Enhancing resilience of interdependent networks by healing
aa r X i v : . [ phy s i c s . s o c - ph ] S e p Enhancing resilience of interdependent networks by healing
Marcell Stippinger ∗ and János Kertész
2, 1, † Department of Theoretical Physics, Budapest University of Technology and Economics,H-1111 Budafoki út 8., Budapest, Hungary Center for Network Science, Central European University, H-1051 Nádor u. 9., Budapest, Hungary
Interdependent networks are characterized by two kinds of interactions: The usual connectivitylinks within each network and the dependency links coupling nodes of different networks. Due to thelatter links such networks are known to suffer from cascading failures and catastrophic breakdowns.When modeling these phenomena, usually one assumes that a fraction of nodes gets damaged inone of the networks, which is followed possibly by a cascade of failures. In real life the initiatingfailures do not occur at once and effort is made replace the ties eliminated due to the failing nodes.Here we study a dynamic extension of the model of interdependent networks and introduce thepossibility of link formation with a probability w , called healing, to bridge non-functioning nodesand enhance network resilience. A single random node is removed, which may initiate an avalanche.After each removal step healing sets in resulting in a new topology. Then a new node fails and theprocess continues until the giant component disappears either in a catastrophic breakdown or in asmooth transition. Simulation results are presented for square lattices as starting networks underrandom attacks of constant intensity. We find that the shift in the position of the breakdown hasa power-law scaling as a function of the healing probability with an exponent close to . Below acritical healing probability, catastrophic cascades form and the average degree of surviving nodesdecreases monotonically, while above this value there are no macroscopic cascades and the averagedegree has first an increasing character and decreases only at the very late stage of the process.These findings facilitate to plan intervention in case of crisis situation by describing the efficiencyof healing efforts needed to suppress cascading failures. PACS numbers: 89.75.Fb, 64.60.aq, 64.60.De, 89.75.DaKeywords: dynamic interdependent networks; critical healing; first and second order percolation transition.
I. INTRODUCTION
Robustness is one of the key issues for network main-tenance and design [1–3]. The representation of complexsystems has been limited to single networks for a longtime [4]. In many cases, however, coupling between sev-eral networks takes place [5, 6]. An important case isthat of interdependency [7, 8] where there are two kindsof links: connectivity and dependency links. An exam-ple of interdependent networks is the ensemble of theInternet and the power supply grid where telecommuni-cation is used to control power plants and electric poweris needed to supply communication devices [7]. Connec-tivity links model the relation of the entities within thesame sector, spanning in the above example a power sup-ply network and a telecommunication network. Depen-dency links depict the basic supplies an entity dependson which are supplied by entities in the other network.If a supplier fails its dependent nodes fail as well. Thesystem is viable if a giant component of interconnectedunits exists in both networks. In the 28 September 2003blackout in Italy it came to evidence that the interdepen-dency of the two networks makes them more vulnerablethan ever thought before [7]. Similar relations occur inthe economics between banks and firms or funds. Banks ∗ [email protected] † [email protected] are related through interbank loans, firms through sup-ply chains and the interdependence comes from loans andsecurities. Inappropriate asset proportions can also leadto global avalanches as seen in the subprime mortgagecrisis [9].Interconnecting similar subsystems used to increase ca-pacity was shown beneficial as long as it does not openpathways to cascades [10]. However, in interdependentnetworks, the aspect of robustness was considered withthe conclusion that broadening the degree distribution ofthe initial networks enhances vulnerability [11]. A cost-intensive intervention to strengthen robustness is to up-grade nodes to be autonomous on some resources [12].Because failures propagate rapidly in infrastructurenetworks, they cannot be stopped by installing backupdevices during the spreading of the damage. but ratherthey require already existing systems. After the cas-cade of failures, damaged devices or elements can be re-placed by new, functioning ones identical to the orig-inals [13]. In contrast to engineered systems, social oreconomic networks are highly responsive and may reactquickly [14, 15]. When a failure occurs considerable ef-fort is made to reorganize the network and rearrange theload of failing elements among functioning ones. The roleof the failing entities is taken over by similar partici-pants. Such processes can be modeled by healing, i.e.,substituting some of the failed elements by new ones.The timescale of an economic crisis is wide enough forthe network to completely restructure itself [15]. So farsuch mechanisms have only been studied for simple net- a ) b ) New linkprob. w AB Beforeattack
