The Cost of Attack in Competing Networks

B. Podobnik,1, 2, 3 D. Horvatic,4 T. Lipic,1 M. Perc,5 J. M. Buldu,6, 7 and H. E. Stanley1

1Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 022152Faculty of Civil Engineering, University of Rijeka, 51000 Rijeka, Croatia

3Zagreb School of Economics and Management, 10000 Zagreb, Croatia4Faculty of Natural Sciences, University of Zagreb, 10000 Zagreb, Croatia

5Faculty of Natural Sciences and Mathematics, University of Maribor, SI-2000 Maribor, Slovenia6Center for Biomedical Technology (UPM), 28223 Pozuelo de Alarcon, Madrid, Spain7Complex Systems Group, Rey Juan Carlos University, 28933 Mostoles, Madrid, Spain

Real-world attacks can be interpreted as the result of competitive interactions between networks,ranging from predator-prey networks to networks of countries under economic sanctions. Althoughthe purpose of an attack is to damage a target network, it also curtails the ability of the attacker,which must choose the duration and magnitude of an attack to avoid negative impacts on itsown functioning. Nevertheless, despite the large number of studies on interconnected networks,the consequences of initiating an attack have never been studied. Here, we address this issue byintroducing a model of network competition where a resilient network is willing to partially weakenits own resilience in order to more severely damage a less resilient competitor. The attackingnetwork can take over the competitor nodes after their long inactivity. However, due to a feedbackmechanism the takeovers weaken the resilience of the attacking network. We define a conservationlaw that relates the feedback mechanism to the resilience dynamics for two competing networks.Within this formalism, we determine the cost and optimal duration of an attack, allowing a networkto evaluate the risk of initiating hostilities.

I. INTRODUCTION

Recent research carried out on competing interactingnetworks [1–6] does not take into account the fact thatreal-world networks often compete not only to survivebut also to take over or even destroy their competitors[7]. For example, in international politics and economics,when one country imposes economic sanctions on an-other, feedback mechanisms can cause the country im-posing the sanctions to also be adversely affected. Thedecision by a wealthier country to keep military spend-ing at a high level long enough to exhaust its poorercompetitor can also contribute to its own exhaustion [8].Similarly, in warfare, any attack depletes the resources ofthe attacking force and can elicit a counter-attack fromthe competing force [9]. Also, in nature, an incursionbetween species can alter the dynamics of predator-preyinteraction [10].

Although, these competing interactions are awidespread real-world phenomenon, current studiesanalyze only effects of attack on attacked networks, butdisregarding its effect on the external attacking network.For example, for both single and interactive networks,existing studies on network robustness report that everynetwork, regardless of the size and architecture, eventu-ally can be destroyed [11–15]. But, what then prevents anetwork from attacking a weaker competitor or, what isthe optimal moment for initiating or ending an attack?In order to identify the factors that inhibit a networkfrom attacking and demolishing a weaker competitorand to determine the optimal moment and durationof an attack, we develop a theoretical framework thatquantifies the cost of an attack by connecting thefeedback mechanisms and resilience dynamics between

two competing dynamic networks with differing levels ofresilience [16, 17].

II. THEORETICAL FRAMEWORK

We introduce a general methodology that can be ap-plied to networks of any size and structure. First, as an il-lustrative example, we describe two competing Barabasi-Albert (BA) networks [18] which we designate networkS and network W. This model differs from the singlenetwork BA model in that the two interconnected net-works have both intra-network and inter-network links[19]. One real-world example of this kind of network in-teraction is firms in an economic network that link withother firms both domestically and abroad.

Using the preferential attachment (PA) rule we gen-erate networks S and W starting with n0 nodes in eachnetwork. At each time step we add a new node thatconnects with mS existing nodes in network S and withmW,S existing nodes in network W, where the probabil-ity of each connection depends on the total node degreesin networks S and W. Similarly, using the PA rule weconnect a new node in network W with mW nodes insidenetwork W and with mS,W = mW,S nodes in network S.

In a broad class of real-world networks, nodes can faileither due to inherent reason [20] or because their func-tionality depends on their neighborhood [20, 21]. Hence,any node in either of the two networks, e.g., a node ni in-side network S with kS neighbors in its own network andkW,S neighbors in network W, can fail at any moment,either internally—independent of other nodes—with aprobability p1 or externally with a probability pW . Nodeni externally fails with a probability p2 when, similar

arX

iv:1

501.

