Identifying symmetries and predicting cluster synchronization in complex networks
Pitambar Khanra, Subrata Ghosh, Prosenjit Kundu, Chittaranjan Hens, Pinaki Pal
CCluster analysis in multilayer networks using eigen vector centrality
Pitambar Khanra , Subrata Ghosh , Prosenjit Kundu ,Chittaranjan Hens , and Pinaki Pal Department of Mathematics, National Institute of Technology, Durgapur 713209, India Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, India and Department of Mathematics, University of Buffalo, State university of New York, Buffalo, USA (Dated: February 16, 2021)The concept of symmetry in multilayer networks and its use for cluster analysis of the networkhave recently been reported in [Rossa et al., Nat. Commun. 11, 1 (2020)]. It has been shown thatclusters in a multilayer network can be determined by finding the symmetry group of the multilayerfrom the symmetry groups of the independent individual layers. However, finding symmetry groupelements of large complex networks consisting of several layers can often be difficult. Here we presenta new mathematical framework involving the structure of the eigen vector centrality (EVC) of theadjacency matrix for cluster analysis in multilayer networks. The framework is based on an analyticalresult showing a direct correspondence between the EVC elements and the clusters of the network.Using this result, cluster analysis is performed successfully for several multilayer networks and ineach case the results are found to be consistent with that obtained using the symmetry group analysismethod. Finally, cluster synchronization in multilayer networks on Sakaguchi-Kuramoto (SK) modelis investigated under the proposed framework. Stability analysis of the cluster synchronization statesare also done using master stability function approach.
PACS numbers: 05.45.Xt, 05.45.Gg, 85.25.Cp, 87.19.lm
I. INTRODUCTION
Natural as well as man made complex systems con-sisting of ensembles of interacting dynamical units areabundant around us. Examples include brain [1], powergrid [2], group of fireflies [3], birds [4] etc. The struc-ture of any such complex system can be mathematicallymodelled as a graph or network whose vertices are theinteracting units and the edges represent the interac-tions. Although some complex systems can be modeledusing simple graphs, there are many real world complexsystems namely power grid network, transportation net-work, neuronal network, social network epidemic spread-ing network [5–11] etc., where different types of interac-tions exist among the set of vertices and it is more realis-tic to model such systems using multilayer networks. Onthe other hand, dynamics of the ensembles are modelledby coupled dynamical systems on top of these networksand the set up is widely used by the researchers for inves-tigating collective phenomena including synchronizationin the network.Synchronization in large dynamical networks encodesthe collective features of several real and man made sys-tems ranging from power grid dynamics to neuronal firing[12–18]. Global synchronization (GS), in which the tra-jectories of oscillators evolve in unison appears due tothe large accessibility of information exchange betweenoscillatory units and suitable master stability functionassociated with the network structure determine the fateof GS [19–21]. However, in many of the real world net-worked systems GS is not attained, rather, they exhibitnetwork structure dependent synchronization, such ascluster (CS) or remote synchronization (RS) [22–36]. InCS (as defined in [24, 27, 37]), a set of oscillators form acoherent cluster i.e. all the trajectories within that clus- ter remain in the same synchronization manifold. Thesalient feature of such synchronization phenomena is thatwithin a cluster, nodes might not be necessarily adjacentin the considered graph. Swarm of animals, or undesiredsynchronous states (within sub units) in power grid areimportant examples of such CS [28]. In graph theoreticperspective, these clusters are the orbits of the graph andthey are the important ingredients of the associated sym-metry group. The stability of such synchronous clusterheavily depends on systematic analysis of the symmetricgroup elements [27].Recently, cluster synchronization in multilayer net-works has been extensively studied [38, 39] where eachlayer may have different oscillatory units. A delicateanalysis of the structural pattern of intra and inter linksin multilayer graph (using computational group theory)leads to predict the cluster states of the coupled nodalunits. Note that, many real world dynamics contain avarying levels of complexity depending on the particularbehavior of interest [10, 40–42], thus be better modeledas multilayer network [7, 43, 44].However, tracing the entire symmetry group of a multi-layer network is not an easy task. It involves finding thesymmetry group elements of the individual layers andparity checking. The process can be quite complex, par-ticularly, when the size of the network is