Edge corona product as an approach to modeling complex simplical networks
aa r X i v : . [ c s . D M ] F e b Edge corona product as an approach tomodeling complex simplical networks
Yucheng Wang, Yuhao Yi, Wanyue Xu and Zhongzhi Zhang,
Member, IEEE
Abstract —Many graph products have been applied to generate complex networks with striking properties observed in real-worldsystems. In this paper, we propose a simple generative model for simplicial networks by iteratively using edge corona product. Wepresent a comprehensive analysis of the structural properties of the network model, including degree distribution, diameter, clusteringcoefficient, as well as distribution of clique sizes, obtaining explicit expressions for these relevant quantities, which agree with thebehaviors found in diverse real networks. Moreover, we obtain exact expressions for all the eigenvalues and their associatedmultiplicities of the normalized Laplacian matrix, based on which we derive explicit formulas for mixing time, mean hitting time and thenumber of spanning trees. Thus, as previous models generated by other graph products, our model is also an exactly solvable one,whose structural properties can be analytically treated. More interestingly, the expressions for the spectra of our model are also exactlydetermined, which is sharp contrast to previous models whose spectra can only be given recursively at most. This advantage makesour model a good test-bed and an ideal substrate network for studying dynamical processes, especially those closely related to thespectra of normalized Laplacian matrix, in order to uncover the influences of simplicial structure on these processes.
Index Terms —Graph product, Edge corona product, Complex network, Random walk, Graph spectrum, Hitting time, Mixing time. ✦ NTRODUCTION C OMPLEX networks are a powerful tool for describingand studying the behavior of structural and dynamicalaspects of complex systems [1]. An important achievementin the study of complex networks is the discovery thatvarious real-world systems from biology to social networksdisplay some universal topological features, such as scale-free behavior [2] and small-world effect [3]. The formerimplies that the fraction of vertices with degree d obeys adistribution of power-law form P ( d ) ∼ d − γ with < γ ≤ .The latter is characterized by small average distance (ordiameter) and high clustering coefficient [3]. In additionto these two topological aspects, a lot of real networks areabundant in nontrivial patterns, such as q -cliques [4] andmany cycles at different scales [5], [6]. For example, spikingneuron populations form cliques in neural networks [7],[8], while coauthors of a paper constitute a clique in scien-tific collaboration networks [9]. These remarkable structuralproperties or patterns greatly affect combinatorial [10], [11],structural [12] and dynamical [13], [14] properties of net-works, and lead to algorithmic efforts on finding nontrivialsubgraphs, e.g., q -cliques [15], [16].In order to capture or account for universal propertiesobserved in practical networks, a lot of mechanisms, ap-proaches, and models were developed in the community ofnetwork science [1]. Currently, there are many importantgraph generation literature [17], [18], [19], graph genera-tors [20], as well as packages [21], for example, NetwrokX . • Yucheng Wang, Yuhao Yi, Wanyue Xu, and Zhongzhi Zhang are with theShanghai Key Laboratory of Intelligent Information Processing, Schoolof Computer Science, Fudan University, Shanghai, 200433, China; andalso with Shanghai Blockchain Engineering Research Center, Shanghai200433, China. (Corresponding author: Zhongzhi Zhang.)(E-mail: [email protected]; [email protected];[email protected]; [email protected]).