0434

8v1

[cs

.SI]

18

Jan

2015

2

a

attacks

attacks

damaged nodes

b

takeover

takeover

FIG. 1: Attacks, failures, takeovers and their cost on the attacking network. In (a), we assume that each node in both the moreresilient (stronger) network S and the less resilient (weaker) network W is described by the same failure probability. Differentnodes spend different times during internal failure—the less opaque a node is, the more time it spends in internal failure. In(b), if a node in the weaker network W remains inactive more than some threshold time, it will be taken over by the strongernetwork S. However, network S pays for this takeover with a reduction of its resilience.

to the Watts model [21], the total fraction of its activeneighbors is less than or equal to a fractional thresholdT which is equal for all nodes in both networks. Thelarger the T value, the less resilient the network. We as-sume that one of the two networks is more resilient thanthe other, distinguishing between strong network S andweak network W. We do so by assigning different frac-tional thresholds to the strong and weak networks, TSand TW , respectively, with TS < TW . As in Ref. [20],we assume that an internally-failed node in network Sor network W recovers from its last internal failure af-ter a period τ . Consecutive failures of the same nodestretch the effective failure times and introduce hetero-geneity into the distribution of inactivity periods. Sincein real-world networks it is dangerous for nodes to be in-active, we allow the strong network to take over nodes inthe weak network when a node ni spends more time ininternal failure than nτ , where n is a constant. Figure 1qualitatively shows the interaction process.

III. RESULTS

We quantify the current collective state of the strongand weak networks in terms of the fraction of activenodes, fS and fW , respectively [20, 22]. We assumethat initially on both networks have internal and exter-nal failure probability values of p1 ≡ pX and p2, respec-tively. Figure 2(a) shows a two-parameter phase dia-gram for each network in which the hysteresis is com-posed of two spinodals separating two collective states,i.e., the primarily “active” and the primarily “inactive.”Figure 2(b) shows that increasing the value of p1 leadsto catastrophic first-order phase transitions in both net-works. When each network recovers (i.e., when p1 is de-

creased to previous values), the fraction of active nodesreturns to an upper state. Nevertheless, the critical pointin the recovery is well beyond the point at which thenetwork collapses. Figure 2(b) also shows (solid line)that the initial choice of parameters makes network Smore resilient to network fluctuations in the value of p1and that the fluctuation needed to initiate the collapseof network S (pS1 ≡ pS1c − pX) is much larger than thefluctuation needed to initiate the collapse of network W(pW1 ≡ pW1c − pX). Furthermore, network W is closer toa critical transition than network S.

Because network S has a higher resilience than net-work W and can more easily withstand fluctuations, Scould induce the collapse of W by increasing p1, but onlyif the fraction of its active links is not dramatically re-duced. Figure 2(b) shows how when network S attacksnetwork W by increasing p1 to ≈ 0.002 the weak net-work becomes abruptly dysfunctional. Figure 2(b) alsoshows that when the values of p1 are reset to their pre-attack levels the collapse of network W is permanent (reddashed line) and, if it ceases its attack, the recovery ofnetwork S is complete and all of its inactive nodes arereactivated (see blue dashed line Figure 2(b)). Similarly,when economic sanctions in a financial system are liftedthe weak economies are not restored but the strong eco-nomics recover after suffering little damage.

Figure 2(c) shows a modified competing network struc-ture in which there are two interconnected Erdos-Renynetworks [23] with inter-network links randomly chosen.Although this structure quantitatively differs from thephase diagram of competing BA networks, the same kindof transition occurs in the random configuration. This in-dicates the generality of these critical transitions in com-peting networks. We obtain similar results when degree-degree correlations are introduced between the links con-

3

0.6 0.7 0.8 0.9 1.0p2

0.000

0.002

0.004

0.006

0.008

p1

a

0.004 0.008

p1

0.0

0.2

0.4

0.6

0.8

1.0

f

pW1c pS1cpx

bNetwork

SW

0.0 0.2 0.4 0.6 0.8 1.0p2

0.0000

0.0005

0.0010

0.0015

0.0020

p1

c

0.4 0.5 0.6 0.7 0.8 0.9 1.0p2

0.000

0.002

0.004

0.006

p1

d

FIG. 2: Attack strategy between two competing networkswith different resilience levels and intra/inter link architec-ture. Shown are fractions of active nodes. The most resilient,strong network S (with TS = 0.3) endangers and partially de-stroys its own nodes by increasing their internal failure prob-ability p1 in order to more severely damage the least resilientnetwork W (with TW = 0.7). Each of S and W has the hys-teresis composed of two spinodals, representing attacking andrecovery phase. The recovery time is τ = 50 and the takeoverand cost mechanisms are disregarded. (a) Attacking strat-egy between two competing BA networks with parameters:mS = mW = 3 and mS,W = mW,S = 2. Strong networkS wants to bring W in parameter space between hysteresesof W (black lines) and S (white lines) where S is predomi-nantly active and W is predominantly inactive (see, b). Darkred (blue) is parameter space where both S and W are active(inactive). (b) For p2 = 0.9, fraction of active nodes in thestrong fS (blue lines) and weak fW (red lines) networks as afunction of the internal failure probability p1. Hysteresis is aresult of increasing p1 from zero to one and then decreasingit back to zero. The increase of p1 accounts for the attacksand the decrease for a repair of the network. (c) Same case as(a) but for two randomly connected competing Erdos-Renyinetworks. (d) Same case as (c) but with an assortative mix-ing in the connection between networks: nodes with degreed1 link, with probability 1/|d1 − d2 + 1|, with nodes in theother network with degree d2.

necting both networks. Figure 2(d) shows nodes in thestrong network linking with nodes in the weak networkonly when they are of similar degree (i.e., “assortativemixing” [24]). As in the other configurations, the bet-ter position of the attacker enables the strong network todestroy the weak one and then return safely to its initialstate.