Figure 1. a) Failures, represented by red dots, affect the nodesone by one in a random order. Whenever a node fails, its coun-terpart, that is, the node in the other network which dependson it, fails as well. In both networks, only the largest con-nected component (LCC) survives. This constraint can causefurther nodes to fail in both networks, which trigger furthershrinking of the LCC, and so on, illustrated by the shadedareas. b) The neighbors of a failing node try to heal the net-work, such that two functioning neighbors of a removed nodeestablish a connectivity link with probability w . works [16–18]. Here we extend the original model [7] ofcascading failures of interdependent networks. After eachremoval, the healing process attempts to bypass the re-moved node with a new connectivity link (see Fig. 1). Inthis paper, we demonstrate how healing acts on interde-pendent networks.The outline of the paper is as follows. In Sec. II wedefine the node failure process in a dynamic way. Weintroduce initial failures one by one to be able to applyhealing at every failure event. Then we relate the originalversion of cascading failures to our model as a special caseand give formulas for comparing the order parameter ofthe two models. The scaling properties of the healing areexplained along with the numeric results in Sec. III. InSec. IV we discuss the properties of the cascades withmicroscopic insight to the model. Finally we concludeour findings in Sec. V. II. THE MODEL
In the standard model of interdependent networks [8]the computer-generated model-system is built up of twotopologically identical networks A and B , e.g., squarelattices of size N = L × L , where each node has connec-tivity links within the same network. In addition, depen-dency links couple between the networks, which are bidi-rectional one-to-one relationships connecting randomlyselected pairs of nodes from the two networks. If any ofthe nodes fails its dependent pair fails too. A node in anynetwork can function only if it is connected to the largestconnected component of that network the node which it a) b) ) d) Figure 2. Part of Network A of a simulated system at p = 0 . at a) no healing ( w = 0 . ) b) below the critical healing ( w =0 . , the average degree stays below ) and c) slightly abovethe critical healing ( w = 0 . ). d) This latter w = 0 . systemis also represented at p = 0 . where one can observe that thenodes get more and more connected and the healing processestablishes links between distant nodes. depends on is also functional, otherwise it fails, i.e., it isremoved from the network.The existence of a macroscopic connected componentin a single network is treated by percolation theory. In theusual case, for a lattice it describes a second-order phasetransition between the phases with and without the exis-tence of a giant component [19]. Adding interdependencyallows cascades of failures to propagate between the twonetworks. The threshold the network can survive withoutcollapse decreases considerably in this setting [8].The collapse due to cascades was shown to be a firstorder transition if the dependency links have unlimitedrange while the transition is of second order if the rangeis less than a critical length r c [8, 20]. Moreover, the firstorder transition has a hybrid character with scaling onone of its sides [21, 22].As mentioned in the Introduction we first introduce adynamic process on the interdependent network model.In the setting of two interdependent networks of generaltopology this dynamic process consists of the repetitionof attacks and relaxations to a rest via cascades. (SeeFig. 1.) Let us suppose that failures affect the nodes oneby one in a random order which defines a timeline. Onetime step is identified with the external attack of onenode. Time is measured by the number of time stepsnormalized by N for systems of different sizes to be com-parable: elapsed time = 1 − p = number of time steps N The externally introduced failure in network A may sep-arate the largest connected component (LCC) into twoor more parts where only the largest one survives. Allthe failed nodes have dependency connections to nodesof the network B causing their failure. Again, the LCCof B may get fragmented and only the largest part sur-vives. This cascading procedure is repeated until no morefailures happen. Of course, our model can easily be gen-eralized to any number of interdependent networks andany density of dependency links.Our aim is to introduce healing into this dynamicmodel. The procedure is as follows: After an externallyintroduced failure (which may cut off a part of theLCC) the healing step follows. Two remaining, function-ing neighbors of a removed node establish a connectivitylink with an independent probability w . (See part b) inFig. 