Effect of the interconnected network structure on the epidemic threshold

Transcription

1 PHYSICAL REVIEW E 8883 Effect of the interconnected network structure on the epidemic threshold Huijuan Wang * Qian Li Gregorio D Agostino 3 Shlomo Havlin 4 H. Eugene Stanley and Piet Van Mieghem Faculty of Electrical Engineering Mathematics and Computer Science Delft University of echnology Delft he Netherlands Center for Polymer Studies and Department of Physics Boston University Boston Massachusetts 5 USA 3 ENEA Casaccia Research Center Via Anguillarese 3 I-3 Roma Italy 4 Department of Physics Bar-Ilan University 59 Ramat-Gan Israel Received March 3; published August 3 Most real-world networks are not isolated. In order to function fully they are interconnected with other networks and this interconnection influences their dynamic processes. For example when the spread of a disease involves two species the dynamics of the spread within each species the contact network differs from that of the spread between the two species the interconnected network. We model two generic interconnected networks using two adjacency matrices A and B in which A is a N N matrix that depicts the connectivity within each of two networks of sie N andban N matrix that depicts the interconnections between the two. Using an N-intertwined mean-field approximation we determine that a critical susceptible-infected-susceptible SIS epidemic threshold in two interconnected networks is / A + B where the infection rate is β within each of the two individual networks and β in the interconnected links between the two networks and A + B is the largest eigenvalue of the matrix A + B. In order to determine how the epidemic threshold is dependent upon the structure of interconnected networks we analytically derive A + B using a perturbation approximation for small and large the lower and upper bound for any as a function of the adjacency matrix of the two individual networks and the interconnections between the two and their largest eigenvalues and eigenvectors. We verify these approximation and boundary values for A + B using numerical simulations and determine how component network features affect A + B. We note that given two isolated networks G and G with principal eigenvectors x and y respectively A + B tends to be higher when nodes i and j with a higher eigenvector component product x i y j are interconnected. his finding suggests essential insights into ways of designing interconnected networks to be robust against epidemics. DOI:.3/PhysRevE.88.8 I. INRODUCION Complex network studies have traditionally focused on single networks in which nodes represent agents and links represent the connections between agents. Recent efforts have focused on complex systems that comprise interconnected networks a configuration that more accurately represents realworld networks []. Real-world power grids for example are almost always coupled with communication networks. Power stations need communication nodes for control and communication nodes need power stations for electricity. When a node at one end of an interdependent link fails the node at the other end of the link usually fails. he influence of coupled networks on cascading failures has been widely studied [3 6]. A nonconsensus opinion model of two interconnected networks that allows the opinion interaction rules within each individual network to differ from those between the networks was recently studied [7]. his model shows that opinion interactions between networks can transform nonconsensus opinion behavior into consensus opinion behavior. In this paper we investigate the susceptible-infectedsusceptible SIS behavior of a spreading virus a dynamic process in interconnected networks that has received significant recent attention [8 ]. An interconnected networks scenario is essential when modeling epidemics because diseases spread across multiple networks e.g. across multiple species or communities through both contact network links within each species or community and interconnected network links * H.Wang@tudelft.nl PACS numbers: k 87.3.Ge 89.. a between them. Dickison et al. [9] study the behavior of susceptible-infected-recovered SIR epidemics in interconnected networks. Depending on the infection rate in weakly and strongly coupled network systems where each individual network follows the configuration model and interconnections are randomly placed epidemics will infect none one or both networks of a two-network system. Mendiola et al. []show that in SIS model an endemic state may appear in the coupled networks even when an epidemic is unable to propagate in each network separately. In this work we will explore how the structural properties of each individual network and the interconnections between them determine the epidemic threshold of two generic interconnected networks. In order to represent two generic interconnected networks we represent a network G with N nodes using an N N adjacency matrix A that consists of elements a ij whichare either one or ero depending on whether there is a link between nodes i and j. Fortheinterconnectednetworksweconsider two individual networks G and G of the same sie N. When nodes in G are labeled from to N and in G labeled from N + tonthetwoisolatednetworksg and G can be presented by a N N matrix A [ A A ]composedoftheir corresponding adjacency matrices A and A respectively. Similarly a N N matrix B [ B B ] represents the symmetric interconnections between G and G.heinterconnected networks are composed of three network components: network A network A andinterconnectingnetworkb. In the SIS model the state of each agent at time t is abernoullirandomvariablewherex i t ifnodei is susceptible and X i t if it is infected. he recovery /3/88/ American Physical Society