A. Mean field theory

Using mean-field theory we analytically describe theattack-and-recovery process between two interconnectednetworks with random regular topologies where all nodeswithin the same network have the same degree. We as-sume that each node in network S is linked with kS nodesin its own network and kW,S nodes in network W. Sim-ilarly, each node in network W is linked with kW nodesinside network W and kS,W nodes in network S. In bothnetworks the fraction of failed nodes is a ≡ 1−f , where fis the fraction of functional nodes. We can approximatethe values of a at each network by

aS = p∗S,1 + pS,2(1− p∗S,1)ES (1)

aW = p∗W,1 + pW,2(1− p∗W,1)EW , (2)

where p∗S,1 ≡ 1 − exp(−pS,1τ) [20] denotes the averagefraction of internally failed nodes and pS,2ES denotes theprobability that a node in network S has externally failed,

ES =

tS∑j=0

j∑i=0

(kS

kS − i

)akS−iS (1− aS)i(

kW,SkW,S − (j − i)

)akW,S−(j−i)W (1− aW )j−i. (3)

Here tS represents the absolute threshold of network Ssimply related to the fractional threshold TS as TS =tS/(kS + kW,S): a node in network S can externally failwith a probability pS,2 only when the number of activeneighbors in both network S and network W is less thanor equal to tS . Similarly, we obtain EW for network W byreplacing S with W, and vice versa, in Eq. (3). Finally,we set network S to be more resilient than network W,by setting tS/(kS + kW,S) < tW /(kW + kS,W ).

The analytical results of Fig. 3(a) indicate that whennetwork S increases the internal failure probability pS,1and so p∗S,1 in an effort to damage network W it alsocauses partial damage to itself. Although it first seemsthat increasing p∗S,1 reduces more active nodes in networkS than in network W, when p∗S,1 > 0.18 the fraction ofactive nodes in network W drops sharply and eventuallyfS > fW . This attack strategy by network S is thuseffective. If p∗S,1 > 0.33, however, network S undergoesa first order transition that leads to collapse, a situationthat network S clearly must avoid.

Inspecting the recovery of the previous internal failureprobability values after the attack we find that the frac-tion of active nodes in both networks exhibit a hysteresisbehaviour. Note that when the transition at p∗S,1 ∼ 0.33is surpassed neither network is able to restore its func-tioning to the levels previous to the attack.

The analytical results indicate that attacking networkS is effective only for certain values of p∗S,1. Thus networkS should increase p∗S,1 only as long as the damage tonetwork W continues to be greater than the damage toitself, i.e., only when ∆aW > ∆aS . Figure 3(b) shows

4

the region in which attacks by network S are effectiveby showing the fraction of failed nodes in both networksin a two-dimensional phase space as the value of p∗S,1 isincreased. Two solid lines with a slope of one indicate theregion in which an attack by network S is effective. Whenthe slope of function aW = f(aS) is greater than one(the region between the two shaded lines), increasing p∗S,1produces more damage in network W than in network Sand is thus an effective attack strategy.

In order to measure the effect of capturing nodes froma competitor network and how takeovers can modify theresilience properties of a network, we design a model inwhich network S is again more resilient than network W(TS < TW ) and where node ni of network W is taken overby network S if its internal failure time exceeds nτ , whereτ is a certain failure time and n a constant. Note thatthe longer a node in network W remains inactive (i.e.,the higher the value of n), the higher the probably thatit will be acquired by network S. Real-world examplesof this mechanism include sick or disabled prey in anecological system [25, 26] or countries whose economicsystems remain in recession for too long a period.

B. Take over and conservation laws

To evaluate the acquisition costs in both networks wedefine network wealth (capital) as proportional to two

