 minus 1)) +1205761

+1205761

119873

sum

119895=1119895 =119894

119886119894119895

119891 (119909119895

(119898 minus 1))

119896119894

+1205762

119873

sum

119895=1119895 =119894

119908119894119895

119891 (119909119895

(119898 minus 1))

119904119894

10038161003816100381610038161003816100381610038161003816100381610038161003816

(3)


119894


119894



119894119895


119895119896

rarr 119908119895119896

+Δ119908119895119896


119895119896

= 119908119894119895

lowast119908119895119896

119904119895






119868(119905)





= 1205762


119877

1198611


1198612


1198611

1198771198771198612



1199011


1



1

= 1205762


119861


strength 119877119878119861


119861

lt 119877119877

1198611

but 119877119878119861

= 119877119877

1198611



119861


119896


119901119896



119904




2


119877

1198611


= 025 and 1205762


2


2


2


2

le 1205761



119870

119861

and119877119878

119861



0 20 40 60 80 1000

02

04

06

08

1Cu

mul

ativ

e fai

lure

pro

port

ion

1205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6


0 10 20 30 40 50 60 70 80 90 1000

010203040506070809

1

R = 2

R = 3

R = 4

R = 5

R = 6

Cum

ulat

ive f

ailu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

01

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762



2


119901

119896

asymp 119905119901

119904


119896

isin [007 008] and 119901119904




119904




1


= 025 and 1205761


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n1205761 = 1205762 = 025

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762



119901

119903

isin [18 20]





1

lt 1205762


119901

119896



0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

011205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step




0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

0071205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step



2



119904


1

gt 1205762

and 1205761

+ 1205762


119901

119896

asymp 119905119901

119904


119896



(119890) = 74

and 119905119904


1

lt 1205762

intentional


1

gt 1205762

hardly does 1205761



0 10 20 30 40 50 60 70 80 90 1000

001002003004005006007008009

011205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

0091205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n

Time step



1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Failu

re p

ropo

rtio

n

Time step

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205762 = 025 R = 5


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Failu

re p

ropo

rtio

n

Time step

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205762 = 025 R = 5



1






119861+119877


119861+119877




0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 1205762 = 025

R = 2

R = 3

R = 4

R = 5

R = 6


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

1205761 = 1205762 = 025

Failu

re p

ropo

rtio



= 1205762


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

0091205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n



= 1205762



119877

119861+119877



119861+119877

=

119877119878

119861+119877


119861


119861


119861+119877



0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055


0 20 40 60 80 1000

0005

001

0015

002

0025

003

0035

004

0045

005

Time step

Failu

re p

ropo

rtio

n

1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055



2





2


2


119861+119877

(1205762


2

(119901119877119861+119877



2



1


lowast

2


isin (0 04] or1205762






2

isin (04 1)



2


2


119901

119904

= 12 lt 119905119901

119896

=

29 (when 1205762

isin (0 04]) or 21 (when 1205762


119904

= 0086 gt 119901119896

= 006 (when 1205762

isin (0 04]) or0072 (when 120576

2


119904

(119890) = 63 lt 119905119896




1


119861+119877

isin [14 24] and 119901119877119861+119877

isin


1

is set as 1205761



1

and 1205762

= 015 or 1205762

= 025 We


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035 1205762 = 065

1205762 = 04

1205762 = 045

1205762 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205761 = 025 R = 5

Failu

re p

ropo

rtio

n

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055



2


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

Failu

re p

ropo

rtio

n

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055



1



1


ge



1


1


120576119878

lowast

1

= 025 and when 1205761



1


lowast

1



0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205762 = 015 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205761 = 065

(a) When 1205762

= 015

0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n(b) When 120576

2



1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n


1



1


119901

119904

= 12


119896

= 17


= 0085 gt 119901119896



6 Conclusions






1


2


1

and 1205762





Acknowledgments


References



















infin











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










67

891

2

3

4

5

Line ALine B

(a) L-space

Line ALine B

5

1

2

3

67

4

98

(b) P-space





119894119895

from stop 119894 to 119895

119902119894119895

= 119886119874119894

119863119895

119891 (119909119894119895

) (1)



119895


)



119894119895




119861+119877


119861


119861

⟩ = 224 and ⟨119896119861+119877







= sum119895

119908119894119895


119861+119877


119861+119877

⟩ = 39500 than that of BSN 119904119861

isin

[64 60918] and ⟨119904119861




k

P(k)

k

1 2 3 4 5 6 7 8

07

06

05

04

03

02

01

0


10minus1

10minus2

10minus3

10minus4

P(k)

k

k

lg(k)

lg(P(k

))

07

06

05

04

03

02

01

01 2 3 4 5 6 7 8 9 10

100

100101



s

s

014

012

01

008

006

004

002

00 1 2 3 4 5 6 7

100102 103 104 105

10minus1

10minus2

10minus3

10minus4

lg(s)