1. https://networkx.github.io/
In recent years, cliques, also called simplicial complexes,have become very popular to model complex networks [15],[22]. Since large real-world networks are usually made upof small pieces, for example, cliques [4], motifs [23], andcommunities [24], graph products are an important andnatural way for modelling real networks, which generatea large graph out of two or more smaller ones. An obviousadvantage of graph operations is the allowance of tractableanalysis on various properties of the resultant compositegraphs. In the past years, various graph products have beenexploited to mimic real complex networks, including Carte-sian product [25], corona product [26], [27], hierarchicalproduct [28], [29], [30], [31], and Kronecker product [32],[33], [34], [35], [36], and many more [37].Most current models based on graph operations eitherfail to reproduce serval properties of real networks or arehard to exactly analyze their spectral properties. For exam-ple, iterated corona product on complete graphs only yieldssmall cycles [26], [27]; while for most networks created bygraph products, their spectra can be determined recursivelyat most. On the other hand, in many real networks [38],[39], such as brain networks [7], [40] and protein-proteininteraction networks [41], there exist higher-order nonpair-wise relations between more than two nodes at a time. Thesehigher-order interactions, also called simplicial interactions,play an important role in other structural and dynamicalproperties of networks, including percolation [42], synchro-nization [43], [44], disease spreading [45], and voter [46].Unfortunately, most models generated by graph productsand generators cannot capture higher-order interactions,and how simplicial interactions affect random walk dynam-ics, i.e., mixing time [47], is still unknown.From a network perspective, higher-order interactionscan be described and modelled by hypergraphs [48], [49].Here we model the higher-order interactions by simplicial complexes [50] generated by a graph product. Althoughboth simplicial complexes and hypergraphs can be appliedfor the modelling and analysis of realistic systems withhigher-order interactions, they differ in some aspects. First,simplicial complexes have a geometric interpretation [51].For example, they can be explained as the result of gluingnodes, edges, triangles, tetrahedra, etc. along their faces.This interpretation for simplicial complexes can be exploitedto characterize the resulting network geometry, such as net-work curvatures [52]. Moreover, a higher-order interactiondescribed by hypergraphs do not require the presence of alllow-order interactions.In this paper, by literately applying edge corona prod-uct [53] first proposed by Haynes and Lawson [54], [55] tocomplete graphs or q -cliques K q with q ≥ , we proposea mathematically tractable model for complex networkswith various cycles at different scales. Since the resultantnetworks are composed of cliques of different sizes, wecall these networks as simplical networks . The networks candescribe simplicial interactions, which have rich structural,spectral, and dynamical properties depending on the pa-rameter q . Thus, they can be used to study the influence ofsimplicial interactions on various dynamics.Specifically, we present an extensive and exact analysisof relevant topological properties for the simplical networks,including degree distribution, diameter, clustering coeffi-cient, and distribution of clique sizes, which reproduce thecommon properties observed for real-life networks. We alsodetermine exact expressions for all the eigenvalues and theirmultiplicities of the transition probability matrix and nor-malized Laplacian matrix. As applications, we further ex-ploit the obtained eigenvalues to derive leading scaling formixing time, as well as explicit expressions for average hit-ting time and the number of spanning trees. The proposedmodel allows for rigorous analysis of structural properties,as previous models generated by graph products. In contrastto previous models for which the eigenvalues for relatedmatrices are given recursively at most, the eigenvalues oftransition probability matrix for our model can be exactlydetermined. This advantage allows to study analyticallyeven exactly related dynamical processes determined by oneor several eigenvalues, for example, mixing time of randomwalks, which gives deep insight into behavior for mixingtime in real-life networks. ETWORK CONSTRUCTION
The network family proposed and studied here is con-structed based on the edge corona product of graphs definedas follows [53], [54], [55], which is a variant of the coronaproduct first introduced by Frucht and Harary [56] of twographs. Let G and G be two graphs with disjoint vertexsets, with the former G having n vertices and m edges.The edge corona G ⊚ G of G and G is a graph obtainedby taking one copy of G and m copies of G , and thenconnecting both end vertices of the i th edge of G to eachvertex in the i th copy of G for i = 1 , , . . . , m .Let K q , q ≥ , be the complete graph with q vertices.When q = 1 , we define K q as a graph with an isolate vertex.Based on the edge corona product and the complete graphs,we can iteratively build a set of graphs, which display the striking properties of real-world networks. Let G q ( g ) , q ≥ and g ≥ , be the network after g iterations. Then, G q ( g ) isconstructed in the following way. Definition 1.