0.0 0.2 0.4 0.6 0.8 1.0

p ∗1

0.0

0.2

0.4

0.6

0.8

1.0

f

aNetwork

SW

0.0 0.2 0.4 0.6 0.8 1.0

aS

0.0

0.2

0.4

0.6

0.8

1.0

aW

C

D

b

FIG. 3: Identifying the optimal parameters for attacking aweak network. Analytical approach. Strong network S (lessvulnerable) uses its own probability of internal failures p∗S,1 tocause damage in weak network W and, unavoidably, inducea partial self-destruction. Model parameters are: kS = 20,kS,W = kW,S = 10, kS = 5, tS = 10, tW = 10, pS,2 =pW,2 = 0.8, and p∗W,1 = 0.05. In (a), fraction of active nodes innetwork S and W, fS = 1−aS and fW = 1−aW , respectively.Strong network S (blue) deliberately initiates its own failures(increasing p∗S,1) to create larger damage in a weak (morevulnerable) network W (red). Note that the fraction of activenodes exhibits a hysteresis behaviour for both networks, witha critical point at pC ≈ 0.33. In (b), we investigate whenS should stop attacking W by increasing its probability ofinternal failure p∗S,1. Shown are the fractions of failed nodes,aS = 1 − fS and aW = 1 − fW . Between points C and D(dashed lines), an increase of p∗S,1 induces more failures in theweaker network, leading to a comparative benefit. Beyondpoint D, the attack is not worthwhile for network S since itsuffers the consequences more intensely than its competitor.

variables: the total number of links in the network—as defined in conservation biology [27, 28]—and the re-silience of the network. Note that if two networks havethe same number of links but different resiliences theirwealth is not equal. Note also that when network S ac-quires a node of degree kW,i from network W the overallresilience of network S decreases because it has acquireda weaker node. Thus network S pays a instantaneous,collective cost through a feedback mechanism that de-creases its resilience from an initial threshold TS to anew threshold T ′S .

One of the important issues in dynamic systems thathave a critical point as an attractor is whether a conser-vation of energy is required in local dynamic interactions[29–31]. To quantify how threshold T ′s changes in com-peting networks, we define a conservation law that relatesthe feedback mechanism to the resilience dynamics as

N 〈kS〉 (T ′S − TS) = kW,i(TW − T ′S). (4)

Here N is the size of the strong network, 〈kS〉 its averagedegree, and kW,i the degree of the node that has beentaken over. Thus, we assume that the more importantthe acquired node (i.e., the larger its degree kW,i), thegreater the cost to the resilience of network S, making it

0 50000 100000

t

0.30

0.32

0.34

0.36

0.38

0.40

T′ S

a

t

∆T′ S

2.0 2.5 3.0 3.5

n

0.0

0.1

0.2

0.3

0.4

0.5

0.6

[%]

btakeovers

T initialS

TfinalS

0.004 0.008

p1

0.0

0.2

0.4

0.6

0.8

1.0

1.2

f

pzpw

p′zp′w

pxpy

p′y

pW1c pS1cpx

cNetwork

SW

FIG. 4: Cost and takeover mechanisms in two competingBA networks. (a) Threshold T ′S of network S as a functionof time for two competing BA networks with n = 2.5 andτ = 50. Fluctuations in the evolution of T ′S are a consequenceof the degree of the acquired node: the higher the degree thehigher the increase of T ′S . (b) Fraction of takeovers and finalthreshold as a function of the time nτ required to acquire anode from the weak network, with τ = 50. As n increases,the number of takeovers decreases to zero. At the same time,the resilience of network S tends to the initial value TS = 0.3.(c) Due to takeovers the fraction of active nodes in the moreresilient network S can increase to values higher than one. Inthis example, network W is irreversibly damaged after p1 isrestored to its initial value.

5

more vulnerable to future attacks. As a result, when apredator (strong) network S increases its size N and itsdegree 〈ks〉, its acquisition cost, T ′S − TS , will decrease.

Here we quantify how threshold T ′S of the stronger net-work changes in competing networks where we assumethat threshold TW of the weaker network does not changebecause every node has the same threshold. The strongernetwork S has the initial number of nodes NS , the aver-age degree 〈kS〉. After a multiple takeovers, where S tookover nodes nw,1, nw,2, ..., nw,n with degrees kw,1, kw,2,..., kw,n, respectively, by using Eq. (4) we obtain

T ′S =(kw,1 + kw,1 + ...+ kw,n)TW +NS〈kS〉TS

NS〈kS〉+ kw,1 + kw,1 + ...+ kw,n. (5)

Figure 4(a) shows that when network S acquires nodesin network W the threshold T ′S of network S is increas-ingly affected as time passes. In this example, a node innetwork W is taken over by network S when the node isin failure state longer than nτ time steps, where n = 2.5and τ = 50. Note that as network S acquires weak nodes,its threshold T ′S increases and it becomes more vulner-able. Figure 4(b) shows the interplay between the timerequired to acquire a node nτ and the threshold T ′S . Notethat as nτ increases, takeovers become increasingly rareand the final threshold of network S approaches its initialresilience, here TS = 0.3.

Figure 4(c) shows that, if the example in Figure 2(b) isextended to include a takeover mechanism, a fraction ofactive nodes fS in network S—measured relative to theinitial number of nodes in each network—reaches valueshigher than one, with a peak at py → pz. Note thatwhen attacks cease (e.g., when, in an economic system,sanctions are lifted) decreasing the value of p1, pz → pw,the fraction of active nodes in network S increases butnetwork W is left irreversibly damaged (see the closedhysteresis p′y → p′z → p′w).