large. Also,the stability calculation of each synchronous cluster de-pends on the decoupling of the independent and depen-dent symmetries of layers [38], thus makes the it moredifficult to simulate.Against this backdrop, we seek an alternative way toidentify all orbits without exploiting the underline sym-metry group of the considered network. In the process,we analytically prove that there is a direct connectionbetween the elements of eigen vector centrality (EVC,eigen vector corresponding to the largest eigen value of a r X i v : . [ n li n . AO ] F e b the considered adjacency matrix [45, 46]) of the supraadjacency matrix and the clusters of the multilayer net-work. Based on this result, we propose a new frameworkfor cluster analysis in multilayer networks. The clustersof different synthetic as well as real multilayer networksdetermined from the proposed framework are found tobe consistent with the ones determined from the sym-metry based analysis [38]. For further confirmation, wehave checked the cluster synchronization emerging fromcoupled phase-lag oscillators (Sakaguchi-Kuramoto dy-namics) [47–54] connected in several syncthetic networksand it is shown the cluster states generated form phasedynamics are perfectly mapped with the EVC elementscalculated from the underlying network. II. MATHEMATICAL FRAMEWORK
We consider multilayer networks consisting of M layersof sizes N , N , . . . , and N M respectively. The intralayer adjacency matrix of the layer σ ( σ = 1 , , . . . , M ), rep-resenting the connectivity among the nodes of the layer,is denoted by the N σ × N σ matrix A σ and the inter-layer connections between two different layers σ and σ ( σ = σ and σ, σ = 1 , . . . , M ) is represented by the N σ × N σ adjacency matrix A σσ . The layer σ ( σ =1 , , . . . , M ) is mathematically described by a connectedgraph G σ ( V σ , E σ ), where V σ and E σ respectively denotethe set of vertices and edges of the layer with | V σ | = N σ .Note that the edge set E σ consists of all ordered pairs( i, j ) ∈ V σ × V σ such that the i th and j th nodes (ver-tices) are connected i.e. E σ ⊆ V σ × V σ . On the otherhand, each pair of layers σ and σ ( σ = σ ) of the mul-tilayer network is described by an interlayer bipartitegraph G σσ ( V σ , V σ , E σσ ), where E σσ ⊆ V σ × V σ . Thesupra-adjacency matrix A of dimension N = P σ N σ forthe multilayer network is then given by A = A A · · ·A A · · · ... ... . . . . (1)We assume that the edges are undirected and, therefore, A σ = ( A σ ) T , A σσ = ( A σσ ) T for σ, σ = 1 , , . . . , M ( σ = σ ) which implies that A = A T . The multilayernetwork can then be described by the graph G ( V , E ),where V = M S σ =1 V σ and E = M S σ,σ =1 E σσ , where E σσ = E σ . We know that the graph G σ has a symmetry if and onlyif it is possible to find a bijective mapping Π σ : V σ → V σ which preserves the adjacency relation of G σ in the layer σ , i.e. the bijection Π σ is an automorphism for G σ . Thismeans there exists a permutation matrix P σ = P σ (Π σ )such that P σ A σ ( P σ ) − = A σ . Collection of all such P σ forms a group with respect to matrix multiplication FIG. 1.
A simple multilayer network diagram consist-ing of two layers.
In the absence of inter-layer connections,symmetry analysis confirms the clusters { (2 3) , (4 7) , (5 6) } inthe layer 1 and { (1 3) , (2 4) } in the layer 2. In the presence ofinterlinks, the orbit { (2 3) } of the layer 1 and { (1 3) } of thelayer 2 are broken. The formation and destruction of clustersin presence (absence) of interlinks are further confirmed byEVC shown in the Figure 2. which the symmetry group of the graph G σ . The sym-metry group of the multilayer network [38] representedby the graph G is formed by the collection of all N × N matrices P = P · · · P · · · . . . P M , (2)where P , P , . . . are the elements of the symmetrygroups of the graphs G , G , . . . respectively and satisfythe conditions P σ A σσ = A σσ P σ and P σ A σ σ = A σ σ P σ , (3) ∀ σ, σ ∈ { , . . . , M } and σ = σ . Now the set of nodes V of the multilayer is partitioned into disjoint invariantsubsets (orbits) by looking at the action of all symme-try elements on V . Each such subset of nodes forms a‘cluster’ of the multilayer network.Therefore, according to the existing theory, determina-tion of the clusters of a multilayer network involves threeessential steps: (1) finding the symmetry groups of theindividual layers, (2) forming the symmetry group of themultilayer network using the conjugacy relations (3) and(3) finding the invariant subsets of the set of nodes ofthe multilayer network. For larger multilayer networksthis algorithm become highly complex. In this backdrop,we propose an alternative mathematical framework fordetermining the clusters of any large multilayer network.The details are given in the following paragraph. FIG. 2.
Identification of cluster nodes of the multilayernetwork shown in figure 1 using eigenvector central-ity (EVC) both in presence and absence of inter-layercoupling.