For g = 0 , G q (0) is the complete graph K q +2 .For g ≥ , G q ( g + 1) is obtained from G q ( g ) and K q by performing edge corona product on them: for everyexisting edge of G q ( g ) , we introduce a copy of thecomplete graph K q and connect all its vertices to bothend vertices of the edge. That is, G q ( g + 1) = G q ( g ) ⊚ K q .Figure 1 illustrates the construction process of G q ( g ) for twoparticular cases of q = 1 and q = 2 . Note that for q = 1 , G q ( g ) is reduced to the pseudofractal scale-free web [57],which only contains triangles but excludes other completegraphs with more than 3 vertices. G (2) G (1) G (0) G (0) G (1) Fig. 1. The first several iterations of G q ( g ) for q = 1 and q = 2 . Let N q ( g ) and M q ( g ) be the number of vertices andnumber of edges in graph G q ( g ) , respectively. Suppose L v ( g ) and L e ( g ) be the number of vertices and the num-ber of edges generated at iteration g . Then for g = 0 , L v (0) = N q (0) = q + 2 and L e (0) = M q (0) = ( q +1)( q +2)2 .For all g ≥ , by Definition 1, we obtain the following tworelations: L v ( g + 1) = qM q ( g ) (1)and L e ( g + 1) = (cid:20) ( q + 1)( q + 2)2 − (cid:21) M q ( g ) , (2)which lead to recursive relationships for N q ( g ) and M q ( g ) as M q ( g + 1) = ( q + 1)( q + 2)2 M q ( g ) (3)and N q ( g + 1) = qM q ( g ) + N q ( g ) . (4)Considering the initial conditions N q (0) = q + 2 and M q (0) = ( q +1)( q +2)2 , the above two equations are solvedto obtain M q ( g ) = (cid:20) ( q + 1)( q + 2)2 (cid:21) g +1 (5) and N q ( g ) = 2 q + 3 (cid:20) ( q + 1)( q + 2)2 (cid:21) g +1 + 2( q + 2) q + 3 . (6)Then, the average degree of vertices in graph G q ( g ) is M q ( g ) /N q ( g ) , which tends to q + 3 when g is large.Therefore, the graph family G q ( g ) is sparse.In addition, inserting Eqs. (5) and (6) into Eqs. (1)and (2) gives L v ( g ) = q h ( q +1)( q +2)2 i g and L e ( g ) = h ( q +1)( q +2)2 − i h ( q +1)( q +2)2 i g for g ≥ , which are helpfulfor the computation in the sequel. TRUCTURAL PROPERTIES
In this section, we study some relevant structural charac-teristics of G q ( g ) , focusing on degree distribution, diameter,clustering coefficient, and distribution of clique sizes. The degree distribution P ( d ) for a network is the probabilityof a randomly selected vertex v has exactly d neighbors.When a network has a discrete sequence of vertex degrees,one can also use cumulative degree distribution P cum ( d ) instead of ordinary degree distribution [1], which is theprobability that a vertex has degree greater than or equalto d : P cum ( d ) = ∞ X d ′ = d P ( d ′ ) . (7)For a graph with degree distribution of power-law form P ( d ) ∼ d − γ , its cumulative degree distribution is alsopower-law satisfying P cum ( d ) ∼ d − ( γ − .For every vertex in graph G q ( g ) , its degree can be ex-plicitly determined. Let d v ( g ) be the degree of a vertex v ingraph G q ( g ) . When v was generated at iteration g v , it has adegree of q + 1 . By construction, for any edge incident with v at current iteration, it will lead to q additional new edgesadjacent to v at the following iteration. Therefore, d v ( g ) = ( q + 1) g − g v +1 . (8)On the other hand, in graph G q ( g ) the degree of all simulta-neously emerging vertices is the same. Then, the numberof vertices with the degree ( q + 1) g − g v +1 is q + 2 and q h ( q +1)( q +2)2 i g v for g v = 0 and g v > , respectively. Proposition 1.