1.) Then the dependent nodes of the removed nodesare removed from the other network. After the propaga-tion of the failure there, again, two functioning neighborsof a removed node establish a connectivity link probabil-ity w . Due to the separation of small components, furtherdamages might propagate back and forth within the net-work, always followed by a healing step. Here, the heal-ing step means that all pairs of neighbors of each failednode is considered as a candidate for a new connectiv-ity link with an independent probability w , then, afterhaving selected the candidates, the connectivity links areestablished simultaneously. The process goes on until nomore separation of components occurs. The healing linksmay change the topology considerably, bridging largerand larger distances as the time goes on (Fig. 2). Oncea critical fraction (1 − p c ) of nodes are removed, a catas-trophic cascade destroys the remaining system.The w = 0 case is simply the dynamic version of thewell studied model of Li et al. . In [8] a fraction (1 − q ) of the original network is destroyed in the first step thenthe size of the giant component after the relaxation ofcascades is traced as a function of q . The important dif-ference between this procedure and ours is that in the ver-sion of Li et al. nodes may be accidentally attacked, whichalready fail in our step-by-step (dynamic) model. Let P ∞ denote the fraction of remaining nodes as a function ofthe fraction of attacked nodes (1 − p ) in the step-by-step model. The number of unattacked but disconnectednodes is [ p − P ∞ ( p )] N . The probability of randomly de-stroying an already disconnected (but not attacked) nodeis ( p − P ∞ ) /p , so the implicit relation between the twoattacking methods is [23] − p ( q ) = Z q − e p − P ∞ ( e p ) e p d e p. (1)Due to the small false target ratio in the random attack,the threshold values of the two models are close. The extrapolated threshold value for the infinite system sizein case w = 0 is p c = 0 . ± . , in good agreementwith the result of Li et al. . III. SCALING WITH THE HEALINGPROBABILITY
The order parameter P ∞ of our model depends notonly on the fraction of attacked nodes but also on thehealing parameter. According to one’s intuition, the datashow that the critical attack (1 − p c ) increases monoton-ically with w .We executed Monte Carlo simulations of our modelwith both periodic and open boundary conditions onsquare lattices starting networks of linear size L = 20 , , , and with , , , and runsrespectively, and we measured that the execution time inour implementation scaled approximately as N . = L . .In the square lattices connectivity links join nodes totheir nearest neighbors within the same network. Depen-dency links were established by first creating the triv-ial mapping between the topologically identical lattices,then randomly shuffling the end of the links. The p c -sare then obtained averaging over the vertical axis: for agiven number of surviving nodes P ∞ ( p, w ) , we averagedthe proportion of nodes − p attacked one-by-one. Fig. 3shows the averaged curves for different values of w . Theshape of the P ∞ ( p, w ) curves suggests the scaling in theform of anisotropic resizing from the S ( p = 1 , P ∞ = 1) point: − P ∞ (1 − p, w ) = 1 − a ( w ) P ∞ (cid:18) − pc ( w ) , (cid:19) (2)which is asymptotically satisfied in the w → limit.In the infinite lattice limit, the initial few attacks al-most surely occur in different parts of the lattice and donot raise cascades, only the attacked points fail, P ∞ ( p ) = p if p ∼ . The unit slope at S with respect to p canbe expressed by differentiation and yields a ( w ) ≡ c ( w ) .Let us express the fraction of unattacked nodes relativeto the threshold without healing: ∆ p = p − p c ≤ .The change in the threshold value ∆ p c ( w ) = p c ( w ) − p c can be identified by the largest ∆ p where P ∞ has aninfinite slope (see Fig. 4): lim ∆ p → ∆ p c ( w )+0 ∂∂ ∆ p P ∞ (1 − p c − ∆ p, w ) = ∞ . Substituting it into (2) yields a ( w ) =(1 − p c − ∆ p c ( w )) / (1 − p c ) . The increase in lifetime, − ∆ p c ( w ) , has a general scaling behavior expressed in − ∆ p ( w ) = h w γ (3)for small w -s, in the range [0 . , . . For the purposeof precise measurement we created simulation data forall system sizes with step size . for w ∈ [0 . , . additional to that shown in Fig. 3. The measurementis hampered by large fluctuations of the small systems,therefore we extrapolated to infinite system size usingstandard finite size scaling [24]. We used both systems . . . . . . . . . . . . . . . . . P ∞ ( p ) ✿ ❢ r ❛ ❝ ✳ ♦ ❢ ❛ ❧✐ ✈ ❡ ♥ ♦ ❞ ❡ s p ✿ ❢r❛❝t✐♦♥ ♥♦❞❡s ♥♦t ❛tt❛❝❦❡❞ ❡①t❡r♥❛❧❧②❤❡❛❧✐♥❣ ♣r♦❜✳ w = 0 . w = 0 . w = 0 . w = 0 . w = 0 . w = 0 . w = 0 . . . . . . . . . . . . . . . . . . ( − P ∞ ( p )) · ( − p c ) / ( − ( p c + h w γ )) (1 − p ) · (1 − p c ) / (1 − ( p c + hw γ )) ✿ s❝❛❧✐♥❣ . . .