First and second rows correspond to the layers- 1 and 2 respectively. The first column ((a) and (b)) showsthe centrality values of the layers in the absence of inter-layercoupling, while that in the presence of the inter-layer cou-pling are shown in the second column ((c) and (d)). Thefilled rectangles and circles of same colors within the shadedelliptical regions show different clusters of the layers. TheEVCs corresponding to the isolated nodes are shown withblack rectangles and circles.
Let X be the eigen vector centrality (EVC) oran eigen vector of the supra adjacency matrix A cor-responding to the largest eigen value λ m (note that allthe eigen values of A are real) and by Peron-Frobenioustheorem [55, 56] we choose X in such a way that all thecomponents of it are strictly positive.Now, we have A X = λ m X . (4)Multiplying both sides of the equation (4) from left byan element P of a symmetry group of G we get, PA X = λ m P X , (5)which implies AP X = λ m P X ( ∵ AP = PA ) . (6)Assuming P X = Y we get ⇒ A Y = λ m Y .Hence, along with X , Y is also an eigenvector of A cor-responding to the same eigenvalue λ m . Hence the set { X , Y } is linearly dependent, which implies that thereexists a real number c such that Y = c X i . e . P X = c X . (7)Since Y is nothing but a rearrangement of the elementsof X , the value of c must be 1. Therefore, we have P X = X . (8) Note that the relation (8) is true for all matrices P be-longing to the symmetry group of G and it says that the eigen vector centrality X remains invariant under theaction of such P .Now since each cluster of the graph G is mapped toitself by the action of all such P , for the equation (8) tobe valid, the components of X corresponding to the nodesof a cluster must be same. Also the components of X corresponding to the nodes of different clusters must bedifferent. A proof of this fact is given using the method ofcontradiction. If possible, let us suppose the componentsof X corresponding to the nodes of different clusters aresame. Then, apart from the symmetry group elementsof the graph G , there will be some more permutationmatrices P for which P X = X . The action of thosematrices on A will not preserve the adjacency relationsand will not satisfy PA = AP . Hence P X (= X ) cannot be an eigen vector of A which is a contradiction.Therefore, the components of X corresponding to thenodes of different clusters must be different. As a result,by looking at the values of the eigen vector centralityone can very easily determine the clusters of a multilayernetwork.For the verification of the above theory we consider amultilayer network consisting of two layers (see Fig.1)having 7 nodes in the first layer and 4 nodes in thesecond layer. In the absence of interlayer connections,we perform the symmetry group analysis as proposedin [24, 38] and identify the clusters in the individuallayers. In the layer 1, the notrivial clusters are givenby { (2 3) , (4 7) , (5 6) } and that in the layer 2, are { (1 3) , (2 4) } . When the layers are connected throughinterlayer links as shown in the Fig. 1, the conjugacyrelation (3) shows that there are 3 nontrivial clusters inthat multilayer network, out of them 2 clusters are fromthe layer 1 ( { (4 7) , (5 6) } ) and 1 cluster is from the layer2 ( { (2 4) } ). Now we calculate the eigen vector centrality(EVC) of the supra adjacency matrix A of the multi-layer network and as predicted by the proposed theory,we find that the values of the eigen vector centrality com-ponents are exactly same in the positions correspondingto a cluster. In absence of the interlinks, the eigen vec-tor centralities (EVC) of the layers 1 and 2 are shownin the Fig. 2(a) and (b) respectively. Clearly, the EVCvalues (exact values are not shown here) in each patchare same : Nodes 2 , , , , , Supplemental Material I . These featuresconform our theory.