2 HUIJUAN WANG et al. PHYSICAL REVIEW E 8883 curing process of each infected node is an independent Poisson process with a recovery rate δ. Eachinfectedagent infects each of its susceptible neighbors with a rate β which is also an independent Poisson process. he ratio τ β/δ is the effective infection rate. A phase transition has been observed around a critical point τ c in a single network. When τ>τ c anonerofractionofagentswillbeinfectedinthe steady state whereas if τ<τ c infectionrapidlydisappears [ 4]. he epidemic threshold via the N-intertwined meanfield approximation NIMFA is τ c A where A is the largest eigenvalue of the adjacency matrix also called the spectral radius [5]. For interconnected networks we assume that the curing rate δ is the same for all the nodes that the infection rate along each link of G and G is β and that the infection rate along each interconnecting link between G and G is β where is a real constant ranging within [ withoutlosinggenerality. We first show that the epidemic threshold for β/δ in interconnected networks via NIMFA is τ c A+B where A + B isthelargesteigenvalueofthematrixa + B. We further express A + Basafunctionofnetworkcomponents A A andb and their eigenvalues and eigenvectors to reveal the contribution of each component network. his is a significant mathematical challenge except for special cases e.g. when A and B commute i.e. AB BA see Sec. IIIA. Our main contribution is that we analytically derive for the epidemic characterier A + Baitsperturbation approximation for small bitsperturbationapproximation for large andcitslowerandupperboundforany as afunctionofcomponentnetworksa A andb and their largest eigenvalues and eigenvectors. Numerical simulations in Sec. IV verify that these approximations and bounds well approximate A + B and thus reveal the effect of component network features on the epidemic threshold of the whole system of interconnected networks which provides essential insights into designing interconnected networks that are robust against the spread of epidemics see Sec. V. Sahneh et al. []recentlystudiedsisepidemicsongeneric interconnected networks in which the infection rate can differ between G and G and derived the epidemic threshold for the infection rate in one network while assuming that the infection does not survive in the other. heir epidemic threshold was expressed as the largest eigenvalue of a function of matrices. Our work explains how the epidemic threshold of generic interconnected networks is related to the properties eigenvalue and eigenvector of network components A A and B without any approximation on the network topology. Graph spectra theory [6] andmodernnetworktheory integrated with dynamic systems theory can be used to understand how network topology can predict these dynamic processes. Youssef and Scoglio [7] have shown that an SIR epidemic threshold via NIMFA also equals /.he Kuramoto synchroniation process of coupled oscillators [8] and percolation [9] also features a phase transition that specifies the onset of a remaining fraction of locked oscillators and the appearance of a giant component respectively. Note that a mean-field approximation predicts both phase transitions at a critical point that is proportional to /.husweexpect our results to apply to a wider range of dynamic processes in interconnected networks. II. EPIDEMIC HRESHOLD OF INERCONNECED NEWORKS In the SIS epidemic spreading process the probability of infection v i t E[X i t] for a node i in interconnected networks G is described by dv i t N N β a ij v j t + β b ij v j t dt j v i t δv i t j via NIMFA where a ij and b ij is an element of matrices A and Brespectively.Itsmatrixformbecomes dvt dt βa + B δiv t βdiagv i ta + BV t. he governing equation of the SIS spreading process on a single network A is dvt dt βa δiv t βdiagv i ta V t whose epidemic threshold has been proven [5] tobe τ c A which is a lower bound of the epidemic threshold []. Hence the epidemic threshold of interconnected networks by NIMFA is τ c A + B which depends on the largest eigenvalue of the matrix A + B. hematrixa + B is a weighted matrix where <.henimfaisanimprovementoverearlierepidemic models [4] in that it takes the complete network topology into account and thus it allows us to identify the specific role of a general network structure on