65 0 .
68 0 . ❤❡❛❧✐♥❣ ♣r♦❜✳ w = 0 . w = 0 . w = 0 . w = 0 . w = 0 . w = 0 . w = 0 . Figure 3. (left)
The fraction P ∞ ( p ) of remaining nodes of the original N = 320 × nodes as a function of the fraction p ofnodes not attacked externally. Note: In order to sharply mark the breakdown, averaging in variable p is done for a given P ∞ ( p ) over simulations. (right) The same curves scaled on each other using relation (3). In (both) parts, plots from the right to theleft correspond to the range w = 0 . to w = 0 . respectively with a step size . , solid lines indicate steps of . . . . . . . . . . . . . . . . . p c ✿ ❝ r ✐ t ✐ ❝ ❛ ❧ ❛ tt ❛ ❝ ❦ P ∞ ✿ s ✐ ③❡ ❜ ❡ ❢ ♦ r ❡❝ ♦ ❧❧ ❛ ♣ s ❡ w ✿ ❤❡❛❧✐♥❣ ♣r♦❜❛❜✐❧✐t② p c ( w ) P ∞ ( p c ( w )+ , w ) Figure 4. p c as a function of w depicts the fraction ofunattacked nodes at the transition for the starting N =320 × system size. P ∞ ( p c ( w )+ , w ) is the giant componentsize just before the transition as a function of w . Its non-zerovalue shows the jump in the first order transition and its zerovalue above w c = 0 . ± . indicates a smooth transition. with periodic and open boundary conditions and mea-sured the finite size fluctuations in p c (approaching fromthe p > p c domain in accord with the hybrid characterof the transition) which yields slightly different scalingexponents within the error tolerance for the two systems(respectively ν p = 1 . ± . and ν o = 1 . ± . ) fromwhich we deduce ν ≈ . . The finite size scaling mea-surements, yielding p c = 0 . ± . , are representedin Fig. 5.The parameters of Eq. (3) are first fitted for each sys-tem size N = L × L = 20 , , , and , thenthe infinite size limit is obtained using /L extrapolation.The systems with periodic and open boundary conditionssimulated at different system sizes collapse well yielding h = 0 . ± . and γ = 1 . ± . for the infinitesize network. . . . . . . . . .
00 0 .
02 0 .
04 0 .
06 0 .
