Inter-similarity between coupled networks
Roni Parshani, Celine Rozenblat, Daniele Ietri, Cesar Ducruet, Shlomo Havlin
aa r X i v : . [ phy s i c s . d a t a - a n ] O c t Inter-similarity between coupled networks
Roni Parshani, Cline Rozenblat,
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Daniele Ietri, Csar Ducruet, and Shlomo Havlin Minerva Center & Department of Physics, Bar-Ilan University, Ramat Gan, Israel Institute of Geography, Lausanne, Switzerland (Dated: October 22, 2018)Recent studies have shown that a system composed from several randomly interdependent net-works is extremely vulnerable to random failure. However, real interdependent networks are usuallynot randomly interdependent, rather a pair of dependent nodes are coupled according to some regu-larity which we coin inter-similarity. For example, we study a system composed from an interdepen-dent world wide port network and a world wide airport network and show that well connected portstend to couple with well connected airports. We introduce two quantities for measuring the levelof inter-similarity between networks (i) Inter degree-degree correlation (IDDC) (ii) Inter-clusteringcoefficient (ICC). We then show both by simulation models and by analyzing the port-airport sys-tem that as the networks become more inter-similar the system becomes significantly more robustto random failure.
Recently, an American Congressional Committee high-lighted the intensified risk in an attack on national infras-tructures, due to the growing interdependencies betweendifferent infrastructures [1]. However, despite the highsignificance and relevance of the subject, only a few stud-ies on interdependent networks exist and these usuallyfocus on the analyses of specific real network data [2–6].The limited progress is mainly due to the absence of theo-retical tools for analyzing interdependent systems. Veryrecent studies [7, 8] present for the first time a frame-work for studying interdependency between networks andshow that such interdependencies significantly increasesthe vulnerability of the networks to random attack. Inthese studies, the dependencies between the networks areassumed to be completely random, i.e., a randomly se-lected node from network A is connected and dependson a randomly selected node from network B and viceversa. Due to the dependencies an initial failure of evena small fraction of nodes from one network can lead to aniterative process of failures that can completely fragmentboth networks.However, the restriction of random interdependenciesis a strong assumption that usually does not occur inmany real interdependent systems. As a first exampleconsider the two infrastructures that are mentioned bothin the committee report [1] and in the studies discussedabove [2, 7, 8]: The Italian power grid and SCADA com-munication networks. A power node depends on a com-munication node for control while a communication nodedepends on a power node for electricity. It is highly un-likely that a central (high degree) communication nodewill depend on a small (low degree) power node. Rather,it is much more common that a central communicationnode depends on a central power station. Moreover, cou-pled networks usually also poses some similarity in struc-ture, for instance, an area that is overpopulated is boundto have many power stations as well as many communica-tion nodes. Another real example is the world wide portand airport networks that we study in this manuscript. We find that well connected ports tend to couple to wellconnected airports therefore supporting our assumptionthat real interdependent networks are usually not ran-domly interdependent.In this Letter we show that inter-similar coupled net-works, i.e. coupled networks in which pairs are coupledaccording to some regularity rather than randomly, aresignificantly more robust to random failure. Moreover,increasing the inter-similarity between the networks leadsto a fundamental change in the networks behavior. Whilerandomly interdependent networks disintegrate in a formof a first order phase transition [7, 8], networks with highlevels of inter-similarity disintegrate in a form of a sec-ond order phase transition. The phase transition occursin the size of the largest connected cluster, P ∞ , of oneof the networks (or both) when a critical fraction q c ofnodes fail (or a critical fraction p c = 1 − q c remains). Forrandomly interdependent networks, when only a fraction p c of nodes remains P ∞ abruptly drops to zero charac-terizing a first order phase transition. For high levels ofinter-similarity we find that P ∞ continuously decreasesat criticality, characterizing a second order transition.Fig. 1 presents simulation results showing the change inthe type of phase transition for increasing levels of inter-similarity.We