P(s)

lg(P(s

))

times104


s

s

103 104 105 106 107

100

10minus1

10minus2

10minus3

10minus4

lg(s)

P(s)

35

3

25

2

15

1

05

00 2 4 6 8 10 12

lg(P(s

))

times105










119909119894

(119905 + 1) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1205761

minus 1205762

) 119891 (119909119894

(119905)) + 1205761

119873

sum

119895=1119895 =119894

119886119894119895

119891 (119909119895

(119905))

119896119894

+1205762

119873

sum

119895=1119895 =119894

119908119894119895

119891 (119909119895

(119905))

119904119894

10038161003816100381610038161003816100381610038161003816100381610038161003816

119894 = 1 2 119873

(2)

where 119909119894


119894119895

)119873times119873


119894119895

= 119886119895119894

= 1 otherwise 119886119894119895

= 119886119895119894

= 0119896119894



119894




1

1205762

isin (0 1) 1205761

+ 1205762



119894



119894



i

k

jwij





119909119894

(119898) = 119877 +

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1205761

minus 1205762

) 119891 (119909119894

(119898 minus 1)) +1205761

+1205761

119873

sum

119895=1119895 =119894

119886119894119895

119891 (119909119895

(119898 minus 1))

119896119894

+1205762

119873

sum

119895=1119895 =119894

119908119894119895

119891 (119909119895

(119898 minus 1))

119904119894

10038161003816100381610038161003816100381610038161003816100381610038161003816

(3)


119894


119894



119894119895


119895119896

rarr 119908119895119896

+Δ119908119895119896


119895119896

= 119908119894119895

lowast119908119895119896

119904119895






119868(119905)





= 1205762


119877

1198611


1198612


1198611

1198771198771198612



1199011


1



1

= 1205762


119861


strength 119877119878119861


119861

lt 119877119877

1198611

but 119877119878119861

= 119877119877

1198611



119861


119896


119901119896



119904




2


119877

1198611


= 025 and 1205762


2


2


2


2

le 1205761



119870

119861

and119877119878

119861



0 20 40 60 80 1000

02

04

06

08

1Cu

mul

ativ

e fai

lure

pro

port

ion

1205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6


0 10 20 30 40 50 60 70 80 90 1000

010203040506070809

1

R = 2

R = 3

R = 4

R = 5

R = 6

Cum

ulat

ive f

ailu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

01

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762



2


119901

119896

asymp 119905119901

119904


119896

isin [007 008] and 119901119904




119904




1


= 025 and 1205761


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n1205761 = 1205762 = 025

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762



119901

119903

isin [18 20]





1

lt 1205762


119901

119896



0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

011205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step




0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

0071205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step



2



119904


1

gt 1205762

and 1205761

+ 1205762


119901

119896

asymp 119905119901

119904


119896



(119890) = 74

and 119905119904


1

lt 1205762

intentional


1

gt 1205762

hardly does 1205761



0 10 20 30 40 50 60 70 80 90 1000

001002003004005006007008009

011205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

0091205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n

Time step



1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Failu

re p

ropo

rtio

n

Time step

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205762 = 025 R = 5


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Failu

re p

ropo

rtio

n

Time step

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205762 = 025 R = 5



1






119861+119877


119861+119877




0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 1205762 = 025

R = 2

R = 3

R = 4

R = 5

R = 6


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

1205761 = 1205762 = 025

Failu

re p

ropo

rtio



= 1205762


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

0091205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n



= 1205762



119877

119861+119877



119861+119877

=

119877119878

119861+119877


119861


119861


119861+119877



0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055


0 20 40 60 80 1000

0005

001

0015

002

0025

003

0035

004

0045

005

Time step

Failu

re p

ropo

rtio

n

1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055



2





2


2


119861+119877

(1205762


2

(119901119877119861+119877



2



1


lowast

2


isin (0 04] or1205762






2

isin (04 1)



2


2


119901

119904

= 12 lt 119905119901

119896

=

29 (when 1205762

isin (0 04]) or 21 (when 1205762


119904

= 0086 gt 119901119896

= 006 (when 1205762

isin (0 04]) or0072 (when 120576

2


119904

(119890) = 63 lt 119905119896




1


119861+119877

isin [14 24] and 119901119877119861+119877

isin


1

is set as 1205761



1

and 1205762

= 015 or 1205762

= 025 We


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035 1205762 = 065

1205762 = 04

1205762 = 045

1205762 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205761 = 025 R = 5

Failu

re p

ropo

rtio

n

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055



2


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

Failu

re p

ropo

rtio

n

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055



1



1


ge



1


1


120576119878

lowast

1

= 025 and when 1205761



1


lowast

1



0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205762 = 015 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205761 = 065