the spreading process. However NIMFA still relies on a mean-field argument and thus approximates the exact SIS epidemics []. III. ANALYIC APPROACH: A + B INRELAION O COMPONEN NEWORK PROPERIES he spectral radius A + Basshowninthelastsection is able to characterie epidemic spreading in interconnected networks. In this section we explore how A + Bisinfluenced by the structural properties of interconnected networks and by the relative infection rate along the interconnection links. Specifically we express A + B asafunctionof the component networks A A andb and their eigenvalues and eigenvectors. For proofs of theorems or lemma see the Appendix. A. Special cases We start with some basic properties related to A + B and examine several special cases in which the relation between A + B andthestructuralpropertiesofnetwork components A A andb are analytically tractable. 8-

3 EFFEC OF HE INERCONNECED NEWORK... PHYSICAL REVIEW E he spectral radius of a subnetwork is always smaller or equal to that of the whole network. Hence Lemma. A + B A max A A. Lemma. A + B B. he interconnection network B forms a bipartite graph. Lemma 3. he largest eigenvalue of a bipartite graph B [ B B ] follows B B B whereb is possibly asymmetric [6]. Lemma 4. When G and G are both regular graphs with the same average degree E[D] andwhenanytwonodesfromg and G respectivelyarerandomlyinterconnectedwithprobability p I theaveragespectralradiusoftheinterconnected networks follows: E[ A + B] E[D] + Np I if the interdependent connections are not sparse. A dense Erdös-Rényi ER random network approaches aregularnetworkwhenn is large. Lemma 4 thus can be applied as well to cases where both G and G are dense ER random networks. If A and B commute thus AB BAthentheeigenvectors of A and B are the same provided that all N eigenvectors are independent [[6] p. 53]. In that case it holds that A + B A + B. his property of commuting matrices makes the following two special cases where A and B are symmetric with orthogonal hence independent eigenvectors analytically tractable. Lemma 5. When A + B [ A A ] + [ I I ] i.e. the interconnected networks are composed of two identical networks where one network is indexed from to N and the other from N + ton withaninterconnectinglink between each so-called image node pair in + i fromthe two individual networks respectively its largest eigenvalue A + B A +. Proof. When A + B [ A A ] + [ I I ] matrices A and B are commuting [ ] A A B BA. A herefore A + B A + B A + B. he network B is actually a set of isolated links. Hence B. Lemma 6. When A + B [ A A ] + [ A A ] its largest eigenvalue A + B + A. Proof. When A + B [ A A ] + [ A A ] matrices A and B are commuting ] A A B [ A BA. herefore A + B A + B + A + A. When A and B are not commuting little can be known about the eigenvalues of A + B given the spectrum of A and of B. ForexampleevenwhentheeigenvalueofA and B are known and bounded the largest eigenvalue of A + B can be unbounded [6]. B. Lower bounds for A + B We now denote matrix A + B to be W. Applyingthe Rayleigh inequality [6] p. 3] to the symmetric matrix W A + B yields W W where equality holds only if is the principal eigenvector of W. heorem 7. he best possible lower bound W of interdependent networks W by choosing as the linear combination of x and ythelargesteigenvectorofa and A respectively is W max A A + A A + ξ A A where ξ x B y. When the lower bound becomes the exact solution W L. When the two individual networks have the same largest eigenvalue A A we have W A + x B y. heorem 8. he best possible lower bound W W by choosing as the linear combination of x and ythelargest eigenvector of A and A respectivelyis W A + B x + A + B y + A + B x A B y + θ where θ A + A x B y. In general W k k W. he largest eigenvalue is lower bounded by W k /k W. heorem 9. Given a vector W s /s W k k is an even integer and <s<k.furthermore W k /k lim k W. 3 /k when Hence given a vector wecouldfurtherimprovethelower bound W k /k by taking a higher even power k. Notethat 8-3

4 HUIJUAN WANG et al. PHYSICAL REVIEW E 8883 heorems 7 and 8 express the lower bound as a function of component networks A A andb and their eigenvalues and eigenvectors which illustrates the effect of component network features on the epidemic characterier W. C. Upper bound for A + B heorem. he largest eigenvalue of interdependent networks W isupperboundedby W max A A + B 4 max A A + B B. 5 his upper bound is reached when the principal eigenvector of B B coincides with the principal eigenvector of A if A A andwhentheprincipaleigenvectorof B B coincides with the principal eigenvector of A if A A. D. Perturbation analysis for small and large In this subsection we derive the perturbation approximation of W forsmallandlarge respectivelyasafunctionof component networks and their eigenvalues and eigenvectors. We start with small cases. he problem is to find the largest eigenvalue sup W of W withtheconditionthat W I When the solution is analytical in weexpress and by aylor expansion as k k k k. k k Substituting the expansion in the eigenvalue equation