develop two measures to asses the level of inter-similarity between interdependent networks. We showthat these measures can also determine the robustness ofcoupled networks. The first quantity, r AB , measures theinter degree-degree correlation (IDDC) between a pair ofdependent nodes. The two networks A and B have adegree distribution of p Ak and p Bk respectively. Similarto assortative mixing in a single network [9], we defineby e jk the joint probability that a dependency link isconnected to an A -node with degree j and to a B -nodewith degree k . For networks with no IDDC, e jk = p Aj p Bk .For networks with IDDC, the level of correlation can bedefined by P jk jk ( e jk − p Aj p Bk ). Normalizing by the max-imum value of IDDC we obtain a general measure, r AB ,in the range − ≤ r ≤
1. The value 1 is achieved fora system with maximum IDDC, the value of zero for noIDDC and a value -1 for a system with maximum antiIDDC. If the two networks have the same degree distribu-tion ( p k = p Ak = p Bk ) the maximum IDDC value is givenby σ q = P k k p k − ( P k kp k ) and we obtain r AB = 1 σ q X jk jk ( e jk − p j p k ) (1)Positive values of r AB indicate that high degree nodesfrom network A tend to couple with high degree nodesfrom network B and vise versa. Negative values of r AB indicate that high degree nodes from network A tendto couple with low degree nodes from network B andvise versa. Randomly interdependent networks corre-spond to the case of r AB = 0. The second measure isthe inter-clustering coefficient (ICC), c AB , that evalu-ates for a pairs of dependent nodes { A j , B j } how manyof the neighbors of A j depend on neighbors of B j andvice versa. Analogous to a single network [10], we definethe local inter clustering coefficient, c Aj of node A j as c Aj = t j k Aj (2)where t j is the number of links connecting the neighborsof A j to the neighbors of B j and k Aj is the degree of A j [11]. Note that c Aj is not equal to c Bj . The global ICCcan be defined as the average of all the local clusteringcoefficient, c A = N P j c Aj . But in this case c A = c B . Wetherefore prefer to define the global clustering as c AB = 1 M X j t j (3)where M is the total number of dependency links be-tween the two networks and 0 ≤ c AB ≤
1. For increasingvalues of c AB more of the neighbors of A i depend onthe neighbors of B i and the two networks become moreinter-similar. For c AB = 1 the two networks must beidentical.The effect of inter-similarity between networks is dra-matically influenced by the network topology. In a casewhere two interdependent networks have a broad degreedistribution, an interdependent pair { A j , B j } can greatlydiffer in their degree. As a result the diversity in thecorrelation between the networks (IDDC) is significantlyincreased. We therefore apply our theory to two impor-tant and very different network topologies. The first isthe Erd˝os - R´enyi (ER) network model [12–14], in whichall links exist with equal probability leading to a Pois-son degree distribution P ( k ) = e −h k i h k i k /k !. The ERnetwork model has become a classic model in randomgraph theory and was intensively studied in the past fewdecades. The other model is that of scale free networks(SF) [15, 16] networks with a broad degree distribution, usually in the form of a power-law, P ( k ) ∼ k − γ with γ >
2. It was found that many real networks are scale-free [15, 16]. p P c=1, r=1 c=0, r=1c=0.4, r=0c=0, r=0 S NO I IDDCICC
FIG. 1: (a) Simulation results for P ∞ , the fraction of nodesremaining in the largest cluster of network B after a randomfailure of a fraction 1 − p of the nodes in network A . The sim-ulations compare four different configurations (see text) oftwo interdependent SF networks with λ = 2 . p for which P ∞ approaches zero is muchsmaller). (b) Simulation results showing the number of it-erations (NOI) in the iterative process of cascading failures,at p c . The NOI is plotted as a function of the similarity(S) between the networks which is measured either by theIDDC (circles), or by the ICC (squares). As the networksbecome more inter-similar the NOI is reduced indicating thatless nodes fail. When measuring the effect of the IDDC onthe NOI the ICC is kept zero. Similar, when the effect of ICCis measured the IDDC is kept zero. The dashed line marksthe region that cannot be properly simulated since the ICCis too high to generate networks with no IDDC. To show the effect of our measures on the robustness ofinter-dependent networks, we compare between the fol-lowing four systems: (i) Two randomly interdependentSF networks ( r AB = 0 and c AB = 0). (ii) Two SF net-works where every pair of dependent nodes { A j , B j } hasthe same degree , k Aj = k Bj ( r AB = 1 and c AB = 0).(iii) Two interdependent SF networks with r AB = 0 and c AB = 0 .