(a) When 1205762

= 015

0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n(b) When 120576

2



1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n


1



1


119901

119904

= 12


119896

= 17


= 0085 gt 119901119896



6 Conclusions






1


2


1

and 1205762





Acknowledgments


References



















infin











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










k

P(k)

k

1 2 3 4 5 6 7 8

07

06

05

04

03

02

01

0


10minus1

10minus2

10minus3

10minus4

P(k)

k

k

lg(k)

lg(P(k

))

07

06

05

04

03

02

01

01 2 3 4 5 6 7 8 9 10

100

100101



s

s

014

012

01

008

006

004

002

00 1 2 3 4 5 6 7

100102 103 104 105

10minus1

10minus2

10minus3

10minus4

lg(s)

P(s)

lg(P(s

))

times104


s

s

103 104 105 106 107

100

10minus1

10minus2

10minus3

10minus4

lg(s)

P(s)

35

3

25

2

15

1

05

00 2 4 6 8 10 12

lg(P(s

))

times105










119909119894

(119905 + 1) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1205761

minus 1205762

) 119891 (119909119894

(119905)) + 1205761

119873

sum

119895=1119895 =119894

119886119894119895

119891 (119909119895

(119905))

119896119894

+1205762

119873

sum

119895=1119895 =119894

119908119894119895

119891 (119909119895

(119905))

119904119894

10038161003816100381610038161003816100381610038161003816100381610038161003816

119894 = 1 2 119873

(2)

where 119909119894


119894119895

)119873times119873


119894119895

= 119886119895119894

= 1 otherwise 119886119894119895

= 119886119895119894

= 0119896119894



119894




1

1205762

isin (0 1) 1205761

+ 1205762



119894



119894



i

k

jwij





119909119894

(119898) = 119877 +

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1205761

minus 1205762

) 119891 (119909119894

(119898 minus 1)) +1205761

+1205761

119873

sum

119895=1119895 =119894

119886119894119895

119891 (119909119895

(119898 minus 1))

119896119894

+1205762

119873

sum

119895=1119895 =119894

119908119894119895

119891 (119909119895

(119898 minus 1))

119904119894

10038161003816100381610038161003816100381610038161003816100381610038161003816

(3)


119894


119894



119894119895


119895119896

rarr 119908119895119896

+Δ119908119895119896


119895119896

= 119908119894119895

lowast119908119895119896

119904119895






119868(119905)





= 1205762


119877

1198611


1198612


1198611

1198771198771198612



1199011


1



1

= 1205762


119861


strength 119877119878119861


119861

lt 119877119877

1198611

but 119877119878119861

= 119877119877

1198611



119861


119896


119901119896



119904




2


119877

1198611


= 025 and 1205762


2


2


2


2

le 1205761



119870

119861

and119877119878

119861



0 20 40 60 80 1000

02

04

06

08

1Cu

mul

ativ

e fai

lure

pro

port

ion

1205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6


0 10 20 30 40 50 60 70 80 90 1000

010203040506070809

1

R = 2

R = 3

R = 4

R = 5

R = 6

Cum

ulat

ive f

ailu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

01

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762



2


119901

119896

asymp 119905119901

119904


119896

isin [007 008] and 119901119904




119904




1


= 025 and 1205761


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n1205761 = 1205762 = 025

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762



119901

119903

isin [18 20]





1

lt 1205762


119901

119896



0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

011205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step




0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

0071205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step



2



119904


1

gt 1205762

and 1205761

+ 1205762


119901

119896

asymp 119905119901

119904


119896



(119890) = 74

and 119905119904


1

lt 1205762

intentional


1

gt 1205762

hardly does 1205761



0 10 20 30 40 50 60 70 80 90 1000

001002003004005006007008009

011205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

0091205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n

Time step



1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Failu

re p

ropo

rtio

n

Time step

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205762 = 025 R = 5


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Failu

re p

ropo

rtio

n

Time step

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205762 = 025 R = 5



1






119861+119877


119861+119877




0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 1205762 = 025

R = 2

R = 3

R = 4

R = 5

R = 6


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

1205761 = 1205762 = 025

Failu

re p

ropo

rtio



= 1205762


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

0091205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n



= 1205762



119877

119861+119877



119861+119877

=

119877119878

119861+119877


119861


119861


119861+119877



0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055


0 20 40 60 80 1000

0005

001

0015

002

0025

003

0035

004

0045

005

Time step

Failu

re p

ropo

rtio

n

1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055



2





2


2


119861+119877

(1205762


2

(119901119877119861+119877



2



1


lowast

2


isin (0 04] or1205762






2

isin (04 1)