The degree distribution of graph G q ( g ) fol-lows a power-law form P ( d ) ∼ d − γ with the powerexponent γ = 2 + ln( q +2)ln( q +1) − ln 2ln( q +1) . Proof:
As shown above, the degree sequence of ver-tices in G q ( g ) is discrete. Thus we can get the degree distri-bution P ( d ) for d = ( q +1) g − g v +1 via the cumulative degreedistribution given by P cum ( d ) = 1 N q ( g ) X τ g v L v ( τ )= (cid:2) ( q + 1)( q + 2) (cid:3) g v +1 + q + 2 (cid:2) ( q + 1)( q + 2) (cid:3) g +1 + q + 2 . (9) From Eq. (8), we derive g v = g + 1 − ln d ln( q +1) . Plugging thisexpression for g v into the above equation leads to P cum ( d ) = 2 ln d ln( q +1) − g − [( q + 1)( q + 2)] − ln d ln( q +1) + g +2 + q + 22 − g − [( q + 1)( q + 2)] g +1 + q + 2= d − ( ln( q +2)ln( q +1) +1 − ln 2ln( q +1) )2 − g − [( q + 1)( q + 2)] g +2 + q + 22 − g − [( q + 1)( q + 2)] g +1 + q + 2 . (10)When g → ∞ , we obtain P cum ( d ) = ( q + 1)( q + 2)2 d − ( ln( q +2)ln( q +1) +1 − ln 2ln( q +1) ) . (11)So the degree distribution follows a power-law form P ( d ) ∼ d − γ with the exponent γ = 2 + ln( q +2)ln( q +1) − ln 2ln( q +1) .It is not difficult to see that the power exponent γ liesin the interval [ ln 2ln 3 + 2 , . Moreover, it is a monotonicallyincreasing function of q : When q increases from to infinite, γ increases from ln 2ln 3 + 2 to . Note that for most real scale-free networks [1], their power exponent γ is in the rangebetween and . In a graph G , where every edge having unit length, ashortest path between a pair of vertices u and v is a pathconnecting u and v with least edges. The distance d ( u, v ) between u and v is defined as the number of edges in sucha shortest path. The diameter of graph G , denoted by D ( G ) ,is the maximum of the distances among all pairs of vertices. Proposition 2.
The diameter D ( G q ( g )) of graph G q ( g ) , is D ( G ( g )) = g + 1 for q = 1 and D ( G q ( g )) = 2 g + 1 for q ≥ . Proof:
For the case of q = 1 , D ( G ( g )) = g + 1 wasproved in [58]. Below we only prove the case of q ≥ .For g = 0 , D ( G q ( g )) = 1 , the statement holds. ByDefinition 1, it is obvious that the diameter of graph G q ( g ) increases at most 2 after each iteration, which means D ( G q ( g )) ≤ g + 1 . In order to prove D ( G q ( g )) = 2 g + 1 , weonly need to show that for q ≥ there exist two vertices in G q ( g ) , whose distance g + 1 . To this end, we alternativelyprove an extended proposition that in G q ( g ) there exist twopairs of adjacent vertices: u and u , u and u , such that d ( u , u ) = d ( u , u ) = d ( u , u ) = d ( u , u ) = 2 g + 1 . Wenext prove this extended proposition by induction on g .For g = 0 , G q (0) , q ≥ , is the complete graph K q +2 .We can arbitrarily choose four vertices as u , u , u , u tomeet the condition. For g ≥ , suppose that the statementholds for G q ( g − , see Fig. 2. In other words, there exist twopairs of adjacent vertices: v and v , v and v in G q ( g − ,with their distances in G q ( g − satisfying d ( v , v ) = d ( v , v ) = d ( v , v ) = d ( v , v ) = 2 g − . For G q ( g − ,let u and u be two adjacent vertices generated by the edgeconnecting v and v at iteration g , and let u and u be twoadjacent vertices generated by the edge connecting v and v at iteration g . Then, by assumption, for the vertex pair u and u in graph G q ( g − , their distance obeys d ( u , u ) =min { d ( v , v ) , d ( v , v ) , d ( v , v ) , d ( v , v ) } + 2 = 2 g + 1 .Similarly, we can prove that in G q ( g − , the distancesof related vertex pairs satisfy d ( u , u ) = d ( u , u ) = d ( u , u ) = 2 g + 1 . v v u u u u v v G q ( g − Fig. 2. Illustrative proof of the extended proposition.