gives A + B k k k k k k k k k where all the coefficients of k on the left must equal those on the right. Performing the products and reordering the series we obtain k A k + B k k k i i k. k k his leads to a hierarchy of equations A k + B k i k k i i. he same expansion must meet the normaliation condition k i or equivalently k k j j j where uv i u iv i represents the scalar product. he normaliation condition leads to a set of equations k k i i 6 i for any k and. Let A A and xy denotethelargesteigenvalue and the corresponding eigenvector of A A respectively. We examine two possible cases: a the nondegenerate case when A > A andbthedegeneratecasewhen A A andthecase A < A isequivalenttothefirst. heorem. For small inthenondegeneratecasethus when A > A W A + x B A I A B x + O3. Note that in A6 B is symmetric and I Aispositive definite and so is B I A B.Hencethissecond-order correction is always positive. heorem. For small when the two component networks have the same largest eigenvalue A A W + + y B I A + xx B y x + x B I A + yy B x y + O 3 where A and x B y. In the degenerate case the first-order correction is positive and the slope depends on B y andx. When A and A are identical the largest eigenvalue of the interdependent networks becomes A + B xx + O. When A [ A A ] and B [ I I ] our result 8 in the degenerate case up to the first order leads to A + B A + whichisanalternateproofoflemma5.when A [ A A ]andb [ A A ] 8 again explains Lemma 6 that A + B + A. Lemma 3. For large thespectralradiusofinterconnected networks is 7 8 A + B B + v Av + O 9 where v is the eigenvector belonging to Band A + B A + B + O. Proof. Lemma 3 follows by applying perturbation theory [3]tothematrixB + AandtheRayleighprinciple[6] which states that v Av A for any normalied vector v such that v v with equality only if v is the eigenvector belonging to the eigenvalue A. 8-4

5 EFFEC OF HE INERCONNECED NEWORK... PHYSICAL REVIEW E IV. NUMERICAL SIMULAIONS In this section we employ numerical calculations to quantify to what extent the perturbation approximation 7 and 8 for small theperturbationapproximation9 for large andtheupper4 and lower bound 3 are close to the exact value W A + B. We investigate the condition under which the approximations provide better estimates. he analytical results derived earlier are valid for arbitrary interconnected network structures. For simulations we consider two classic network models as possible topologies of G and G :itheerdös-rényi ER random network [4 6] andii thebarabási-albert BA scale-free network [7]. ER networks are characteried by a binomial degree distribution Pr[D k] N k p k p N k wheren is the sie of the network and p is the probability that each node pair is randomly connected. In scale-free networks the degree distribution is given by a power law Pr[D k] ck such that N k ck and 3inBAscale-free networks. In numerical simulations we consider N N. Specifically in the BA scale-free networks m 3andthe corresponding link density is p BA.6. We consider ER networks with the same link density p ER p BA.6. A coupled network G is the union of G and G whicharechosen from the above-mentioned models together with random interconnection links with density p I theprobabilitythatany two nodes from G and G respectivelyareinterconnected. Given the network models of G and G and the interacting link density p I we generate interconnected network realiations. For each realiation we compute the spectral radius W its perturbation approximation 7 and 8 for small theperturbationapproximation9 for large and the upper bound 4 and lower bound 3 for any. We compare their averages over the coupled network realiations. We investigate the degenerate case G G wherethelargesteigenvaluesofg and G are the same and the nondegenerate case where G G respectively. A. Nondegenerate case We consider the nondegenerate case in which G is a BA scale-free network with N m 3 G is an ER random network with the same sie and link density p ER p BA.6 and the two networks are randomly interconnected with link density p I.Wecomputethelargest average eigenvalue E[ W] and the average of the perturbation approximations and bounds mentioned above over interconnected network realiations for each interconnection link density p I [.5.4] such that the average number of interdependent links ranges from N 4 N NN to 4N and for each value that ranges from to with step sie.5. For a single BA scale-free network where the power exponent β 3 >.5 the largest eigenvalue is + o d max where d max is the maximum degree in the network [8]. he spectral radius of a single ER random graph is close to the average degree N p ER when the network is not sparse. When p I G max G ER G BA G BA > G ER. he perturbation approximation is expected to be close to the exact W onlyfor and.howeverasshowninfig.atheperturbationapproximation for small approximates Wwellforarelative large range of especiallyforsparserinterconnectionsi.e. for a smaller interconnection density p I.Figureb shows that the exact spectral radius W isalreadyclosetothe large perturbation