2


2


119901

119904

= 12 lt 119905119901

119896

=

29 (when 1205762

isin (0 04]) or 21 (when 1205762


119904

= 0086 gt 119901119896

= 006 (when 1205762

isin (0 04]) or0072 (when 120576

2


119904

(119890) = 63 lt 119905119896




1


119861+119877

isin [14 24] and 119901119877119861+119877

isin


1

is set as 1205761



1

and 1205762

= 015 or 1205762

= 025 We


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035 1205762 = 065

1205762 = 04

1205762 = 045

1205762 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205761 = 025 R = 5

Failu

re p

ropo

rtio

n

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055



2


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

Failu

re p

ropo

rtio

n

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055



1



1


ge



1


1


120576119878

lowast

1

= 025 and when 1205761



1


lowast

1



0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205762 = 015 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205761 = 065

(a) When 1205762

= 015

0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n(b) When 120576

2



1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n


1



1


119901

119904

= 12


119896

= 17


= 0085 gt 119901119896



6 Conclusions






1


2


1

and 1205762





Acknowledgments


References



















infin











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119909119894

(119905 + 1) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1205761

minus 1205762

) 119891 (119909119894

(119905)) + 1205761

119873

sum

119895=1119895 =119894

119886119894119895

119891 (119909119895

(119905))

119896119894

+1205762

119873

sum

119895=1119895 =119894

119908119894119895

119891 (119909119895

(119905))

119904119894

10038161003816100381610038161003816100381610038161003816100381610038161003816

119894 = 1 2 119873

(2)

where 119909119894


119894119895

)119873times119873


119894119895

= 119886119895119894

= 1 otherwise 119886119894119895

= 119886119895119894

= 0119896119894



119894




1

1205762

isin (0 1) 1205761

+ 1205762



119894



119894



i

k

jwij





119909119894

(119898) = 119877 +

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1205761

minus 1205762

) 119891 (119909119894

(119898 minus 1)) +1205761

+1205761

119873

sum

119895=1119895 =119894

119886119894119895

119891 (119909119895

(119898 minus 1))

119896119894

+1205762

119873

sum

119895=1119895 =119894

119908119894119895

119891 (119909119895

(119898 minus 1))

119904119894

10038161003816100381610038161003816100381610038161003816100381610038161003816

(3)


119894


119894



119894119895


119895119896

rarr 119908119895119896

+Δ119908119895119896


119895119896

= 119908119894119895

lowast119908119895119896

119904119895






119868(119905)





= 1205762


119877

1198611


1198612


1198611

1198771198771198612



1199011


1



1

= 1205762


119861


strength 119877119878119861


119861

lt 119877119877

1198611

but 119877119878119861

= 119877119877

1198611



119861


119896


119901119896



119904




2


119877

1198611


= 025 and 1205762


2


2


2


2

le 1205761



119870

119861

and119877119878

119861



0 20 40 60 80 1000

02

04

06

08

1Cu

mul

ativ

e fai

lure

pro

port

ion

1205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6


0 10 20 30 40 50 60 70 80 90 1000

010203040506070809

1

R = 2

R = 3

R = 4

R = 5

R = 6

Cum

ulat

ive f

ailu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

01

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762



2


119901

119896

asymp 119905119901

119904


119896

isin [007 008] and 119901119904




119904




1


= 025 and 1205761


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n1205761 = 1205762 = 025

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762



119901

119903

isin [18 20]





1

lt 1205762


119901

119896



0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

011205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step




0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

0071205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step



2



119904


1

gt 1205762

and 1205761

+ 1205762


119901

119896

asymp 119905119901

119904


119896



(119890) = 74

and 119905119904


1

lt 1205762

intentional


1

gt 1205762

hardly does 1205761



0 10 20 30 40 50 60 70 80 90 1000

001002003004005006007008009

011205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

0091205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n

Time step



1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Failu

re p

ropo

rtio

n

Time step

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205762 = 025 R = 5


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Failu

re p

ropo

rtio

n

Time step

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205762 = 025 R = 5



1






119861+119877


119861+119877




0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 1205762 = 025

R = 2

R = 3

R = 4

R = 5

R = 6


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

1205761 = 1205762 = 025

Failu

re p

ropo

rtio



= 1205762


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

0091205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n



= 1205762



119877

119861+119877



119861+119877

=

119877119878

119861+119877


119861


119861


119861+119877



0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055


0 20 40 60 80 1000

0005

001

0015

002

0025

003

0035

004

0045

005

Time step

Failu

re p

ropo

rtio

n

1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055



2





2


2


119861+119877

(1205762


2

(119901119877119861+119877



2



1


lowast

2


isin (0 04] or1205762






2

isin (04 1)