C. Threshold diversity in competing networks

Thus far we have studied competing interconnectednetworks in which there is only one threshold charac-terizing each network. However, in real-world intercon-nected networks commonly the functionality of a nodein a given network is not equally sensitive on the neigh-bours in its own and the other network. To this end,we assume that node ni in network S can externally failwith probability p2 if the fraction of the active neighborsof node ni in network S is equal to or lower than somethreshold TS , or if the fraction of the active neighborsof node ni in network W is equal to or lower than somethreshold TW,S . We similarly define external failure inthe less resilient network W by replacing threshold TSwith TW . The functioning of each node is thus depen-dent on its neighbors in network S and network W, butwith different sensitivities—different resilience to exter-nal fluctuations (see Supplementary Material).

Figure 5(a) shows, for a given set of parameters, atwo-parameter phase diagram of competing networks, amodel that incorporates the threshold separation for ex-ternal failure but excludes takeover and feedback mech-anisms. This model resembles that in Fig. 2 but uti-lizes different configurations. Suppose network S sponta-neously activates at time t0 but, due to differences in thevariables characterizing network S and network W, initi-ates a substitution mechanism, not a takeover. Thus eachtime node ni in network W spends a time period in an in-active mode that exceeds the substitution time—e.g., inecology, a time period without food—ni is replaced by anew node from network S. Figure 5(b) shows the fractionof active nodes in each network calculated relative to theinitial number of nodes at time t0. Fractions of activenodes of both networks exhibit a catastrophic disconti-nuity (a phase flip) at t ≈ 2, 000, which is characteristicof a first-order transition. Since both networks are in-

0.0 0.2 0.4 0.6 0.8 1.0p2

0.000

0.005

0.010

0.015

p1

a

0 500 1000 1500 2000 2500 3000

t

0.2

0.4

0.6

0.8

1.0

f

bNetwork

SW

0.0000 0.0002 0.0004 0.0006

p1

0.0

0.2

0.4

0.6

0.8

1.0

f

C'

C''

c

1800 1900 2000 2100 2200

t

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

∆(fI/f II)

d

FIG. 5: Quantification of the optimal attack duration.Model parameters are: mS = 6,mW = 3, mS,W = mW,S = 2.Similar as in Fig. 4 but now with two thresholds used fordefining resilience of each network. External failure thresh-olds are TW = 0.7, TS = 0.3, and TS,W = 0.4. In (a), wefind the phase diagram in model parameters (p1, p2) whereeach network has its own hysteresis. The takeover mecha-nism is disregarded and recovery time is τ = 50. In (b), foreach network we show the time evolution of the fractions ofactive nodes with the takeover mechanism included where weuse a takeover period of 1.5τ , p1 = 0.0004, and p2 = 0.6.At some point, S creates larger damage in a weak (less re-silient) network W than to itself. (c) Related with (b), asa result of the attack, both networks are damaged, but Wis damaged more. We also show how the fractions changewith decreasing p1 during the recovery phase. (d) Early-warning signal for determining when the attack should bestopped, defined as the change in the ratio between two frac-tions, fI(t + ∆t)/fII(t + ∆t) − fI(t)/fII(t), where ∆t = 20.The attack should be stopped when the indicator reaches themaximum.

6

terdependent, substituting nodes from the less resilientnetwork W affects the functionality of network S evenmore dramatically than that shown in Fig. 2. Thus be-yond some threshold we expect that additional weakeningof network W will also permanently damage network S.This demonstrates how dangerous an attacking strategycan be for an attacker in a system of interdependent net-works, e.g., between countries that are at the same timecompetitors and economics partners.

Figure 5(c) shows that when the attacks and substitu-tions cease, the fractions of active nodes in network S andnetwork W reach points C ′ and C ′′, respectively. If theprobability of internal failure p1 spontaneously decreasesduring the recovery period, because of network interde-pendence the functionality of network S is not substan-tially improved. The triumph of network S over networkW has its price. In ecology, for example, although thepopulation of each species tends to increase, a dominancestrategy is risky, e.g., the extinction of a key species cantrigger, through a cascade mechanism [14, 32], the ex-tinction of many other species [33].

Figure 5(d) shows the change in the ratio between thefraction of active nodes in network S and network W as afunction of time. This ratio can serve as an early-warningmechanism [34] that indicates when attacks should bestopped. Optimally, the stopping time for attacks willbe when the ratio reaches its maximum.

Finally, Fig. 6(a) shows that when the feedback mech-anism (the cost of taking over) defined in Eq. (4) isincluded, the fraction of active nodes in each networkexhibits an even richer discontinuous behaviour than inFig. 5(c), where the cost was excluded. After 50,000steps, because of the decrease in network S’s resilienceafter each substitute, the final fraction of active nodes innetwork S is substantially smaller than the correspond-ing fraction in Fig. 5(c) (i.e., when the cost is excluded).At the same time, Fig. 6(b) shows that an increase in thetakeover time nτ decreases the fraction of substitutes.