From Eq. (6), the number of vertices N q ( g ) ∼ h ( q +1)( q +2)2 i g +1 . Thus, the diameter D ( G q ( g )) of G q ( g ) scales logarithmically with N q ( g ) , which means that thegraph family G q ( g ) is small-world. Clustering coefficient [3] is another crucial quantity charac-terizing network structure. In a graph G = G ( V , E ) withvertex set V and edge set E , the clustering coefficient C v ( G ) of a vertex v with degree d v is defined [3] as theratio of the number ǫ v of edges between the neighboursof v to the possible maximum value d v ( d v − / , that is C v ( G ) = ǫ v d v ( d v − . The clustering coefficient C ( G ) of thewhole network G is defined as the average of C v ( G ) over allvertices: C ( G ) = |V| P v ∈V C v ( G ) .For graph G q ( g ) , the clustering coefficient for all verticesand their average value can be determined explicitly. Proposition 3.
In graph G q ( g ) , the clustering coefficient C v ( G q ( g )) of any vertex with degree d v ( g ) is C v ( G q ( g )) = q + 1 d v ( g ) . (12) Proof:
By Definition 1, when a vertex v was createdat iteration g v , its degree and clustering coefficient are q + 1 and 1, respectively. In any two successive iterations t and t − ( t ≤ g ), its degrees increases by a factor of q as d v ( t ) =( q + 1) d v ( t − . Moreover, once its degree increases by q ,then the number of edges between its neighbors increases by q ( q + 1) / . Then, in network G q ( g ) , the clustering coefficient C v ( G q ( g )) of vertex v with degree degree d v ( g ) is C v ( G q ( g )) = q ( q +1)2 + d v ( g ) − q − q q ( q +1)2 d v ( g )( d v ( g ) − = q + 1 d v ( g ) , (13)as claimed by the Proposition.Thus, in graph G q ( g ) , the clustering coefficient of anyvertex is inversely proportional to its degree, a scalingobserved in various real-world networked systems [59]. Proposition 4.
For all g ≥ , the clustering coefficient of G q ( g ) is C ( G q ( g )) = h ( q +1) ( q +2)2 i g +1 + q + 4 q + 4 q +4 q +5( q +1)( q +3) h ( q +1) ( q +2)2 i g +1 + ( q +2)( q +4 q +5) q +3 ( q + 1) g . (14) Proof:
By using Proposition 3, the quantity C ( G q ( g )) can be calculated by C ( G q ( g )) = 1 N q ( g ) g X g v =0 L v ( g v ) · q + 1 d v ( g ) ! = 1 N q ( g ) ( ( q + 2)( q + 1) g + g X g v =1 q (cid:20) ( q + 1)( q + 2)2 (cid:21) g v q + 1( q + 1) g − g v +1 ) = h ( q +1) ( q +2)2 i g +1 + q + 4 q + 4 q +4 q +5( q +1)( q +3) h ( q +1) ( q +2)2 i g +1 + ( q +2)( q +4 q +5) q +3 ( q + 1) g . (15) This finishes the proof.From Proposition 4, we can see that the clustering coeffi-cient of graph G q ( g ) is very high. For large g , the clusteringcoefficient G q ( g ) converges to a large constant as lim g →∞ C ( G q ( g )) = q + 4 q + 3 q + 4 q + 5 . (16)Thus, similarly to the degree exponent γ , clustering coeffi-cient C ( G q ( g )) is also dependent on q , with large q corre-sponding to large C ( G q ( g )) . When q → ∞ , the clusteringcoefficient of the graph tends to . It is apparent that graph G q ( g ) contains many cliques assubgraphs. Let N k ( G q ( g )) denote the number of k -cliquesin graph G q ( g ) . Since graph G q (0) is a q + 2 complete graph,the maximum clique size in it is q + 2 . Then in G q (0) thenumber N k ( G q (0)) of k -cliques is the combinatorial number C kq +2 = ( q +2)! k !( q +2 − k )! for k = 2 , , . . . , q + 2 , and is for k >q + 2 . For graph G q ( g ) with g ≥ , the number of 2-cliquesequals the number of edges, while for cliques with size morethan 2, we have the following proposition. Proposition 5.