2


2


119901

119904

= 12 lt 119905119901

119896

=

29 (when 1205762

isin (0 04]) or 21 (when 1205762


119904

= 0086 gt 119901119896

= 006 (when 1205762

isin (0 04]) or0072 (when 120576

2


119904

(119890) = 63 lt 119905119896




1


119861+119877

isin [14 24] and 119901119877119861+119877

isin


1

is set as 1205761



1

and 1205762

= 015 or 1205762

= 025 We


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035 1205762 = 065

1205762 = 04

1205762 = 045

1205762 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205761 = 025 R = 5

Failu

re p

ropo

rtio

n

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055



2


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

Failu

re p

ropo

rtio

n

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055



1



1


ge



1


1


120576119878

lowast

1

= 025 and when 1205761



1


lowast

1



0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205762 = 015 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205761 = 065

(a) When 1205762

= 015

0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n(b) When 120576

2



1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n


1



1


119901

119904

= 12


119896

= 17


= 0085 gt 119901119896



6 Conclusions






1


2


1

and 1205762





Acknowledgments


References



















infin











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119868(119905)





= 1205762


119877

1198611


1198612


1198611

1198771198771198612



1199011


1



1

= 1205762


119861


strength 119877119878119861


119861

lt 119877119877

1198611

but 119877119878119861

= 119877119877

1198611



119861


119896


119901119896



119904




2


119877

1198611


= 025 and 1205762


2


2


2


2

le 1205761



119870

119861

and119877119878

119861



0 20 40 60 80 1000

02

04

06

08

1Cu

mul

ativ

e fai

lure

pro

port

ion

1205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6


0 10 20 30 40 50 60 70 80 90 1000

010203040506070809

1

R = 2

R = 3

R = 4

R = 5

R = 6

Cum

ulat

ive f

ailu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

01

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762



2


119901

119896

asymp 119905119901

119904


119896

isin [007 008] and 119901119904




119904




1


= 025 and 1205761


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n1205761 = 1205762 = 025

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762



119901

119903

isin [18 20]





1

lt 1205762


119901

119896



0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

011205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step




0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

0071205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step



2



119904


1

gt 1205762

and 1205761

+ 1205762


119901

119896

asymp 119905119901

119904


119896



(119890) = 74

and 119905119904


1

lt 1205762

intentional


1

gt 1205762

hardly does 1205761



0 10 20 30 40 50 60 70 80 90 1000

001002003004005006007008009

011205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

0091205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n

Time step



1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Failu

re p

ropo

rtio

n

Time step

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205762 = 025 R = 5


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Failu

re p

ropo

rtio

n

Time step

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205762 = 025 R = 5



1






119861+119877


119861+119877




0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 1205762 = 025

R = 2

R = 3

R = 4

R = 5

R = 6


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

1205761 = 1205762 = 025

Failu

re p

ropo

rtio



= 1205762


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

0091205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n



= 1205762



119877

119861+119877



119861+119877

=

119877119878

119861+119877


119861


119861


119861+119877



0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055


0 20 40 60 80 1000

0005

001

0015

002

0025

003

0035

004

0045

005

Time step

Failu

re p

ropo

rtio

n

1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055



2





2


2


119861+119877

(1205762


2

(119901119877119861+119877



2



1


lowast

2


isin (0 04] or1205762






2

isin (04 1)



2


2


119901

119904

= 12 lt 119905119901

119896

=

29 (when 1205762

isin (0 04]) or 21 (when 1205762


119904

= 0086 gt 119901119896

= 006 (when 1205762

isin (0 04]) or0072 (when 120576

2


119904

(119890) = 63 lt 119905119896




1


119861+119877

isin [14 24] and 119901119877119861+119877

isin


1

is set as 1205761



1

and 1205762

= 015 or 1205762

= 025 We


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035 1205762 = 065

1205762 = 04

1205762 = 045

1205762 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205761 = 025 R = 5

Failu

re p

ropo

rtio

n

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055



2


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

Failu

re p

ropo

rtio

n

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055



1



1


ge



1


1


120576119878

lowast

1

= 025 and when 1205761



1


lowast

1



0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205762 = 015 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205761 = 065

(a) When 1205762

= 015

0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n(b) When 120576

2



1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n


1



1


119901

119904

= 12


119896

= 17


= 0085 gt 119901119896



6 Conclusions






1


2


1

and 1205762





Acknowledgments


References



















infin











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 1000

02

04

06

08

1Cu

mul

ativ

e fai

lure

pro

port

ion

1205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6


0 10 20 30 40 50 60 70 80 90 1000

010203040506070809

1

R = 2

R = 3

R = 4

R = 5

R = 6

Cum

ulat

ive f

ailu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

01

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762



2


119901

119896

asymp 119905119901

119904


119896

isin [007 008] and 119901119904




119904




1


= 025 and 1205761


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n1205761 = 1205762 = 025

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762



119901

119903

isin [18 20]





