yup kim kyung hee university

1The 2nd KIAS Conference on Statistical Physics (2006)

Yup KimKyung Hee University

Conserved Mass Aggregation and Lamb-lion Problem on complex

networks

CollaborationsSoon-Hyung Yook, Sungchul Kwon, Sungmin Lee

ReferencesS. Kwon, S. Lee and Y. Kim, PRE 73, 056102 (2006)S. Lee, S. Yook and Y. Kim, Submitted to PRE, cond-mat/0603647 S. Lee, S. Yook and Y. Kim, Submitted to PRL


Outline• Condensation phase transition on complex networks

– Symmetric Conserved mass aggregation (SCMA) model– SCMA model on complex networks– Mass distribution of a node with degree k, m(k)– Existence of infinite aggregation– Finite sized results for random walks (RWs) on scale-free netw

orks (SFNs)• Lamb-lion problem on complex networks• Application

– Peer-to-Peer network – Propose an efficient algorithm

• Conclusions


ijji mmm,0m

1mm,1mm jjii

Condensation phase transition

Diffusion

Chipping

Diffusion with unit rate :

Chipping with rate :

fluid phase Condensed phase

Examples : clouds, colloidal suspensions,polymer gels, aerosols, river networks

- Symmetric Conserved-mass aggregation (SCMA) model

S. N. Majumdar et al, J. Stat. Phys. 99, 1 (2000)

Diffusion tends to aggregate masses.

Chipping tends to split masses.

(j is one of nns to i)


For , competition between diffusion and chipping

→ phase transitions from condensed phase into fluid phase.

0

0

diffusion-dominant ( ) : aggregation on a site

chipping-dominant ( ) : masses scattered over entire lattice.

(zero-range process : ZRP)

Zero Range Process (ZRP)

A particle jumps out of the site at the rate

and hops to a site with the Probability .

A condensed phase, which a finite fraction of

total particles condenses on a single site,

arises or not according to , .

Braz. J. Phys. 30, 42 (2000)

Hopping

Jumping


Phase diagramOrder parameter : P(m) = mass distribution of a single site

= Probability that a site has mass m in the steady state.

Fluid phase

Condensed

phase

condensatem

ycriticalitatm

phasefluidinemP mm

~

~

~)(*/

2/5

11

c

theoryfieldMeanm

P(m)


Diffusion with unit rate :Chipping with rate ω :

ω= 0 : complete condensation on a node ω = ∞ : Zero-range process with constant chipping rate → Condensation always exists on scale free networks with ; J. D. Noh et al, Phys. Rev. Lett. 94, 198701 (2005).

ijji mmm,0m 1mm,1mm jjii

2

- SCMA model on complex networks

Degree distribution

scale-free networks (SFNs)

What about 0 < ω < ∞ case on SFNs ?What is the effect of underlying topology like SFNs on condensation transitions ?


(1) Random and scale free networks ofN = # of nodes = 10000, K = # of links = <k>N/2 = 20000

<k> = average degree = 4 in our simulations (a) = Random net. (RN) (b) = SFN of

3

3.4


Phase diagram : RNs (a) and SFN of (b)3.4

3.4:)2(33.2

RN:)3(38.2

The same type of condensation transition as those on regular lattices.(SFN with )

But the critical line depends on network structures.

3


(2) SFNs of(a) (b) (c) expected phase diagram

In limit, it is practically impossible to show the existence of the condensation. (Consideration of Diffusive Capture Process or Lamb-Lion Problem on the networks).

30.3 4.2

0


Total mass of nodes with degree k = kM

In a certain run of simulation,

)4.2(3

1

By diffusion, the aggregate diffuses around networks and the dominant hub is not the node at which the condensate is located unlike ZRP.


- Average mass of a node with degree k ,

At , it was shown that complete condensation always takes place on SFNs. What about on RNs ?

For , the behavior in the condensed phase ?

km

0

3

1

PRL 94, 198701

(2005)

ZRP on SFN


0

),(m

k kmPmM

kPmPkmP )(),(

= the probability of finding a random walker on degree k in k-space

kP


T

Condensedphase

Fluidphase

Lamb-lion problem

- For the existence of an infinite condensate, the two masses should aggregate again in finite time interval.