For g ≥ , we have N k ( G q ( g )) = h ( q +1)( q +2)2 i g +1 − ( q +1)( q +2)2 − q + 2)! k !( q + 2 − k )! , (17)for k = 3 , , . . . , q + 2 . And N k ( G q ( g )) = 0 , for k > q + 2 . Proof:
The proposition is naturally satisfied in graph G q (0) . Thus, we only need to prove the proposition for g ≥ . By definition, when g ≥ , G q ( g ) is obtained from G q ( g − by introducing a new q -complete graph for every edge.Then, all the k -cliques in G q ( g ) can be partitioned into twoparts: (i) the k -cliques in G q ( g − , and (ii) the k -cliques thatcontain at least one newly introduced vertex.For part (i), the number of k -cliques is N k ( G q ( g − .For part (ii), since every newly introduced vertex is onlyconnected to other vertices in the q + 2 compete graphgenerated by an edge of G q ( g − , any k -clique containthis new vertex must be a subgraph of this q + 2 competegraph. The number of new q + 2 compete graphs equalsthe number M q ( g − of edges in G q ( g − , and in everynew q + 2 complete graph, the number of k -cliques is thecombinatorial number C kq +2 for k ≤ q + 2 . Since in everynew q + 2 complete graph, there are only two old vertices,each of its k -clique subgraph with k ≥ includes at leastone new vertex. Thus, for part (ii) the number of k -cliques can be calculated by M q ( g − C kq +2 for ≤ k ≤ q + 2 , andis obviously 0 for k > q + 2 .Combining the above results, we have that for g ≥ , N k ( G q ( g )) = N k ( G q ( g − M q ( g − C kq +2 , (18)for ≤ k ≤ q + 2 , and N k ( G q ( g )) = N k ( G q ( g − for k > q + 2 . Together with M q ( g −
1) = h ( q +1)( q +2)2 i g , C kq +2 = ( q +2)! k !( q +2 − k )! , and the initial values for G q (0) , theabove recursive relation is solved to obtain the proposition. PECTRA OF PROBABILITY TRANSITION MATRIXAND NORMALIZED L APLACIAN MATRIX
Let A g = A ( G q ( g )) denote the adjacency matrix of graph G q ( g ) , the entries A g ( i, j ) of which are defined as follows: A g ( i, j ) = 1 if the vertex pair of i and j is adjacent in G q ( g ) by an edge denoted by i ∼ j , or A g ( i, j ) = 0 otherwise.The vertex-edge incident matrix R g = R ( G q ( g )) of graph G q ( g ) is an N q ( g ) × M q ( g ) matrix, the entries R g ( v, e ) ofwhich are defined in the following way: R g ( v, e ) = 1 if vertex v is incident to edge e , and R g ( v, e ) = 0 oth-erwise. The diagonal degree matrix of G q ( g ) is D g = D ( G q ( g )) = diag { d ( g ) , d ( g ) , . . . , d N q ( g ) ( g ) } , where the i thnonzero entry is the degree d i ( g ) of vertex i in graph G q ( g ) .The Laplacian matrix L g = L ( G q ( g )) of graph G q ( g ) is L g = D g − A g . The transition probability matrix of G q ( g ) ,denoted by P g = P ( G q ( g )) , is defined by P g = D − g A g , withthe ( i, j ) th element P g ( i, j ) = 1 /d i ( g ) representing the tran-sition probability for a walker going from vertex i to vertex j in graph G q ( g ) . Matrix P g is asymmetric, but is similar tothe normalized adjacency matrix ˜ A g ( G q ( g )) = ˜ A g of graph G q ( g ) defined by ˜ A g = D − g A g D − g , since ˜ A g = D − g P g D g .By definition, the ( i, j ) th entry of matrix ˜ A g is ˜ A g ( i, j ) = A g ( i,j ) √ d i ( g ) √ d j ( g ) . Thus, matrix ˜ A g is real and symmetric, andhas the same set of eigenvalues as the transition probabilitymatrix P g . For