1

lt 1205762


119901

119896



0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

011205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step




0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

0071205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step



2



119904


1

gt 1205762

and 1205761

+ 1205762


119901

119896

asymp 119905119901

119904


119896



(119890) = 74

and 119905119904


1

lt 1205762

intentional


1

gt 1205762

hardly does 1205761



0 10 20 30 40 50 60 70 80 90 1000

001002003004005006007008009

011205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

0091205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n

Time step



1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Failu

re p

ropo

rtio

n

Time step

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205762 = 025 R = 5


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Failu

re p

ropo

rtio

n

Time step

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205762 = 025 R = 5



1






119861+119877


119861+119877




0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 1205762 = 025

R = 2

R = 3

R = 4

R = 5

R = 6


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

1205761 = 1205762 = 025

Failu

re p

ropo

rtio



= 1205762


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

0091205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n



= 1205762



119877

119861+119877



119861+119877

=

119877119878

119861+119877


119861


119861


119861+119877



0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055


0 20 40 60 80 1000

0005

001

0015

002

0025

003

0035

004

0045

005

Time step

Failu

re p

ropo

rtio

n

1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055



2





2


2


119861+119877

(1205762


2

(119901119877119861+119877



2



1


lowast

2


isin (0 04] or1205762






2

isin (04 1)



2


2


119901

119904

= 12 lt 119905119901

119896

=

29 (when 1205762

isin (0 04]) or 21 (when 1205762


119904

= 0086 gt 119901119896

= 006 (when 1205762

isin (0 04]) or0072 (when 120576

2


119904

(119890) = 63 lt 119905119896




1


119861+119877

isin [14 24] and 119901119877119861+119877

isin


1

is set as 1205761



1

and 1205762

= 015 or 1205762

= 025 We


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035 1205762 = 065

1205762 = 04

1205762 = 045

1205762 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205761 = 025 R = 5

Failu

re p

ropo

rtio

n

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055



2


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

Failu

re p

ropo

rtio

n

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055



1



1


ge



1


1


120576119878

lowast

1

= 025 and when 1205761



1


lowast

1



0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205762 = 015 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205761 = 065

(a) When 1205762

= 015

0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n(b) When 120576

2



1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n


1



1


119901

119904

= 12


119896

= 17


= 0085 gt 119901119896



6 Conclusions






1


2


1

and 1205762





Acknowledgments


References



















infin











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n1205761 = 1205762 = 025

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n

1205761 = 1205762 = 025

Time step



= 1205762



119901

119903

isin [18 20]





1

lt 1205762


119901

119896



0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

011205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step




0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

0071205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step



2



119904


1

gt 1205762

and 1205761

+ 1205762


119901

119896

asymp 119905119901

119904


119896



(119890) = 74

and 119905119904


1

lt 1205762

intentional


1

gt 1205762

hardly does 1205761



0 10 20 30 40 50 60 70 80 90 1000

001002003004005006007008009

011205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

0091205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n

Time step



1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Failu

re p

ropo

rtio

n

Time step

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205762 = 025 R = 5


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Failu

re p

ropo

rtio

n

Time step

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205762 = 025 R = 5



1






119861+119877


119861+119877




0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 1205762 = 025

R = 2

R = 3

R = 4

R = 5

R = 6


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

1205761 = 1205762 = 025

Failu

re p

ropo

rtio



= 1205762


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

0091205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n



= 1205762



119877

119861+119877



119861+119877

=

119877119878

119861+119877


119861


119861


119861+119877



0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055


0 20 40 60 80 1000

0005

001

0015

002

0025

003

0035

004

0045

005

Time step

Failu

re p

ropo

rtio

n

1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055



2





2


2


119861+119877

(1205762


2

(119901119877119861+119877



2



1


lowast

2


isin (0 04] or1205762






2

isin (04 1)



2


2


119901

119904

= 12 lt 119905119901

119896

=

29 (when 1205762

isin (0 04]) or 21 (when 1205762


119904

= 0086 gt 119901119896

= 006 (when 1205762

isin (0 04]) or0072 (when 120576

2


119904

(119890) = 63 lt 119905119896




1


119861+119877

isin [14 24] and 119901119877119861+119877

isin


1

is set as 1205761



1

and 1205762

= 015 or 1205762

= 025 We


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035 1205762 = 065

1205762 = 04

1205762 = 045

1205762 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205761 = 025 R = 5

Failu

re p

ropo

rtio

n

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055



2


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

Failu

re p

ropo

rtio

n

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055



1



1


ge



1


1


120576119878

lowast

1

= 025 and when 1205761



1


lowast

1



0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205762 = 015 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205761 = 065