IV. SUMMARY

In conclusion, we have presented a theoretical frame-work based on resilience, competition, and phase tran-sitions to introduce a cost-of-attack concept that relatesfeedback mechanisms to resilience dynamics defined us-ing a linear conservation law. Our model for competingnetworks can be applied across a wide range of human ac-tivities, from medicine and finance to international rela-tions, intelligence services, and military operations. Thisability to measure network resilience and its cost is cru-cial because every weakening of the resilience reduces theprobability of network survival under future attacks. Forexample, in socio-economic systems a network approachto dominating other countries economically can be as ef-fective as using military weapons. Interdependent linksestablished between countries during prosperous timescan facilitate sanctions (intentional fluctuations) that are

0 10000 20000 30000 40000 50000

t

0.0

0.2

0.4

0.6

0.8

1.0

f

aNetwork

SW

1.5 2.0 2.5 3.0

n

0.0

0.2

0.4

0.6

0.8

1.0

[%]

btakeovers

T initialS

TfinalS

FIG. 6: Evaluating the effect of the cost mechanism. Modelparameters are: mS = mW = 3, mS,W = mW,S = 2, p1 =0.0004 p2 = 0.6. (a) The cost of the attacking strategy withtakeover mechanism additionally decreases network resilience.The fraction of active nodes exhibits more discontinuities thanin the case where the cost of an attack was excluded (Fig. 5).This is a consequence of the larger change in the resilience ofS due to inclusion of the cost mechanism. (b) The fractionof takeovers and threshold T ′S of the stronger network S as afunction of takeover time nτ , with τ = 50.

used as a weapon when more resilient countries try toovercome less resilient countries. They also can facilitatethe global propagation of economic recessions (sponta-neous fluctuations). During long economic crises theseinterdependent links can become fatal for less resilientcountries. In addition to the intentional fluctuationscharacteristic of human societies, our methodology canalso be applied to a broad class of complex systems inwhich spontaneous fluctuations occur, from brain func-tioning to ecological habitats and climate fluctuations[27, 33, 35–40].

V. METHODS

A. Resilience dynamics in finance

Here we demonstrate that the resilience dynamics in fi-nance can be presented in terms of a conservation (linear)law, where the threshold is controlled by an asset-debtratio. Recall that in our model a node externally failswith a certain probability when the total fraction of itsactive neighbors is equal to or lower than the fractionalthreshold Th. The larger the Th value, the less resilientthe network. In quantifying the impact of a perturba-tion (attack) on the network, we shall demonstrate thatthe more severe the attack, the larger the impact on thenetwork resilience.

Suppose a bank ni has an interbank asset ABi investedequally in each of its ki neighboring banks. Bank ni hasalso some asset Ai, considered as a stochastic variable.Following Refs. [19, 41], we define a bank to be solvent(active) when (1 − φ)ABi + Ai − LBi > 0, i.e., when thebank’s assets exceed its liabilities, LBi . Here φ representsthe fraction of inactive neighboring banks that ni canwithstand and still function properly. Note that this φ isrelated to threshold Th as Th = 1 − φ, since we assume

7

that the number of incoming links is equal to the numberof outgoing links, ABi = LBi . The larger the Ai value, themore stable the bank. Suppose that for each bank thereis a linear dependence between ABi and Ai on one sideand the network degree k—e.g., that ABi = ki and Ai =0.5ki. Then a bank is inactive when 1 − Th = φ = 0.5(Th = 0.5) or when at least 50% of its neighboring banksare inactive. Let us assume next that Ai increases dueto an external perturbation Ai = 0.5 + ε. Then the newthreshold is equal to

T ′h = 0.5− ε/ki, (6)

or

ki(T′h − Th) = −ε. (7)

If the external perturbation is negative (positive), oralternatively, the asset increases (decreases) (Eq. (6)),the threshold increases (decreases) and the resilience de-creases (increases). The larger the number of neighbors(ki), the smaller the change in the network resilience.Note that if the external perturbation attacking one nodeis shared not only by its neighbors but by the entire net-work with N nodes, then ki is replaced by N〈ki〉 as inEq. (4) in the paper. Note that if we replace the lineardependence between assets and degree with a non-linear(power-law) dependence, e.g., ABi = kαi , where α is aconstant parameter, we obtain a similar relationship toEq. (7), kαi (T ′h − Th) = −ε.