III. CLUSTER SYNCHRONIZATION INFRUSTRATED KURAMOTO DYNAMICS
Now using the proposed theory, we proceed to inves-tigate cluster synchronization in multilayer network ofphase oscillators in presence of phase lag (Sakaguchi-Kuramoto model) [47, 49, 51, 57–59]. In the netwrok, thephase θ σi of node i (= 1 , . . . , N σ ) in layer σ (= 1 , . . . , M )of the multilayer network is governed by the coupled dif-ferential equations˙ θ σi = ω σi + (cid:15) σ N σ X j =1 A σij sin( θ σj − θ σi − α σ )+ X σ = σ (cid:15) σσ N σ X j =1 A σσ ij sin( θ σ j − θ σi ) , (9)where A σ = [ A σij ] and ω σi respectively denote the adja-cency matrix and natural frequency of the i th oscillatorin layer σ , while (cid:15) σ and α σ denote the uniform couplingstrength and phase-lag parameters respectively for layer σ . (cid:15) σσ and A σσ ij indicate interlayer coupling strengthand the ij th element of the interlayer coupling matrix re-spectively between the layers σ and σ ( σ = σ ).The presence of lag parameter ( α ) in the system usu-ally prevent the global synchronization except for somespecial choice of natural frequencies[50]. However, it hasbeen found that in an appropriate region of the parame-ter space, the system is known to exhibit cluster synchro-nization [24] or remote synchronization [60, 61]. We willfocus here the emergence of CS in a network of phase-lag oscillators. To measure the cluster synchronizationin the network, we introduce the cluster order parameter R σ cluster for all clusters present in layer σ defined by R σ cluster = 1 ℘ σ ℘ σ X k =1 R σk , (10)where ℘ σ is the total number of clusters (including trivialand non-trivial clusters) and R σk is the order parameterof the k th cluster in layer σ which is again defined as R σk e i ( ψ σ − α σ ) = 1 N σk X j ∈ v σk e i ( θ σj − α σ ) , (11)where N σk is the number of nodes in the k th cluster and v σk is the set of nodes involved in the cluster k of layer σ . R kσ = 1 quantifies a perfectly synchronized k th clusterand 0 ≤ R kσ < ω i = 1). For this choice of naturalfrequencies, the system exhibits cluster synchronizationfor wide range of α (0 ≤ α ≤ . (cid:15) σ = 1 , ( σ = 1 , (cid:15) = 1. In the absence of inter layer coupling( (cid:15) = 0) the time evolution of the phases of the firstlayer will satisfy the relations θ ( t ) = θ ( t ) , θ ( t ) = θ ( t ) , and θ ( t ) = θ ( t ) , (12)which shows four distinct groups of phases namely { θ } , { θ , θ } , { θ , θ } , and { θ , θ } . On the other hand, thesecond layer shows two distinct groups of phases namely { θ , θ } and { θ , θ } satisfying the relations θ ( t ) = θ ( t ) , and θ ( t ) = θ ( t ) . (13)These dynamical results are fully consistent with theEVC elements described in the Fig. 2 (a) and (b). In pres-ence of interlayer coupling ( (cid:15) = 1), the system showsthree nontrivial clusters along with five trivial clusters(also see figures 2(c) and (d)) satisfying θ ( t ) = θ ( t ) , θ ( t ) = θ ( t ) , and θ ( t ) = θ ( t ) . (14)Apart from the choice of identical frequencies we havealso considered the degree correlated frequencies: ω i = d i .The numerical integration of the system is performedusing the the fourth order Runge-Kutta (RK4) methodwith time step 0 .
01. We now numerically integrate themodel Eqn. (9) for identical frequencies ( ω i = 1) aswell as degree-correlated frequency ( ω i = d i , d i is thedegree of i th node). After discarding the transients,we calculate the cluster order parameter ( R σ cluster )of each layer using the relation (10). The phase lagparameter α σ is varied in the range [0 ,
2] with step size α σ = 0 .
01. The top panel of the Fig. 3 shows the timeaveraged order parameters as a function of α . The redand cyan stars connected by solid lines represent thecluster order parameters of the first and second layersrespectively for uniform natural frequency distribution,while the empty blue and magenta circles connected bysolid lines show the cluster order parameters of the firstand second layers respectively in the degree-frequencycorrelated environment. As mentioned earlier, thecluster synchronization persists in the layers as long as R σ cluster = 1 and clusters are broken when R σ cluster < α ( α c ∼ .
12 for ω i = 1 and α c ∼ .
52 for ω i = d i ). Time series of thephases simulated from the model Eq. (9) with uniformfrequency distribution as ω i = 1 and α σ = 0 . Supplimental Material I . Further, we check thestability of the cluster states by computing transverselyapunov exponents (TLE) from the master stabilityfunction given by
FIG. 3.
Cluster order parameter and transeverse Lya-punov exponent.
Panel (a) indicates order parameter ofthe clusters ( R , ) for the two layers as a function of phaselag parameter α and panel (b) indicates the transversal Lya-punov exponent (Λ max ) calculated for the network shown inFig. (1) to check the stability of the clusters. Red line indi-cates the case of uniform frequency distribution and blue lineindicates the degree-frequency correlated case. All the figuresare generated with a fixed intralayer coupling strength (cid:15) σ = 1and interlayer coupling strength (cid:15) σσ = 1. ˙ δθ σi = J f ( S σu ) δθ σi + (cid:15) σ N σ X j =1 A σσij J H σσ ( S σv , S σu ) δθ σj δθ σi + X σ = σ (cid:15) σσ N σ X j =1 A σσ ij J H σσ ( S σ v , S σu ) δθ σ j δθ σi , where J f , J H σσ and J H σσ denote the jacobian corre-sponding to f , the intralayer coupling function H σσ andthe interlayer coupling function H σσ . Q σσ and Q σσ re-spectively denote the intralayer and interlayer quotientmatrices. The details of the derivation of the master sta-bility function has been provided in the SupplementalMaterial section I.