approximation at least for >8. As depicted in Fig. thelowerbound3 and upper bound 4 are sharp i.e. close to W forsmall. helower and upper bounds are the same as W when. For large thelowerboundbetterapproximates Wwhenthe interconnections are sparser. Another lower bound B W i.e. Lemma is sharp for large asshowninfig.3 especially for sparse interconnections. We do not illustrate the lower bound because the lower bound 3 is always sharper or equally good. he lower bound B considersonly the largest eigenvalue of the interconnection network B and ignores the two individual networks G and G.hedifference E[ W] BA-ER simulation p I.5 approximation p I.5 simulation p I.5 approximation p I.5 simulation p I. approximation p I. simulation p I. approximation p I. simulation p I.4 approximation p I.4 E[ W] BA-ER simulation p I.5 approximation p I.5 simulation p I.5 approximation p I.5 simulation p I. approximation p I. simulation p I. approximation p I. simulation p I.4 approximation p I.4 5 a b 9 FIG.. Color online A plot of W as a function of for both simulation results symbol and its a perturbation approximation 7 for small dashed line and b perturbation approximation 9 for large dashed line. he interconnected network is composed of an ER random network and a BA scale-free network both with N and link density p.6 randomly interconnected with density p I.All the results are averages of realiations. 8-5

6 HUIJUAN WANG et al. PHYSICAL REVIEW E 8883 E[ W] 5 4 BA-ER simulation p I. 5 lower bound p I.5 simulation p I. 5 lower bound p I. 5 simulation p I. lower bound p I. simulation p I. lower bound p I. simulation p I. 4 lower bound p I.4 E[ W] BA-ER simulation p I. 5 upper bound p I.5 simulation p I. 5 upper bound p I.5 simulation p I. upper bound p I. simulation p I. upper bound p I. simulation p I. 4 upper bound p I a b 6 8 FIG.. Color online Plot W as a function of for both simulation results symbol and its a lower bound 3 dashed line and b upper bound 4 dashed line. he interconnected network is composed of an ER random network and a BA scale-free network both with N and link density p.6 randomly interconnected with density p I. All the results are averages of realiations. W B v Av + O accordingtothelarge perturbation approximation is shown in Fig. 3 to be larger for denser interconnections. It suggests that G and G contribute more to the spectral radius of the interconnected networks when the interconnections are denser in this nondegenerate case. For large the upper bound is sharper when the interconnections are denser or when p I is larger as depicted in Fig. b. hisisbecause B W B + max A A. When the interconnections are sparse Wisclosetothelowerbound Bandhencefarfrom the upper bound. Most interdependent or coupled networks studied so far assume that both individual networks have the same number of nodes N and that the two networks are interconnected randomly by N one-to-one interconnections or by a fraction q of the None-to-one interconnections where <q [67]. hese coupled networks correspond to our sparse interconnec- E[ W] BA-ER simulation p I.5 E[ B] p I.5 simulation p I.5 E[ B] p I.5 simulation p I. E[ B] p I. simulation p I. E[ B] p I. simulation p I.4 E[ B] p I FIG. 3. Color online Plot W as a function of for both simulation results symbol and its lower bound B dashed line. he interconnected network is composed of an ER random network and a BA scale-free network both with N and link density p.6 randomly interconnected with density p I. All the results are averages of realiations tion cases where p I when Biswellapproximatedby the perturbation approximation for both small and large.he spectral radius Wincreasesquadraticallywith for small asdescribedbythesmall perturbation approximation. he increase accelerates as increases and converges to a linear increase with with slope B. Here we show the cases in which G G andtheinterconnectionsaresparseasinmost real-world networks. However all the analytical results can be applied to arbitrary interconnected network structures. B. Degenerate case We assume the spectrum [9] to be a unique fingerprint of alargenetwork.wolargenetworksofthesamesieseldom have the same largest eigenvalue. Hence most interconnected networks belong to the nondegenerate case. Degenerate cases mostly occur when G and G are identical or when they are both regular networks with the same degree. We consider two degenerate cases where both network G and G are ER random networks or BA scale-free networks. Both ER and BA networks lead to the same observations. Hence as an example we show the case in which both G and G are BA scale-free networks of sie N and both are randomly interconnected with density p I [.5.4] as in the nondegenerate case. Figure 4a shows that the perturbation analysis well approximates W forsmall especially when the interconnection density is small. When the interconnections are dense the small perturbation approximation performs better in the degenerate case i.e. is closer to W thaninnondegeneratecases[seefig.a]. Similar to the nondegenerate case Fig. 4b illustrates that the exact spectral radius Wisclosetothelarge perturbation approximation even since 8. Similarly Fig. 5 shows that both the lower and upper bound are sharper for small. helowerboundbetterapproximates W forsparserinterconnectionswhereastheupperbound better approximates W fordenserinterconnections. hus far we have examined the cases where G G and the interconnections are sparse as is the case in most real-world networks. However if both G and G are dense 8-6