(a) When 1205762

= 015

0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n(b) When 120576

2



1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n


1



1


119901

119904

= 12


119896

= 17


= 0085 gt 119901119896



6 Conclusions






1


2


1

and 1205762





Acknowledgments


References



















infin











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

011205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step




0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

0071205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055

Failu

re p

ropo

rtio

n

Time step



2



119904


1

gt 1205762

and 1205761

+ 1205762


119901

119896

asymp 119905119901

119904


119896



(119890) = 74

and 119905119904


1

lt 1205762

intentional


1

gt 1205762

hardly does 1205761



0 10 20 30 40 50 60 70 80 90 1000

001002003004005006007008009

011205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

0091205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n

Time step



1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Failu

re p

ropo

rtio

n

Time step

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205762 = 025 R = 5


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Failu

re p

ropo

rtio

n

Time step

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205762 = 025 R = 5



1






119861+119877


119861+119877




0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 1205762 = 025

R = 2

R = 3

R = 4

R = 5

R = 6


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

1205761 = 1205762 = 025

Failu

re p

ropo

rtio



= 1205762


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

0091205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n



= 1205762



119877

119861+119877



119861+119877

=

119877119878

119861+119877


119861


119861


119861+119877



0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055


0 20 40 60 80 1000

0005

001

0015

002

0025

003

0035

004

0045

005

Time step

Failu

re p

ropo

rtio

n

1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055



2





2


2


119861+119877

(1205762


2

(119901119877119861+119877



2



1


lowast

2


isin (0 04] or1205762






2

isin (04 1)



2


2


119901

119904

= 12 lt 119905119901

119896

=

29 (when 1205762

isin (0 04]) or 21 (when 1205762


119904

= 0086 gt 119901119896

= 006 (when 1205762

isin (0 04]) or0072 (when 120576

2


119904

(119890) = 63 lt 119905119896




1


119861+119877

isin [14 24] and 119901119877119861+119877

isin


1

is set as 1205761



1

and 1205762

= 015 or 1205762

= 025 We


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035 1205762 = 065

1205762 = 04

1205762 = 045

1205762 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205761 = 025 R = 5

Failu

re p

ropo

rtio

n

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055



2


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

Failu

re p

ropo

rtio

n

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055



1



1


ge



1


1


120576119878

lowast

1

= 025 and when 1205761



1


lowast

1



0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205762 = 015 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205761 = 065

(a) When 1205762

= 015

0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n(b) When 120576

2



1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n


1



1


119901

119904

= 12


119896

= 17


= 0085 gt 119901119896



6 Conclusions






1


2


1

and 1205762





Acknowledgments


References



















infin











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 60 70 80 90 1000

001002003004005006007008009

011205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n

Time step


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

0091205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n

Time step



1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Failu

re p

ropo

rtio

n

Time step

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205762 = 025 R = 5


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Failu

re p

ropo

rtio

n

Time step

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205762 = 025 R = 5



1






119861+119877


119861+119877




0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 1205762 = 025

R = 2

R = 3

R = 4

R = 5

R = 6


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

1205761 = 1205762 = 025

Failu

re p

ropo

rtio



= 1205762


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

0091205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n



= 1205762



119877

119861+119877



119861+119877

=

119877119878

119861+119877


119861


119861


119861+119877



0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055


0 20 40 60 80 1000

0005

001

0015

002

0025

003

0035

004

0045

005

Time step

Failu

re p

ropo

rtio

n

1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055



2





2


2


119861+119877

(1205762


2

(119901119877119861+119877



2



1


lowast

2


isin (0 04] or1205762






2

isin (04 1)



2


2


119901

119904

= 12 lt 119905119901

119896

=

29 (when 1205762

isin (0 04]) or 21 (when 1205762


119904

= 0086 gt 119901119896

= 006 (when 1205762

isin (0 04]) or0072 (when 120576

2


119904

(119890) = 63 lt 119905119896




1


119861+119877

isin [14 24] and 119901119877119861+119877

isin


1

is set as 1205761



1

and 1205762

= 015 or 1205762

= 025 We


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035 1205762 = 065

1205762 = 04

1205762 = 045

1205762 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205761 = 025 R = 5

Failu

re p

ropo

rtio

n

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055



2


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

Failu

re p

ropo

rtio

n

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055



1



1


ge



1


1


120576119878

lowast

1

= 025 and when 1205761



1


lowast

1



0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205762 = 015 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205761 = 065