The bottom panel of the FIG. 3shows the variation of the maximum transverse Lyapunovexponents (Λ max ) with the changes of α . The solid redand blue curves show the variations of Λ max with α for identical frequencies and degree-frequency correlation re-spectively. From the figure, we can see that the Λ max become positive from negative by crossing the horizon-tal line exactly at the same values of α where the clus-ter order parameters of the layers becomes less than 1which confirms that the cluster synchronization state ex-hibited by the system is stable. For further confirmationwe have designed several synthetic multilayer networks.The details of the results (number of clusters present inthe graph) are given in the Table I. Note that for each ofthe networks listed in the table we have also determinedthe clusters using the symmetry group analysis and foundthat the results perfectly match with the ones obtainedusing the proposed theory. The networks and stability ofeach clusters are described in the Supplemental Ma-terial section II . IV. CONCLUSIONS
We have proposed a new mathematical framework forcluster analysis in multilayer networks. We have analyt-ically proved a direct relation between the eigen vectorcentrality elements of the supra adjacency matrix andthe clusters (orbits) of the network. In this framework,extracting cluster information of a network is a singlestep method. For finding the clusters of a network, onehas to only find the EVC of the supra adjacency matrixand look at its elements. Thus, cluster analysis of anynetwork is drastically simplified. Several multilayer net-works have been analyzed using the proposed frameworkfor identifying the clusters and we find that the resultsare consistent with the ones obtained from the symmetrygroup analysis.We further used the framework for investigating thecluster synchronization in Sakaguchi-Kuramoto model onmultilayer networks. The results are found to be con-sistent with the clusters determined from the EVC. Weexpect that the method based on the EVC of the adja-cency matrix will be applicable for finding orbital clusterin any kind of network. Thus, will facilitate cluster anal-ysis in a range of problems including epileptic seizure,power-grid failure, social network analysis, investigationof brain dynamics etc. in the field of complex systems.
V. ACKNOWLEDGEMENTS
CH is supported by INSPIRE-Faculty grant (Code:IFA17-PH193). [1] B. M. Adhikari, C. M. Epstein, and M. Dhamala, Phys-ical Review E , 030701 (2013).[2] S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, andS. Havlin, Nature , 1025 (2010).[3] J. Buck, The Quarterly review of biology , 265 (1988). [4] A. Attanasi, A. Cavagna, L. Del Castello, I. Giardina,A. Jelic, S. Melillo, L. Parisi, O. Pohl, E. Shen, andM. Viale, Journal of The Royal Society Interface ,20150319 (2015). Network Type No. of Nontrivial Clusters
No. of Node ( | V | ) and Edges ( | E | ) Without Interlayer With Interlayer1 st layer 2 nd layer Interlayer 1 st layer 2 nd layer 1 st layer 2 nd layer N | V | = 7, | E | = 9 | V | = 4, | E | = 5 5 3 2 2 1 N | V | = 11, | E | = 14 | V | = 11, | E | = 13 11 4 3 2 2 N | V | = 11, | E | = 14 | V | = 11, | E | = 13 11 4 3 3 3 N | V | = 34, | E | = 72 | V | = 34, | E | = 78 33 5 4 3 4 N | V | = 100, | E | = 99 | V | = 100, | E | = 103 100 19 20 19 19 TABLE I.