graph G q ( g ) , its normalized Laplacian matrix ˜ L g ( G q ( g )) = ˜ L g is defined by ˜ L g = I g − ˜ A g , where I g is the N q ( g ) × N q ( g ) identity matrix.In the remainder of this section, we will studythe full spectrum of transition probability matrix P g and normalized Laplacian matrix ˜ L g for graph G q ( g ) .For i = 1 , , · · · , N q ( g ) , let λ i ( g ) = λ i ( G q ( g )) and σ i ( g ) = σ i ( G q ( g )) denote the N q ( g ) eigenvalues of ma-trices P g and ˜ L g , respectively. Let Λ g and Σ g denotethe set of eigenvalues of matrices P g and ˜ L g , respec-tively, that is Λ g = { λ ( g ) , λ ( g ) , . . . , λ N q ( g ) ( g ) } and Σ g = { σ ( g ) , σ ( g ) , . . . , σ N q ( g ) ( g ) } . It is obvious that forall i = 1 , , · · · , N q ( g ) , the relation λ i ( g ) = 1 − σ i ( g ) holds. Moreover, the eigenvalues of matrices P g and ˜ L g can be listed in a nonincreasing (or nondecreasing) orderas: λ ( g ) ≥ λ ( g ) ≥ . . . ≥ λ N q ( g ) ( g ) ≥ − and σ ( g ) ≤ σ ( g ) ≤ · · · ≤ σ N q ( g ) ( g ) ≤ .The one-to-to correspondence λ i ( g ) = 1 − σ i ( g ) between λ i ( g ) and σ i ( g ) , for all i = 1 , , · · · , N q ( g ) , indicates thatif one determines the eigenvalues of matrix P g , then theeigenvalues of matrix ˜ L g are easily found. Lemma 1.
For λ = − q +1 and λ = q − q +1 , λ is an eigenvalue of P g +1 if and only if ( q + 1) λ − q is an eigenvalue of P g , and the multiplicity of λ of P g +1 , denoted by m g +1 ( λ ) , isthe same as the multiplicity of eigenvalue ( q + 1) λ − q of P g , denoted by m g (( q + 1) λ − q ) , i.e. m g +1 ( λ ) = m g (( q +1) λ − q ) . Proof:
Let V g +1 be the set of vertices in graph G q ( g +1) . It can be looked upon the union of two disjoint sets V g and V ′ g +1 = V g +1 \V g , where V ′ g +1 includes all the newlyintroduced vertices by the edges in G q ( g ) . For all vertices in V g +1 , we label those in V g from 1 to N q ( g ) , while label thevertices V ′ g +1 from N q ( g ) + 1 to N q ( g + 1) . In the followingstatement, we represent all the vertices by their labels.Let y = ( y , y , . . . , y N q ( g +1) ) ⊤ denote the eigenvectorof eigenvalue λ of matrix P g +1 , where the component y i corresponds to vertex i in G q ( g + 1) . Then, λ y = P g +1 y . (19)By construction, for any two adjacent old vertices u and v in V g , there are q vertices newly introduced by theedge connecting u and v , which are denoted by h , h , . . . , h q . These q vertices, together with u and v form acomplete graph of q + 2 vertices. Moreover, each vertex h i in set { h , h , . . . , h q } is exactly connected to u , v , and othervertices in { h , h , . . . , h q } excluding h i itself. Then the rowin Eq. (19) corresponding to vertex h i , i = 1 , , . . . , q , can bewritten as λ y h i = N q ( g +1) X j =1 P g +1 ( h i , j ) y j = 1 d h i ( g + 1) X j ∼ h i y j = 1 q + 1 ( y u + y v + y h + . . . + y h i − + y h i +1 + . . . + y h q ) , (20)Adding q +1 y h i to both sides of the above equation yields (cid:18) λ + 1 q + 1 (cid:19) y h i = 1 q + 1 y u + y v + q X j =1 y h j , (21)for all i = 1 , , . . . , q . Therefore, for λ = − q +1 , y h = y h = . . . = y h q . (22)Combining Eqs. (21) and (22), we can derive that, for λ = q − q +1 y h i = 1( q + 1) λ − q − y u + y v ) (23)holds for i = 1 , , . . . , q . According to Eq. (19), we