7 EFFEC OF HE INERCONNECED NEWORK... PHYSICAL REVIEW E E[ W] BA-BA simulation p I.5 approximation p I.5 simulation p I.5 approximation p I.5 simulation p I. approximation p I. simulation p I. approximation p I. simulation p I.4 approximation p I.4 E[ W] BA-BA simulation p I.5 approximation p I.5 simulation p I.5 approximation p I.5 simulation p I. approximation p I. simulation p I. approximation p I. simulation p I.4 approximation p I a b 9 FIG. 4. Color online A plot of W as a function of for both simulation results symbol and its a perturbation approximation 8 for small dashed line and b perturbation approximation 9 for large dashed line. he interconnected network is composed of two identical BA scale-free networks with N and link density p.6 randomly interconnected with density p I. All the results are averages of realiations. ER random networks and if the random interconnections are also dense the upper bound is equal to W i.e. W G + B seelemma4.equivalentlythe difference W B isaconstant G G independent of the interconnection density p I. In both the nondegenerate and degenerate case W is well approximated by a perturbation analysis for a large range of especiallywhentheinterconnectionsaresparse.he lower bound 3 and upper bound 4 are sharper for small. Most real-world networks are sparsely interconnected where our perturbation analysis better approximates Wforalarge range of andthuswellrevealstheeffectofcomponent network structures on the epidemic characterier W. V. CONCLUSION We study interconnected networks that are composed of two individual networks G and G andinterconnectinglinks represented by adjacency matrices A A andbrespectively. We consider SIS epidemic spreading in these generic coupled networks where the infection rate within G and G is β the infection rate between the two networks is β andthe recovery rate is δ for all agents. Using a NIMFA we show that the epidemic threshold with respect to β/δ is τ c A+B where A [ A A ]istheadjacencymatrixofthetwoisolated networks G and G.helargesteigenvalue A + B can thus be used to characterie epidemic spreading. his eigenvalue A + Bofafunctionofmatricesseldomgives the contribution of each component network. We analytically express the perturbation approximation for small and large lower and upper bounds for any of A + Basafunction of component networks A A andb and their largest eigenvalues and eigenvectors. Using numerical simulations we verify that these approximations or bounds approximate well the exact A + B especially when the interconnections are sparse as is the case in most real-world interconnected E[ W] BA-BA simulation p I.5 lower bound p I.5 simulation p I.5 lower bound p I.5 simulation p I. lower bound p I. simulation p I. lower bound p I. simulation p I.4 lower bound p I.4 E[ W] BA-BA simulation p I.5 upper bound p I.5 simulation p I.5 upper bound p I.5 simulation p I. upper bound p I. simulation p I. upper bound p I. simulation p I.4 upper bound p I.4 4 a b 6 8 FIG. 5. Color online Plot W as a function of for both simulation results symbol and a its lower bound 3 dashed line and b upper bound 4 dashed line. he interconnected network is composed of two identical BA scale-free networks N and link density p.6 randomly interconnected with density p I. All the results are averages of realiations. 8-7

8 HUIJUAN WANG et al. PHYSICAL REVIEW E 8883 networks. Hence these approximations and bounds reveal how component network properties affect the epidemic characterier A + B. Note that the term x B y contributes positively to the perturbation approximation 8 and the lower bound 3 of A + B wherex and y are the principal eigenvector of network G and G.hissuggeststhatgiven two isolated networks G and G theinterconnectednetworks have a larger A + B orasmallerepidemicthresholdif the two nodes i and j with a larger eigenvector component product x i y j from the two networks respectively are interconnected. his observation provides essential insights useful when designing interconnected networks to be robust against epidemics. he largest eigenvalue also characteries the phase transition of coupled oscillators and percolation. Our results apply to arbitrary interconnected network structures and are expected to apply to a wider range of dynamic processes. ACKNOWLEDGMENS We wish to thank ONR Grants No. N and No. N DRA Grants No. HDRA and No. HDRA NSF Grant No. CMMI 59 the European EPIWORK MULIPLEX CONGAS Grant No. FP7-IC MOIA Grant No. JLS-9-CIPS-AG-C-6 and LINC projects the Deutsche Forschungsgemeinschaft DFG the Next Generation Infrastructure Bsik and the