Table for different multilayer network typologies. | V | and | E | denote the number of nodes and edges presentin the graph. Here non trivial clusters mean at least two nodes are synchronized. In Supplemental Material, section I ,the participating nodes of each cluster are marked with different color. The stability analysis of each cluster is also describedin the
Supplemental Material, section I .[5] C. Granell, S. G´omez, and A. Arenas, Physical reviewletters , 128701 (2013).[6] V. Nicosia, P. S. Skardal, A. Arenas, and V. Latora,Physical review letters , 138302 (2017).[7] M. M. Danziger, I. Bonamassa, S. Boccaletti, andS. Havlin, Nature Physics , 178 (2019).[8] C. I. Del Genio, J. G´omez-Garde˜nes, I. Bonamassa, andS. Boccaletti, Science Advances , e1601679 (2016).[9] D. Soriano-Pa˜nos, Q. Guo, V. Latora, and J. G´omez-Garde˜nes, Physical Review E , 062311 (2019).[10] M. De Domenico, C. Granell, M. A. Porter, and A. Are-nas, Nature Physics , 901 (2016).[11] D. Soriano-Pa˜nos, L. Lotero, A. Arenas, and J. G´omez-Garde˜nes, Physical Review X , 031039 (2018).[12] S. Strogatz, Sync: The emerging science of spontaneousorder (Penguin UK, 2004).[13] A. Arenas, A. D´ıaz-Guilera, J. Kurths, Y. Moreno, andC. Zhou, Physics reports , 93 (2008).[14] F. A. Rodrigues, T. K. D. Peron, P. Ji, and J. Kurths,Physics Reports , 1 (2016).[15] J. G´omez-Gardenes, Y. Moreno, and A. Arenas, Physicalreview letters , 034101 (2007).[16] F. Dorfler and F. Bullo, SIAM Journal on Control andOptimization , 1616 (2012).[17] A. E. Motter, S. A. Myers, M. Anghel, and T. Nishikawa,Nature Physics , 191 (2013).[18] P. Ashwin, S. Coombes, and R. Nicks, The Journal ofMathematical Neuroscience , 2 (2016).[19] L. M. Pecora and T. L. Carroll, Physical Review Letters , 2109 (1998).[20] L. M. Pecora and T. L. Carroll, Physical review letters , 821 (1990).[21] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchroniza-tion: a universal concept in nonlinear sciences , Vol. 12(Cambridge university press, 2003).[22] T. Dahms, J. Lehnert, and E. Sch¨oll, Physical ReviewE , 016202 (2012).[23] P. S. Skardal, E. Ott, and J. G. Restrepo, Physical Re-view E , 036208 (2011).[24] V. Nicosia, M. Valencia, M. Chavez, A. D´ıaz-Guilera,and V. Latora, Physical review letters , 174102(2013).[25] F. Sorrentino and E. Ott, Physical Review E , 056114(2007).[26] C. R. Williams, T. E. Murphy, R. Roy, F. Sorrentino,T. Dahms, and E. Sch¨oll, Physical review letters , 064104 (2013).[27] L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E.Murphy, and R. Roy, Nature communications , 4079(2014).[28] F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, T. E.Murphy, and R. Roy, Science advances , e1501737(2016).[29] M. Lodi, F. Della Rossa, F. Sorrentino, and M. Storace,Scientific reports , 1 (2020).[30] A. Bergner, M. Frasca, G. Sciuto, A. Buscarino, E. J.Ngamga, L. Fortuna, and J. Kurths, Physical Review E , 026208 (2012).[31] F. Sorrentino and L. Pecora, Chaos: An InterdisciplinaryJournal of Nonlinear Science , 094823 (2016).[32] L. V. Gambuzza and M. Frasca, Automatica , 212(2019).[33] A. B. Siddique, L. Pecora, J. D. Hart, and F. Sorrentino,Physical Review E , 042217 (2018).[34] Y. Wang, L. Wang, H. Fan, and X. Wang, Chaos: AnInterdisciplinary Journal of Nonlinear Science , 093118(2019).[35] B. Karakaya, L. Minati, L. V. Gambuzza, and M. Frasca,Physical Review E , 052301 (2019).[36] L. Zhang, A. E. Motter, and T. Nishikawa, Physicalreview letters , 174102 (2017).[37] Y. S. Cho, T. Nishikawa, and A. E. Motter, Physicalreview letters , 084101 (2017).[38] F. Della Rossa, L. Pecora, K. Blaha, A. Shirin, I. Klick-stein, and F. Sorrentino, Nature communications , 1(2020).[39] F. Sorrentino, L. M. Pecora, and L. Trajkovic,IEEE Transactions on Network Science and Engineering(2020).[40] R. Pastor-Satorras and A. Vespignani, Physical reviewletters , 3200 (2001).[41] R. Pastor-Satorras, C. Castellano, P. Van Mieghem, andA. Vespignani, Reviews of modern physics , 925 (2015).[42] S. Chakravartula, P. Indic, B. Sundaram, and T. Killing-back, PloS one , e0178975 (2017).[43] M. De Domenico, A. Sol´e-Ribalta, E. Cozzo, M. Kivel¨a,Y. Moreno, M. A. Porter, S. G´omez, and A. Arenas,Physical Review X , 041022 (2013).[44] J. Aguirre, R. Sevilla-Escoboza, R. Guti´errez, D. Papo,and J. Buld´u, Physical review letters , 248701 (2014).[45] P. Pradhan, C. Angeliya, and S. Jalan, Physica A: Statis-tical Mechanics and its Applications , 124169 (2020). [46] M. Newman, Networks: an introduction (Oxford univer-sity press, 2010).[47] Y. Kuramoto,