B. Resilience dynamics in interconnected networks

For two interconnected networks we focus on a meanfield approximation of the level of external and internalfailures between nodes. Every node has an internal fail-ure probability p1 [20]—assumed, for reasons of simplic-ity, to be the same in both networks, p1,S = p1,W = p1.If each node in network S has kS links with nodes inits own network and kW,S links with nodes in networkW, here we define that there must be more than mS

nodes in network S and mW,S nodes in network W ifthe nodes in S are to function properly. In contrast, ifthe number of inactive nodes in network S is ≤ mS , theprobability that the node in network S will externallyfail is p2. Similarly, if the number of inactive nodes innetwork W is ≤ mW,S , the probability that the node innetwork S will externally fail is p2. For simplicity reasonp2,S = p2,W = p2. We denote the time averaged fractionsof failed nodes in network S and network W as aS andaW , respectively. Using combinatorics, we calculate theprobability that a node in network S will have a criticallydamaged neighborhood among its neighbours in S to be

ES =∑mS

j=0

(kSkS−j

)akS−jS (1 − aS)j . Similarly, we calcu-

late the probability that a node in network S will havea critically damaged neighborhood among its neighbours

in W to be EW =∑mW,S

j=0

(kW,S

kW,S−j)akW,S−jW (1−aW )j . The

probability that a node will fail externally due to failures

in network S (network W) is p2ES (p2E

W ). If we denotethe internal failures in network S by A and the externalfailures by B, and the external failures in network W byC, then the probability that a randomly chosen node inS will fail is a1 ≈ P (A) + P (B) + P (C)− P (A)(P (B) +P (C))−P (B)P (C)+P (A)P (B)P (C). If we assume thatA, B, and C are not mutually exclusive, but interdepen-dent events, finally we obtain

aS ≈ p∗1 + p2(ES + EW )− p∗1p2(ES + EW ) (8)

− p2p2ESEW + p∗1p2p2E

SEW . (9)

where p∗1 = 1 − exp(−p1τ), and as in Ref. [20] node jrecovers from an internal failure after a time period τ .Similarly, from the above Equation we obtain the frac-tion of failed nodes in network W (either internally orexternally failed) aW , by interchanging S and W.

C. Coupled BA interdependent networks withequal connectivity but different threshold

For the coupled interdependent BA networks withequal connectivity but different threshold, where TS <TW , Fig. 7 shows that for each of two interconnected net-works the fraction of active nodes simultaneously jumpsfrom a stable state to another one. Phase flipping is ob-tained by setting the network close to a critical pointthat is reported for a single network in Ref. [20]. Sincethe threshold in network S is substantially smaller thanthe fraction in network W, and so S is more resilient thannetwork W, the fraction of active nodes in network S islarger than the fraction of active nodes in network W.Thus the volatile phase-flipping in network functional-ity is more dangerous for network W than for networkS. Approximately, the fractions, as a function of time,can model the populations of preys and predators and so

0 5000 10000 15000 20000

t

0.4

0.5

0.6

0.7

0.8

0.9

1.0

f

NetworkSW

FIG. 7: Coupled interdependent networks with equal con-nectivity but different threshold. For the parameters weuse: τ = 50, p1 = 0.0012, p2 = 0.48, mS = mW = 3,mS,W = mW,S = 2, TS = 0.3, TW = 0.7. The larger theTh value, the less resilient the network. We obtain that bothnetworks exhibit the same hysteresis. The parameters are setin a part of phase space to enable phase flipping between ac-tive and inactive states. However, due to different thresholds,network S is more resilient than network W—the average frac-tion of active nodes in network S is practically always largerthan in network W.

8

be related to the periodic solutions of the Lotka-Volterra (predator-prey) equations [38].

[1] Bastolla, U. et al. The architecture of mutualistic net-works minimizes competition and increases biodiversity.Nature 458, 1018-1020 (2009).

[2] Rohr, R. P., Saavedra, S. & Bascompte, J. On the struc-tural stability of mutualistic systems. Science 345, 416(2014).

[3] Aguirre, J., Papo, D. & Buldu, J. M. Successful strategiesfor competing networks. Nat. Phys. 9, 230-234 (2013).

[4] D’Souza, R. M. Complex network: A winning strategy.Nat. Phys. 9, 212-213 (2013).

[5] Kivela, M., Arenas, A., Barthelemy, M., Gleeson, J.P.,Moreno, Y. & Porter, M. Multilayer networks. J. Com-plex Networks 2, 203-271 (2014).

[6] Scholtes, I. et al. Causality-driven slow-down and speed-up of diffusion in non-Markovian temporal networks. Nat.Commun. 5, 5024 (2014).

[7] Thebault, E. & Fontaine, C. Stability of ecological com-munities and the architecture of mutualistic and trophicnetworks. Science 329, 853-856 (2010).

[8] Richardson, L. F. The Mathematical Psychology of War.Nature 135, 830-831 (1935).

[9] Shakarian, P., Lei, H. & Lindelauf, R. Power GridDefense Against Malicious Cascading Failure. arXiv1401.1086v1 (2014).