Israel Science Foundation for financial support. APPENDIX: PROOFS. Proof of Lemma 4 In any regular graph the minimal and maximal node strength are both equal to the average node strength. Since the largest eigenvalue is lower bounded by the average node strength and upper bounded by the maximal node strength as proved below in Lemma 4 a regular graph has the minimal possible spectral radius which equals the average node strength. When the interdependent links are randomly connected with link density p I the coupled network is asymptotically a regular graph with average node strength E[D] + Np I ifp I is a constant. Lemma 4. For any N N weighted symmetric matrix W E[S] W max s r where s r N j w rj is defined as the node strength of node r and E[S] istheaveragenodestrengthoverallthenodesin graph G. Proof. he largest eigenvalue follows: x Wx sup x x x when matrix W is symmetric and the maximum is attained if and only if x is the eigenvector of W belonging to W. For any other vector y x it holds that y Wy y y.bychoosing the vector y u... we have N N i j N w ij N N s i E[S] i where w ij is the element in matrix W and E[S]istheaverage node strength of the graph G. heupperboundisproved by the Gerschgorin circle theorem. Suppose component r of eigenvector x has the largest modulus. he eigenvector can be always normalied such that x x x x N where x j forallj. Equatingcomponentr on both sides of the eigenvalue equation Wx x gives W N j w rj x j N w rj j x j N w rj s r j when none of the elements of matrix W are negative. Since any component of x may have the largest modulus W max s r.. Proof of heorem 7 We consider the N vector as [ C x C y ]the linear combination of the principal eigenvector x and y of the two individual networks respectively where x x y y C + C suchthat andcompute [ ][ ] W [C x C y A B C x ] B A C y C x A x + C y A y + C C x B y C A + C A + C C ξ where ξ x B y. By Rayleigh s principle W W W. We could improve this lower bound by selecting as the best linear combination C and C ofx and y. Let L be the best possible lower bound W via the optimal linear combination of x and y. hus L max C C +C A + C A + C C ξ. We use the Lagrange multipliers method and define the Lagrange function as C A + C A + C C ξ µ C + C where µ is the Lagrange multiplier. he maximum is achieved at the solutions of C C A + C ξ C µ C C A + C ξ C µ µ C + C. Note that C C + C C / L µ which leads to µ L. Hence the maximum L is achieved at the solution of C A + C ξ C L C A + C ξ C L 8-8

9 EFFEC OF HE INERCONNECED NEWORK... PHYSICAL REVIEW E that is his leads to A L ξ det. ξ A L L A + A A A + + ξ A + A + A A + A A + ξ A A max A A + A A + ξ A A. he maximum is obtained when ] ±[ A L A + A L x A L A + A L y. 3. Proof of heorem 8 By Rayleigh s principle W W W.Weconsider as linear combination [ C x C y ]ofx and y. he lower bound [ W [C x C y ] A + B B A B + B A A B + B A A + B B ][ ] C x C y C x A x + C y A y + C x B B x + C y B B y + C C x A B + B A y C A + C A + C C θ + C B x + C B y where θ A + A x B y.let L be the best possible lower bound W via the optimal linear combination C and C ofx and y. hus L max C C +C A + C A + C C θ + C B x + C B y. We use the Lagrange multipliers method and define the Lagrange function as C A + C A + C C θ + C B x + C B y µ C + C where µ is the Lagrange multiplier. he maximum is achieved at the solutions of C C A + C A + A x B y + C B x C µ C C A + C A + A x B y + C B y C µ µ C + C which lead to C C + C C / L µ. Hence the maximum L is achieved at the solution of C A + C θ + C B x C L C A + C θ + C B y C L that is det A + B x L θ θ A + B y. L his leads to L A + B x + A + B y L + A + B x A + B y θ. 8-9

10 HUIJUAN WANG et al. PHYSICAL REVIEW E 8883 Hence A + B x + A + B y L + A + B x + A + B y 4 A + B x A + B y θ A + B x + A + B y A + B x + A B y + θ which is obtained when C θ θ + L A B x C L A B x θ + L A B x. 4. Proof of heorem 9 Any vector of sie N with m can be expressed as a linear combination of the eigenvectors... N of matrix W m where N i c i. Hence Hence W s N i N i N i N i c i i c i i N i c i i N i c i k i s c i W s i c i s i i N c s i i k i W k /k lim k W. According to Lyapunov s inequality E[ X s ] /s E[ X t ] /t when <s<t.akingpr[x i ] c i wehave N c s i i s i N ci i s i E[ X s ] /s E[ X k ] /k since k is even and k>s>. N c k i i k i. 5. Proof of heorem [ ] W max [x y x ]A + B x x+y y y max x x+y y [ ] [ ] [x y x ]A + [x y y x ]B y max x A x + y A y x x+y y [ ] + max [x y x ]B x x+y y y max A A + B. he inequality is due to the fact that the two terms are maximied independently. he second term B x B y + y B x max x x+y y max x B y x x+y y is equivalent to the system of equations or B y Bx B y Bx x x + y y B B y B y B B x B x x x + y y which is to find the maximum eigenvalue or more precisely the positive square root of the symmetric positive matrix B B B max x B B x x. his actually proves Lemma 3 the property B B B ofabipartitegraphb. 6. Proof of heorem he explicit expression up to the second order reads A + B O O O 3. A 8-