Chemical oscillations, waves, and turbu-lence (Courier Corporation, 2003).[48] P. Khanra, P. Kundu, P. Pal, P. Ji, and C. Hens, Chaos:An Interdisciplinary Journal of Nonlinear Science ,031101 (2020).[49] P. Khanra, P. Kundu, C. Hens, and P. Pal, PhysicalReview E , 052315 (2018).[50] P. Kundu, C. Hens, B. Barzel, and P. Pal, EPL (Euro-physics Letters) , 40002 (2018).[51] M. Brede and A. C. Kalloniatis, Physical Review E ,062315 (2016).[52] P. Kundu and P. Pal, Chaos: An Interdisciplinary Jour-nal of Nonlinear Science , 013123 (2019).[53] M. Lohe, Automatica , 114 (2015). [54] E. Omel’chenko and M. Wolfrum, Physical review letters , 164101 (2012).[55] O. Perron, Mathematische Annalen , 248 (1907).[56] G. Frobenius, F. G. Frobenius, F. G. Frobenius, F. G.Frobenius, and G. Mathematician, (1912).[57] P. Kundu, P. Khanra, C. Hens, and P. Pal, PhysicalReview E , 052216 (2017).[58] J. A. Acebr´on, L. L. Bonilla, C. J. P. Vicente, F. Ri-tort, and R. Spigler, Reviews of modern physics , 137(2005).[59] P. Khanra and P. Pal, Chaos, Solitons & Fractals ,110621 (2021).[60] V. Vlasov and A. Bifone, Scientific reports , 1 (2017).[61] V. Vuksanovi´c and P. H¨ovel, NeuroImage , 1 (2014). upplementary Material of “Cluster analysis in multilayer networks using eigen vectorcentrality” Pitambar Khanra , ∗ Subrata Ghosh , ∗ Prosenjit Kundu , Chittaranjan Hens , and Pinaki Pal Department of Mathematics, National Institute of Technology, Durgapur 713209, India Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India and Department of Mathematics, University of Buffalo, State university of New York, Buffalo, USA ∗ Equal contribution a r X i v : . [ n li n . AO ] F e b I. ADJACENCY MATRICES CORRESPONDING TO THE FIG. 1 OF THE MAIN TEXT
The adjacency matrices of the network shown in the Fig. 1 of the main text in the absence (presence) of interlinkare described below. In the absence of interlayer links, the red shaded matrix ( A , Eq. 1) describes the network of thelayer I and blue shaded matrix describes the network of the layer II ( A , Eq. 2). On the other hand, in the presence ofinterlayer links, the supra adjacency matrix A describes the multilayer network shown in the Fig. 1 of the main text.Note that the white shaded portion in the matrix A of Eq. (3) corresponds to the interlayer connections between thelayers. Eigen vector centralities (EVC) corresponding to each matrices are also shown in the equations (1) - (3). TheEVC components associated with each nontrivial cluster have been shown with same colors (other than gray), whilethe gray colored EVC components are associated with trivial or single node clusters. The color coding used in theEVC components are consistent with the color coding of the Fig. 2 of the main text. • Adjacency matrices associated with the layers I and II in the absence of interlayer links and corresponding EVC: A = , EVC = . . . . . . . (1) A = , EVC = . . . . (2) • Supra adjacency matrix in the presence of interlayer links and corresponding EVC: A = , EVC = . . . . . . . . . . . , (3) FIG. 1.
Time series of thenetwork described in Fig.(1) of main text both inthe presence and absenceof interlayer interaction .First and second rows corre-spond to the layers 1 and 2respectively. The first col-umn ((a) and (c)) shows thetime series in absence of inter-layer coupling, while that inthe presence of the inter-layercoupling are shown in the sec-ond column ((b) and (d)).