- If not, unit mass continuously chips off from the infinite aggregation, which will finally disappear.

Fluid phase

Condensed phase

Condensed phase

(no Fluid phase)

Or

: survival probability)(tS

: average life time

- Existence of infinite aggregation

finite


10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 105 10610-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

=2.4

N

v(t)/

N

t/N

N=1000 N=10000 N=100000 N=1000000

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 10510-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

=3.0

Nv(t

)/N

t/N

N=1000 N=10000 N=100000 N=1000000

NtNT vsat ~~~For any

vN : The number of visited distinct sites of a random walker

satT : The saturation time

T : The average life time

- Finite sized results for RWs on SFNs

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 10510-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

=4.3

N

v(t)/

N

t/N

N=1000 N=10000 N=100000 N=1000000


10-1 100 101 102 10310-1

100

N=1000 N=10000 N=100000 N=1000000

R/ (

log(

N))

0.97

t / (log(N))0.97

=4.3

10-1 100 101 102 103 104

1

N=1000 N=10000 N=100000 N=1000000

R /

(log(

N))

0.6

5

t / (log(N))0.65

=3.0

100 101 102 103 104 105

1

2

3

4

5

6

N=1000 N=10000 N=100000 N=1000000

R /

(log(

log(

N))

)0.4

5

t / (log(log(N)))0.45

=2.4

R: the distance between two random walkers (the shortest path)

N: the number of nodes


Static trap

Hub effect

random walker to random walker random walker

No!!

On networks

On regular lattice

Yes

?

Diffusive capture process = lamb-lion problem


Lamb-lion problemThe diffusion-controlled reactions, in which diffusing

particlesare immediately converted to a product if a pair of

them meetstogether, have many physical applications.Examples : electron trapping and recombination, wetting, melting,exciton fusion, and commensurate-incommensurate transitions

Among these examples, dynamic properties of wetting, melting, and commensurate-

incommensurate transition are known to be related to the diffusive capture process,

whose kinetics can be simplified by lamb-lion problem (diffusing preys-predators model).

P.L.Krapivsky and S.Redner J.Phys.A 29, 5347 (1996)

What is the survival probability of a diffusing lamb which is hunted by hungry lions?

On regular lattice

PRB 39, 889 (1989), JSP 34, 667 (1984), PRB 29, 239 (1984), JPA 21, L89 (1988)

Diffusion-controlled reaction

First passagephenomena of RWs

Survival probability of a diffusing lamb


The major searching engines, such as Google, use general

random walking robots along the links between hyper-texts

to collect information of each web page. The searching algorithm can be mapped to the

system of a diffusing particle to find an immobile

absorbing particle.

One of interesting applications can be found in searching information over the Internet.

If the lamb meets the lion, the lamb is captured.

At each time step, a lamb and lions

take random walks simultaneously.

Initially a lamb and lions are randomly

distributed to the nodes on the networks.

Our model

Korean Phys. Soc. 48, S202 (2006)

Degree distribution


We measure and on LSFNs with various and network size .


We measure the average life time and the survival probability on TSFNs.


Origin of long-living tail of for

The data explicitly shows that lamb-lion

with corresponds to the long

surviving tail. In the used networks, the

explicitly demonstrates that the

lamb and the lion are in different

branches.


Relation between degrees and capture events

We measure the number of captures

occurring at a node with degree .

PRL 92, 118701 (2004).

Assume ( : the model dependent parameter satisfying )

increases as for . Determined 's from the data in (a) and (b) are for LSFN and

for TSFN.


Relation between degrees and capture events

Assume ( : the model dependent parameter satisfying )

increases as for . Determined 's from the data in (a) and (b) are for LSFN and

for TSFN. provides the topological origin of the gathering behavior of

randomwalks at hubs. This implies that the lamb and the lion have a strong

tendency to move intobig hubs.


Complex Network

Lamb

Lion

Information packet

Query packet

Implementing an efficient searching algorithm is the key to a better performance of P2P protocol design.

P2P systems are distributed systems in which nodes exchange files directly with each other.

We apply results of our study on diffusive capture process to the searching algorithm to find file in the Peer-to-Peer (P2P) file-sharing networks.