11 EFFEC OF HE INERCONNECED NEWORK... PHYSICAL REVIEW E he ero-order expansion is simply A. he problem at ero order becomes to find the maximum of In the nondegenerate case Hence A A. max A xa x xx A. A [x ] where the first N elements of are x and the rest N elements are all eros. Let us look at the first-order correction. Imposing the identity for the first-order expansion in A gives A + B +. A Furthermore we impose the normaliation condition to [see 6] which leads to. A3 he first-order correction to the principal eigenvector is orthogonal to the ero order. Plugging this result in A A + B + A + B that is B. A4 Since x and B [ B B ] the first-order correction in this nondegenerate case is null. Equation A allows us to calculate also the first-order correction to the eigenvector A + B A I B. A IisinvertibleoutofitskernelA I that is the linear space generated by andsinceb we have I A B. A5 Let us look for the second-order correction. Imposing the identification of the second-order term of A we obtain A + B + +. Projecting this vectorial equation on provides the secondorder correction to A + B + + A + B + B B. Substituting A5 gives B I A B A6 which can be further expressed as a function of the largest eigenvalue and eigenvector of individual network A A or their interconnections B.Since we have B B B x B x B I A B I A B x I A B x x I A B I A B x x I A B I A B x x B I A B x which finishes the proof. 7. Proof of heorem he ero-order correction [ x y ] of the principal eigenvector of W is a vector of sie N with the first N elements denoted as vector x and the last N elements denoted as y.similarly [ x y ]. In the degenerate case the solution of the ero-order expansion equation A can be any combination of the principal eigenvector x and y of the two individual networks: x c x y c y c + c and A A. he first-order correction of the largest eigenvalue in the nondegenerate case A4 holds as well for the degenerate case B A7 which is however nonero in the degenerate case due to the structure of and is maximied by the right choice of c and c.hus Wmax c c A + B + O A + max c c c c B yx + B xy + O A + B yx + B xy + O A + xb y + O where c c is maximum when c c /. Hence [ x y ] [ x y ]. One may also evaluate the second-order correction of the largest eigenvalue. he following results we derived in the nondegenerate case hold as well for the degenerate case B A + B +. 8-

12 HUIJUAN WANG et al. PHYSICAL REVIEW E 8883 he second equation allows us to calculate the first-order correction to the principal eigenvector: I A B I. A8 his linear equation has solution when B I is orthogonal to the kernel of the adjoint matrix of I A where the kernel is defined as Ker I A {v : I Av }. First we are going to prove that B I is orthogonal to the kernel v.weassumethatthelargesteigenvalueisunique thus differs from the second largest eigenvalue in each single network A and A asobservedinmostcomplexnetworks. In this case A x x A y y. he kernel of the matrix I A is the linear space generated by x and y ax v. Combining A7 wehave by v B I a B x y + bx B y. herefore the solution of in A8 exists. Secondly we will prove that all solutions of lead to the same.anytwosolutionsof differ by a vector in Ker I A andcanbedenotedbyforexample x y andẑ x +ax y by confinedbythenormaliation condition A3: x x + y y x x + ax + y y + by which leads to a b.he and ˆ corresponding to the two solutions B y x are equal since B x y ˆ B y x + ax y ay B x ˆ + a [ y B x x B y ]. herefore all solutions of lead to the same second-order correction to the eigenvalue and we are allowed to select any specific solution. We choose one solution by imposing the orthogonality of x with x and y with y.equationa8 in components reads I A x B y x I A y B x y. We could replace I A by I A + x x and replace I A by I A + y y since x is orthogonal with x and y is orthogonal with y : I A + x x x B y x I A + y y y B x y. his allows us to calculate algebraically. he first-order correction to the principal eigenvector is x I A + x x B y x y I A + y y B x y. he second-order correction of the largest eigenvalue follows: B y I A + x x B y x I A + y y B x y B x which can be expressed as a function of the principal eigenvector x and y of each single network y B I A + xx B y x + x B I A + yy B x y. A9 [] S. V. Buldyrev R. Parshani G. Paul H. E. Stanley and S. Havlin Nature London [] E. A. Leicht and R. M. D Soua arxiv: [3] R. Parshani S. V. Buldyrev and S. Havlin Phys. Rev. Lett [4] J. Gao S. V. Buldyrev H. E. Stanley and S. Havlin Nature Physics 8 4. [5] D. Zhou H. E. Stanley G. D Agostino and A. Scala Phys. Rev. E [6] X. Huang S. Shao H. Wang S. V. Buldyrev H. E. Stanley and S. Havlin Europhys. Lett [7] Q. Li L. A. Braunstein H. Wang J. Shao H. E. Stanley and S. Havlin J. Stat. Phys [8] S. Funk and V. A. A. Jansen Phys. Rev. E [9] M. Dickison S. Havlin and H. E. Stanley Phys. Rev. E [] A. Saumell-Mendiola M. Ángeles Serrano and M. Boguñá Phys. Rev. E [] F. D. Sahneh C. Scoglio and F. N. Chowdhury in Proceedings of the 3 American Control Conference Washington DC IEEE 3 pp

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