4, which is the maximum ICC we were ableto obtain without inserting IDDC. (iv) Two identicalinterdependent SF networks ( r AB = 1 and c AB = 1).Fig.1(a) presents P ∞ , the fraction of nodes remaining inthe largest cluster of network B after a random failureof a fraction 1 − p of network’s A nodes. The Figureshows that high IDDC even with no ICC or high ICCeven with no IDDC, significantly increases the fractionof failing nodes (smaller p c ) that will fragment the sys-tem ( P ∞ = 0), indicating that the system is more robust.Moreover, for high IDDC or ICC the jump in the sizeof P ∞ that characterizes a first order phase transitionchanges to a gradual decrease identified with a secondorder transition. Fig.1(b) provides additional supportfor our claim that inter-similarity increases the robust-ness of inter-dependent networks. The number of iter-ations (NOI) in the process of cascading failures at p c for a network of size N , scales with N / for randomlyinterdependent networks [7] and is equal to 1 for iden-tical networks. The figure shows that indeed when theinter-similarity is increased either via the IDDC measureor via the ICC measure, the NOI decreases respectively.Next, we study a real interdependent system composedfrom the world wide port network and the world wideairport network. The airport network is composed from1767 airports and records the majority of the air trafficaround the world. The port network is composed from1076 ports and records the flow of commodities aroundthe world. Previous studies have shown [17–20] thatdifferent transportation systems that are located in thesame city (or area) depend on each other through theircommon influence on the economic prosperity of that city.In terms of our two networks, the evolvement of an air-port in a city will lead to an increase in air-traffic thatin tern will result in economic prosperity. The prosper-ity of that city will have a positive effect on the evolve-ment and increase of traffic to that city’s port and viseversa. Accordingly, for our mapping we assume that aport depends on a nearby airport and vise versa. How-ever, since the networks are not of the same size we firstrenormalized the networks so that they corresponds tothe model presented in [7, 8]. We first match betweenpairs of ports and airports with the minimal distancebetween them under the condition that a port only de-pends on one airport and vise versa. The remaining air-ports that do not depend on any port are merged withthe closest airport such that the new renormalized nodeincludes the accumulated traffic of these airports. A sim-ilar process is applied to the port network. At the endof this process we obtain two networks both of size 992that are coupled based on geographical location (GL).We find that for the GL interdependent port-airport net-works the coupling between the networks is not random,the IDDC parameter is r AB = 0 . r AB → P ∞ of the port network for an in-creasing fraction of failing nodes in the airport network(similar results are obtained when the initial nodes failfrom the port network). The figure compares betweentwo configurations of the port-airport system: (i) Thenetworks are randomly coupled. (ii) The networks arecoupled based on geographical location (GL). The re-sults support our theory that a systems with high IDDCis more robust to random attack ( P ∞ is larger) and thatthe phase transition changes from first to second orderas the networks become more inter-similar. However,since each of the networks has a very high average de- gree (within each network the nodes are well connected)and as a result the networks are very hard to fragment,we have made the reasonable assumption that even if75 percent of the traffic to a certain port (or airport) isdisabled that port becomes non functional. p p GLRandom FIG. 2: The sea-air interdependent system is composed froma world wide port network and a world wide airport network.The simulations present the fraction of nodes remaining inthe largest cluster of the airport network, P ∞ , after a frac-tion p of the ports in the port network are randomly removed(similar results are obtained for the opposite case). The re-sults are compared between two different configurations, (i)(Squares) The networks are randomly coupled. (ii) (Circles)The networks are coupled according to geographic locations(GL), i.e., an airport depends on the nearest port and viceversa. When the networks are coupled by GL the system issignificantly more robust to random failure (the value of p forwhich P ∞ approaches zero is much smaller). Until now we have shown the critical effect of inter-similarity on the robustness of a system composed frominterdependent networks. But what is the effect of thelocal properties within each of the networks on the ro-bustness of the interdependent system? Here we showthat the degree-degree correlation (DDC) [9] within anetwork that has only a minor effect on the robustness ofsingle networks, greatly effects the robustness of an in-terdependent system. While for single networks a higherDDC usually slightly increases the robustness of the net-work for an interdependent system a higher DDC signifi-cantly increases the vulnerability of the system. In Fig 3we demonstrate the effect for the case of ER networks.For ER networks that are characterized by a very nar-row degree distribution, the DDC is expected to have avery limited effect on a single network. But, when twosuch ER networks with high DDC become randomly in-terdependent the effect of the DDC becomes dramatic,as shown in Fig 3.After showing that real coupled networks are indeedinter-similar, we present a mechanism for generatinginter-similar coupled networks. The model we presentcan be regarded as a generalization of the Barab´asi-Albert (BA) preferential attachment model [15, 16] totwo interdependent networks, that naturally incorporatesinter degree-degree correlations between the nodes of the p p DDC =0DDC =0.945DDC =0DDC =0.945 FIG. 3: Simulation results for ER networks with h k i = 3 . two networks. According to the BA model, a single net-work with an initial set of m randomly connected nodesis grown by adding on each step a new node that is con-nected to m different nodes from the already existingnetwork. The probability of the new node to connectto a specific node is proportional to that node’s degree.Generalizing the model to two interdependent networks A and B , we start with two initial sets of nodes m A and m B of the same size. The two sets are each internallyrandomly connected and in addition each node from m A is randomly connected to one node in m B . On each step t , a pair of dependent nodes { A t , B t } are added to thenetworks, A t to network A and B t to network B , indepen-dently, according to the preferential attachment model.Since the two nodes are added independently, there is nocorrelation between the neighbors of node A t in network A and the neighbors of B t in network B . This processmimics a natural process of two interdependent growingnetwork. In terms of our initial example of a power net-work and a communication network, at different timesnew developing areas are populated and connected to in-frastructures. Every such area adds a pair of dependentnodes, a power node and a communication node. Eventhough A t and B t are differently connected within eachnetwork, because of the preferential attachment processthe fact that they were added at the same time signifi-cantly increases the probability that they have a similardegree. When simulating a system of two interdependentnetworks according to our generalized BA model we ob-tained two SF networks with λ = 3 as obtained by theBA model for a single SF network [15, 16]. We also ob-tain a very high level of IDDC ( r AB = 0 .