The time evolution of the phases of the Kuramoto-Sakaguchi model on the multilayer network corresponding tothe supra adjacency matrix A is shown in the figure 1 for α σ = 0 . , (cid:15) σ = 1 and (cid:15) σσ = 1. From the figure, thesynchronized clusters consistent with the proposed theory is clear. II. STABILITY ANALYSIS OF CLUSTER SYNCHRONIZED STATE
The model Eq. (9) of the main text can be written as,˙ θ σi = f ( θ σi ) + (cid:15) σ N σ X j =1 A σij H σσ ( θ σj , θ σi ) + X σ = σ (cid:15) σσ N σ X j =1 A σσ ij H σσ ( θ σ j , θ σi ) , (4)where H σσ denotes the underlying coupling function between the nodes of layer σ and H σσ denotes the couplingfunction between the layers σ, σ . In our case we have taken sinusoidal coupling function with phase frustration inintralayer coupling.Using the EVC framework, layer σ can be partitioned into ℘ σ number of orbital clusters namely C σ , C σ , . . . , C σ℘ σ .Let us define a ℘ σ × ℘ σ intralayer quotient matrix Q σσ such that for each pair of σ - clusters ( C σu , C σv ), Q σσuv = X j ∈C σv A σij ; ∀ i ∈ C σu , u, v = 1 , , . . . , ℘ σ , (5)and the corresponding ℘ σ × ℘ σ interlayer quotient matrix Q σσ can be written in the form such that for each pairof clusters, C σu will be from the layer σ and the second cluster C σ v will be selected from layer σ ( σ = σ ) and is given by, Q σσ uv = X j ∈C σ v A σσ ij ; ∀ i ∈ C σu , u = 1 , , . . . , ℘ σ & v = 1 , , . . . , ℘ σ . (6)Let S σu indicates the synchronized manifold for the clusters C σu in layer σ , u = 1 , , . . . , ℘ σ . Thus the time evolutionequation for the synchronized manifold can be written as,˙ S σu = f ( S σu ) + (cid:15) σ ℘ σ X v =1 Q σσuv H σσ ( S σv , S σu ) + X σ = σ (cid:15) σσ ℘ σ X v =1 Q σσ uv H σσ ( S σ v , S σu ) , (7)where σ = 1 , , . . . , M and u = 1 , , . . . , ℘ σ . Every node θ σi of layer σ is mapped into one synchronized manifold S σu , u = 1 , , . . . , ℘ σ such that θ σi ≡ S σu , ∀ i ∈ C σu . Now considering small perturbations δθ σi around the synchronizedmanifold, the phase becomes θ σi = S σu + δθ σi , ∀ i ∈ C σu . Then the variational equation corresponding to Eq. 4 is,˙ δθ σi = J f ( S σu ) δθ σi + (cid:15) σ N σ X j =1 A σσij J H σσ ( S σv , S σu ) δθ σj δθ σi + X σ = σ (cid:15) σσ N σ X j =1 A σσ ij J H σσ ( S σ v , S σu ) δθ σ j δθ σi , (8)where J f , J H σσ and J H σσ denote the jacobian corresponding to f , the intralayer coupling function H σσ and theinterlayer coupling function H σσ . Equation (8) is the required master stability function to check the stability of thecluster states. Now we calculate the maximal lyapunov exponent (Λ max ) of the clusters along the transverse directionusing the Eq. (7). (-ve) value of that tranversal lyapunov exponent indicates the stable region for the clusters andthe (+ve) value indicates the unstable region. We have calculated the the stability of cluster states of coupled phase-frustrated oscillators for five different networks shown in Fig. 2 f-j. The networks and associated clusters are shown inFig. 2 a-e. In the table 1 of the main text, the number of nodes, links and clusters of each network are given explicitly.Note that all the cluster nodes are identified with eigen vector centrality of the supra-adjacency matrix A . Using thestability analysis of coupled phase oscillators we have identified a wide range of α where the synchrous cluster statesare stable (Fig. 2 f-j). FIG. 2.
Multilayer Networks and cluster stability diagram . (a-e) Five synthetic muti layer networks ( N ,.., ) are drawnusing gephi[1] and edited with inkscape[2]. (a) N has three clusters :two red, two blue in upper layer and two green in lowerlayer. The stability of all the clusters are shown in (f) as a function phas-lag ( α ) parameter. The coupling is fixed at (cid:15) = 1.The cluster order parameter R , of each layer deviates from 1 around α c ∼ . α where Λ max changes it sign. The network has 11 nodes and total 14 edges. (b) and (g) N . Total nodes : 22 andnumber of links: 27. Each of the layer has two clusters: blue and green in the upper layer where as two red and two orangenodes male cluster in the lower layer. The transverse exponent becomes positive at α c ∼ . R , isdisturbed form the unit value. (c) and (h) N . Total nodes : 22 and number of links: 27. The system has five clusters. In theupper layer, there is three clusters: four red nodes participate in a cluster. Two magenta and two green nodes participate intwo different clusters. In the lower layer, four blue nodes, two apple green nodes and two brick red nodes participate to formthree separate clusters. Here the critical phase lag value α c ∼ .
47. (d) and (i) N . 68 nodes and 150 links. Total 7 clusters(green, violet and red in upper layer, blue, brick red, cyan and apple green in the lower layer) exist in the system. The clusterslose their stability at α c ∼ .
55. (e) and (j) N . 200 nodes and 202 links. 38 clusters exist in the network. The cluster in thesystem loses its stability at α c ∼ .
74. All the coupling strength is fixed for simulation at 1, i.e. (cid:15) σ = 1 , ( σ = 1 ,
2) and (cid:15) = 1.= 1.