Application


Each node forwards the received query packets to all of its nearest neighbors until the pre-assigned time-to-live (TTL) becomes 0.

Flooding based algorithm (FB) n-random walker algorithm (n-RW)

The node who want to search a file produces n query packets. Each querying packet takes random walks along the P2P connections until the pre-assigned TTL becomes 0.

FB causes significant traffic congestion. For example, the P2P traffic consumes 60-70% of European Operators’ bandwidth.

n-RW algorithm can cause long waiting time if there are a few requested files in the network and they are located at the node with small number of connections.

(http://www.theregister.co.uk/2003/10/14 /edonkey_rides_like_the_wind/)

s

T

s

query packet

T


In general, the degree distribution of P2P networks follows the power law, with , or highly skewed fat-tailed distributions.(=> Expect attracting hubs)

=> exists effective attractor (Hubs)

Degree distribution of P2P network

L.A.Adam, R.M.Lukose, B.Huberman, & A.R.Puniyani, PRE 64, 46135 (2001)M.Ripeanu, I.Foster & A.Iamnitchi, IEEE Internet Computing Journal 6, 50 (2002)Stutzbach, D. & Rejaie, R. In Global Internet Symposium, 127 Mar. (2005)

We expect two main benefits by using new algorithm.1) the amount of traffic is constant and much less than

FB algorithm2) provide more scalable searching time than n-RW

algorithm


ii) Independently, a randomly chosen node sends out one query packet to find

a specific file. (The query packet also takes random walks.) iii) If the query packet meets an information packet which has the requested file

name in its list, then the IP information in the information packet is sent back

to the requesting node. iv) And then the query packet is discarded but the information

packets continue random walks for the next query.

i) Each node sends out an information packet (names of files + IP address). (Each of these packets takes random walks along the P2P

connections.)

One query event

s

query packetinformation packet

n-lion and lamb algorithm (NLL)

Propose an efficient algorithm


The inset shows the time evolution of obtained from a single run of simulation of FB.

The local maximum exceeds which is 2,000 times larger than the traffic generated

by NLL.

The average traffic of each algorithm.

FB generates around 50 times more traffic than NLL on the average.

At the moment of occurring such large amount of traffic, FB would consume the most of the bandwidth of the Internet and cause severe traffic congestion over the network. However, NLL always guarantees a constant level of traffic, which is much less than that of FB and comparable to that of n-RW.


: the number of available files on a network : the average searching time

(More scalable searching time than n-RW)

The average searching time of NLL on SFNs is, at least, 10 times faster than n-RW on SFNs.

The average searching times satisfy

The difference between NLL and 2-RW, , increases as .

Since the probability that two random walks visits a node with degree is proportional to , the hub can collect all information packets.

increases almost linearly for n-RW

. However, NLLgrows as for small , but

seems to be less than 0.5 or

becomes saturated to a fixed finite value

for large .

hub


ConclusionOn RN and SFNs with , CMA model undergoes the same typeof condensation transitions as one dimensional lattice.(The critical line depends on the underlying network structures.)

On SFNs with , an infinite aggregation with exponentiallydecaying background mass distribution always takes place for anynonzero density. (no phase transitions)

On LSFNs,

On TSFNs,


The lamb and the lion have a strong tendency to move into big hubs. By numerical simulations, we verify that our new searching algorithm can

drastically decrease the traffic congestion compared to FB algorithm and

can provide more scalable average searching time than n-RW algorithm

and comparable with FB algorithm.The hubs spontaneously play a very similar role of the

directory serversin structured P2P networks. However, we expect the one of

the advantagesof using NLL algorithm can be in reducing large amount of

systemresources for the directory server to store and handle the

huge centralizedinformation packet, because most of information packets

stay on thedominant hubs and their nearest neighbors for a

considerable amount oftime. Therefore, the information on the nearest neighbors of

the hubs at agiven time is easily accessed through the hubs by random

walks withoutstoring all information at the hubs.


Static trap

Hub effect

NTNT vsat ~~~NT ~ 0 NT

T

sat

)1(

3

If)(tSThen is finite.

SFNs

random walker to random walker random walker

No!!

On networks

On regular lattice

Yes

?

)(tS : survival probability

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