08 0 . p c ✿ ❝ r ✐ t ✐ ❝ ❛ ❧ ❢ r ❛ ❝ t ✐ ♦ ♥ ♦ ❢ ♥ ♦ ❞ ❡ s L − /ν ✿ s❝❛❧❡❞ s②st❡♠ ❧❡♥❣t❤♣❡r✐♦❞✐❝ ❜❝✳♦♣❡♥ ❜❝✳ Figure 5. The standard deviation of the critical attack fraction − p c was used to obtain the length scaling exponents ν p =1 . ± . and ν o = 1 . ± . for periodic (filled) and open(void symbols) boundary conditions as described in Sec. III.Then p c on the is plotted against L − /ν giving good collapsefor the infinite system size. IV. CASCADES CHANGE TOPOLOGY
We call cascades all events involving more nodes thanthe attacked one and its dependency counterpart. Thesize (number of nodes involved compared to the startinglattice size) of typical cascades is small up to the pointof breakdown.The healing dynamics changes the network topologyand the average degree as well. Fig. 6 allows us to de-scribe a transition: below a critical healing threshold w c we find a sharp breakdown in the number of survivingnodes. The critical healing is defined as the lowest w forwhich the P ∞ ( p ) function does not have an infinite slope.In our simulation we observe w c = 0 . ± . . For w > w c also there is no macroscopic cascade and P ∞ ( p ) goes smoothly to zero in a second-order transition as p decreases (see also Fig. 4). Figure 6. (left and inset)
The average degree on the horizontalaxis as a function of the fraction of dead nodes on the verticalaxis for the starting N = 320 × system size. The averagedegree remains constant for w c = 0 . ± . . Plotted linesfrom the right to the left correspond to the range w = 0 . to w = 0 . respectively with a step size . , solid linesindicate steps of . . The shaded areas represent . stan-dard deviations in the left part. In the inset, shaded areas areonly plotted for solid lines and represent . standard devia-tion. (right) The fraction P ∞ ( p ) of failing nodes as a functionof the fraction p of nodes not attacked externally using thesame averaging as in Fig. 3. Above w c = 0 . ± . thereis no breakdown. The healing performed by the k neighbors introduces w (cid:0) k (cid:1) new links on average. A rough mean-field estimateof w c is the healing probability, which conserves the av-erage degree in the initial settings, leading to w c (cid:0) k (cid:1) = k (each link joins nodes). As the square lattice has k = 4 ,the result is w mean-field c = 1 / . According to the left plotin Fig. 6 we find that the average degree k = 4 changesleast through the simulation for w c = 0 . ± . , whichagrees well with the critical healing determined from the P ∞ curves [25]. The change in the topology along withthe trend of the average degree can be observed in Fig. 2.Below the critical healing w c the average degree is mono-tonically decreasing function of − P ∞ and the connectiv-ity links remain local, conserving the disordered lattice- like topology. Thorough inspection shows that all simula-tions end with a cascade wiping out all of the remainingnetwork at p c ( w ) . Above w c the healing promotes theformation of densely connected regions and connectivitylinks begin to join distant nodes. We remark that in theterminal stage the defined dynamics removes all nodesand links in both cases. In summary, the difference isthat for w < w c the process terminates with a macro-scopic cascade, while for w > w c there is no macroscopiccascade. In the latter case the average degree increasesuntil it has to decrease due to the small number of re-maining nodes. V. CONCLUSIONS
We examined the consequences of healing by edge for-mation in interdependent networks under random at-tack. We found that the increase in resilience of thenetwork, measured in the number of survived attacks,has power-law scaling with the probability w of healing.By establishing new random links in the neighborhoodof the failed nodes, we delayed the collapse of the net-work through the hindering of cascades. We found thatit is possible to completely suppress macroscopic cascad-ing failures for healing probabilities higher than a criti-cal value w c ; we demonstrated that this critical healingprobability keeps the average degree of the nodes closeto the initial value while the network topology changes.By analyzing healing efficiency, these findings can aid inthe development of intervention strategies for crisis situ-ations. The presented model contains a number of unre-alistic features, like the starting lattice, the unboundedrange and the high density of dependency links and thenon-locality of the healing links. Further studies shouldclarify the role of these simplifications. VI. ACKNOWLEDGEMENTS
This work was partially supported by the EuropeanUnion and the European Social Fund through projectFuturICT.hu (Grant No.: TAMOP-4.2.2.C-11/1/KONV-2012-0013). JK thanks MULTIPLEX, Grant No. 317532.Thanks are due to Éva Rácz for her help at the earlystage of this work and to Michael Danziger for a criticalreading of the manuscript. [1] R. Cohen, K. Erez, D. ben Avraham, and S. Havlin,Phys. Rev. Lett. , 4626 (2000).[2] D. S. Callaway, M. E. J. Newman, S. H. Strogatz, andD. J. Watts, Phys. Rev. Lett. , 5468 (2000).[3] R. Albert, H. Jeong, and A. L. Barabasi,Nature , 378 (2000).[4] M. E. J. Newman, Networks: An Introduction (OxfordUniversity Press, 2010).[5] M. Kivelä, A. Arenas, M. Barthelemy, J. P. Gleeson,Y. Moreno, and M. A. Porter, pre-print (2013), arXiv:1309.7233 [soc-ph].[6] D. Y. Kenett, J. Gao, X. Huang, S. Shao, I. Vodenska,S. V. Buldyrev, G. Paul, H. E. Stanley, and S. Havlin, in
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