[10] Scheffer, M. et al. Catastrophic shifts in ecosystems. Na-ture 413, 591-596 (2001).

[11] Albert, R., Jeong, H. & Barabasi, A. L. Error and at-tack tolerance of complex networks. Nature 406, 378-382(2000).

[12] Cohen, R., Erez, K., ben-Avraham, D. & Havlin, S. Re-silience of the internet to random breakdowns. Phys. Rev.Lett. 85, 46264628 (2000).

[13] Reis, S. D. S. et al. Avoiding catastrophic failure in cor-related networks of networks, Nat. Phys. 10, 762-767(2014).

[14] Buldyrev, S. V. et al. Catastrophic Cascade of Failures inInterdependent Networks. Nature 464, 1025-1028 (2010).

[15] Dorogovtsev, S. N. & Goltsev, A. V. Critical phenom-ena in complex networks. Rev. Mod. Phys. 80, 1275-1335(2008).

[16] May, R. Networks and webs in ecosystems and financialsystems. Phil. Trans. Soc. A 371, 20120376 (2013).

[17] Downing, A. S., van Nes, E. H., Mooij, W. M. & Scheffer,M. The Resilience and Resistence of an Ecosystem to aCollapse of Diversity. PloS One 7, e46135 (2012).

[18] Barabasi, A. -L. & Albert, R. Emergence of scaling inrandom networks. Science 286, 509-512 (1999).

[19] Podobnik, B., Horvatic, D., Dickison, M. & Stanley, H.E. Preferential attachment in the interaction between dy-namicly generated interdependent networks, EPL 100,50004 (2012).

[20] Majdandzic, A. et al. Dynamic Networks with Sponta-neous Recovery. Nat. Phys. 10, 34-38 (2014).

[21] Watts, D. J. A simple model of global cascades on ran-dom networks. Proc. Natl. Acad. Sci. USA 99, 5766-5771

(2002).[22] Podobnik, B. et al. Network Risk and Forecasting Power

in Phase-Flipping Dynamic Networks. Phys. Rev. E 89,042807 (2014).

[23] Erdos, P. & Renyi, A. On random graphs. I. Publ. Math.6, 290-297 (1959).

[24] Newman, M. E. J. Assortative mixing in networks. Phys.Rev. Lett. 89, 208701 (2002).

[25] Errington, P.L. Predation and vertebrate populations. Q.Rev. Biol. 21, 144-177 (1946).

[26] Genovart, M. et al. The young, the weak and the sick: ev-idence of natural selection by predation. PloS One. 5(3),e9774 (2010).

[27] Hunter, M. L. Fundamentals of conservation biology. (Oxford: Blackwell Science, 1996).

[28] Costanza, R. et al.. The value of the world’s ecosystemservices and natural capital. Nature 387, 253-260 (1997).

[29] Bak, P., Tang, C. & Wiesenfeld, K. Self-organized criti-cality: an explanation of 1/f noise. Phys. Rev. Lett. 59,381-384 (1987).

[30] Noel, P. A., Brummitt, C. D. & D’Souza, R. M. Control-ling Self-organizing Dynamics on Networks Using Modelsthat Self-Organize. Phys. Rev. Lett. 111, 078701 (2013).

[31] Markovic, D. & Gros, C. Power laws and self-organizedcriticality in theory and nature. Phys. Rep. 536, 41-74(2014).

[32] Mold, J. W. & Stein, H. F. The cascade effect in the clin-ical care of patients. New England Journal of Medicine314, 512-514 (1986).

[33] Estes, J. A., Duggins, D.O. & Rathbun, G. B. The ecol-ogy of extinctions in kelp forest communities. Conserva-tion Biology 3, 252-264 (1989).

[34] Dakos, V. et al. Slowing down as an early warning signalfor abrupt climate change. Proc. Natl. Acad. Sci. USA105, 14308-14312 (2008).

[35] Adger, W. N. et al. Social-ecological resilience to coastaldisasters. Science 309, 1036-1039 (2005).

[36] Thompson, J. N. The geographic mosaic of coevolution.(University of Chicago Press, 2005).

[37] De Lange, H. J., Sala, S., Vighi, M. & Faber, H. J. Eco-logical vulnerability in risk assessment. A review and per-spectives: Science of the total Environment 408, 3871-3879 (2010).

[38] Nowak, M. A. & Highfield, R. SuperCooperators: Altru-ism, Evolution, and Why We Need Each Other to Suc-ceed. (Simon and Schuster, 2011).

[39] Vespignani, A. Modelling dynamical processes in complexsocio-technical systems Nat. Phys. 8, 32-39 (2012).

[40] Battiston, S., Caldarelli, G., Georg, C., May, R. &Stiglitz J. Complex derivatives. Nat. Phys. 9, 123-125(2013).

[41] Gai, P. & Kapadia, S. Contagion in Financial Networks.Proc. R. Soc. A 466, 2401 (2010).