(a) When 1205762

= 015

0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n(b) When 120576

2



1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n


1



1


119901

119904

= 12


119896

= 17


= 0085 gt 119901119896



6 Conclusions






1


2


1

and 1205762





Acknowledgments


References



















infin











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 1205762 = 025

R = 2

R = 3

R = 4

R = 5

R = 6


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

1205761 = 1205762 = 025

Failu

re p

ropo

rtio



= 1205762


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

0081205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

0091205761 = 1205762 = 025

Time step

R = 2

R = 3

R = 4

R = 5

R = 6

Failu

re p

ropo

rtio

n



= 1205762



119877

119861+119877



119861+119877

=

119877119878

119861+119877


119861


119861


119861+119877



0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055


0 20 40 60 80 1000

0005

001

0015

002

0025

003

0035

004

0045

005

Time step

Failu

re p

ropo

rtio

n

1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055



2





2


2


119861+119877

(1205762


2

(119901119877119861+119877



2



1


lowast

2


isin (0 04] or1205762






2

isin (04 1)



2


2


119901

119904

= 12 lt 119905119901

119896

=

29 (when 1205762

isin (0 04]) or 21 (when 1205762


119904

= 0086 gt 119901119896

= 006 (when 1205762

isin (0 04]) or0072 (when 120576

2


119904

(119890) = 63 lt 119905119896




1


119861+119877

isin [14 24] and 119901119877119861+119877

isin


1

is set as 1205761



1

and 1205762

= 015 or 1205762

= 025 We


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035 1205762 = 065

1205762 = 04

1205762 = 045

1205762 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205761 = 025 R = 5

Failu

re p

ropo

rtio

n

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055



2


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

Failu

re p

ropo

rtio

n

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055



1



1


ge



1


1


120576119878

lowast

1

= 025 and when 1205761



1


lowast

1



0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205762 = 015 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205761 = 065

(a) When 1205762

= 015

0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n(b) When 120576

2



1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n


1



1


119901

119904

= 12


119896

= 17


= 0085 gt 119901119896



6 Conclusions






1


2


1

and 1205762





Acknowledgments


References



















infin











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055


0 20 40 60 80 1000

0005

001

0015

002

0025

003

0035

004

0045

005

Time step

Failu

re p

ropo

rtio

n

1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055



2





2


2


119861+119877

(1205762


2

(119901119877119861+119877



2



1


lowast

2


isin (0 04] or1205762






2

isin (04 1)



2


2


119901

119904

= 12 lt 119905119901

119896

=

29 (when 1205762

isin (0 04]) or 21 (when 1205762


119904

= 0086 gt 119901119896

= 006 (when 1205762

isin (0 04]) or0072 (when 120576

2


119904

(119890) = 63 lt 119905119896




1


119861+119877

isin [14 24] and 119901119877119861+119877

isin


1

is set as 1205761



1

and 1205762

= 015 or 1205762

= 025 We


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035 1205762 = 065

1205762 = 04

1205762 = 045

1205762 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205761 = 025 R = 5

Failu

re p

ropo

rtio

n

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055



2


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

Failu

re p

ropo

rtio

n

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055



1



1


ge



1


1


120576119878

lowast

1

= 025 and when 1205761



1


lowast

1



0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205762 = 015 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205761 = 065

(a) When 1205762

= 015

0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n(b) When 120576

2



1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n


1



1


119901

119904

= 12


119896

= 17


= 0085 gt 119901119896



6 Conclusions






1


2


1

and 1205762





Acknowledgments


References



















infin











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205761 = 025 R = 5

1205762 = 015

1205762 = 025

1205762 = 035 1205762 = 065

1205762 = 04

1205762 = 045

1205762 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205761 = 025 R = 5

Failu

re p

ropo

rtio

n

1205762 = 015

1205762 = 025

1205762 = 035

1205762 = 045

1205762 = 055



2


0 10 20 30 40 50 60 70 80 90 1000

01

02

03

04

05

06

07

08

09

1

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

Failu

re p

ropo

rtio

n

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055



1



1


ge



1


1


120576119878

lowast

1

= 025 and when 1205761



1


lowast

1



0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205762 = 015 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205761 = 065

(a) When 1205762

= 015

0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n(b) When 120576

2



1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n


1



1


119901

119904

= 12


119896

= 17


= 0085 gt 119901119896



6 Conclusions






1


2


1

and 1205762





Acknowledgments


References



















infin











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

Cum

ulat

ive f

ailu

re p

ropo

rtio

n1205762 = 015 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

1205761 = 065

(a) When 1205762

= 015

0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n(b) When 120576

2



1


0 10 20 30 40 50 60 70 80 90 1000

001

002

003

004

005

006

007

008

009

Time step

1205762 = 025 R = 5

1205761 = 015

1205761 = 025

1205761 = 035

1205761 = 045

1205761 = 055

Failu

re p

ropo

rtio

n


1



1


119901

119904

= 12


119896

= 17


= 0085 gt 119901119896



6 Conclusions






1


2


1

and 1205762





Acknowledgments


References



















infin











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1


2


1

and 1205762





Acknowledgments


References



















infin











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article cascading failures in weighted...

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