can alsoexpress the rows corresponding to components y u and y v .For the row associated with component y u , we have λ y u = N q ( g +1) X j =1 P g +1 ( u, j ) y j = 1 d u ( g + 1) X j ≤ N q ( g ) j ∼ u y j + X j>N q ( g ) j ∼ u y j . (24) By Definition 1, for an old vertex u , all its adjacent vertices in V ′ g +1 are introduced by the edges between u and its neigh-boring vertices in V g . Thus, combining Eqs. (23) and (24),we derive λ y u = 1 d u ( g + 1) X j ≤ N q ( g ) j ∼ u y j + X j ≤ N q ( g ) j ∼ u q ( y u + y j )( q + 1) λ − q − . (25)Considering d u ( g + 1) = ( q + 1) d u ( g ) , Eq. (25) can be recastas (cid:18) ( q + 1) λ − q ( q + 1) λ − q − (cid:19) y u = 1 d u ( g ) X j ≤ N q ( g ) j ∼ u (cid:18) q ( q + 1) λ − q − (cid:19) y j . (26)When λ = − q +1 and λ = q − q +1 , the above equation issimplified as [( q + 1) λ − q ] y u = 1 d u ( g ) X j ≤ N q ( g ) j ∼ u y j . = N q ( g ) X j =1 P g ( u, j ) y j , (27)which implies if y = ( y , y , . . . , y N q ( g ) , . . . , y N q ( g +1) ) ⊤ isan eigenvector of matrix P g +1 associated with eigenvalue λ ,then ˜ y = ( y , y , . . . , y N q ( g ) ) ⊤ is an eigenvector of matrix P g associated with eigenvalue ( q + 1) λ − q .On the other hand, suppose that ˜ y =( y , y , . . . , y N q ( g ) ) ⊤ is an eigenvector of matrix P g associated with eigenvalue ( q + 1) λ − q , then y = ( y , y , . . . , y, . . . , y N q ( g +1) ) ⊤ is an eigenvector ofmatrix P g +1 associated with eigenvalue λ if and onlyif its components y i , i = N q ( g ) + 1 , N q ( g ) + 2 , . . . , N q ( g + 1) , can be expressed by Eq. (23). Thus, thenumber of linearly independent eigenvectors of λ isthe same as that of ( q + 1) λ − q . Since both P g and P g +1 are normal matrices, which are diagonalizable,the multiplicity of λ (or ( q + 1) λ − q ) is equal to thenumber of its linearly independent eigenvectors. Hence, m g +1 ( λ ) = m g (( q + 1) λ − q ) .Lemma 1 indicates that except λ = − q +1 and q − q +1 , alleigenvalues λ of matrix P g +1 can be derived from thoseof matrix P g . However, it is easy to check that both − q +1 and q − q +1 are eigenvalues of matrix P g +1 . Moreover, theirmultiplicities can be determined explicitly. The followinglemma gives the multiplicity of − q +1 , while the multiplicityof q − q +1 will be provided later. Lemma 2.
The multiplicity of − q +1 as an eigenvalue ofmatrix P g +1 is ( q − M q ( g ) + N q ( g ) , i.e. m g +1 ( − q +1 ) =( q − M q ( g ) + N q ( g ) . Proof:
Let y = ( y , y , . . . . . . , y N q ( g +1) ) ⊤ be an eigen-vector associated with eigenvalue − q +1 of matrix P g +1 .Then, − q + 1 y = P g +1 y . (28) For an edge e x , x = 1 , . . . , M q ( g ) , in graph G q ( g ) withend vertices u and v , at iteration g + 1 , it will generate q vertices h , h , . . . , h q in V ′ g +1 . Then, the row in Eq. (28)corresponding to vertex h i , i = 1 , , . . . , q , can be expressedby − q + 1 y h i = N q ( g +1) X j =1 P g +1 ( h i , j ) y j = 1 q + 1 ( y u + y v + y h + . . . + y h i − + y h i +1 + . . . + y h q ) , (29)which is equivalent to q X i =1 y h i = − ( y u + y v ) . (30)On the other hand, the row in Eq. (28) corresponding tovertex u can be expressed as − q + 1 y u = 1 d u ( g + 1) X j ≤ N q ( g ) j ∼ u y j + X j>N q ( g ) j ∼ u y j . (31)Note that Eq. (30) holds for every pair of adjacent vertices ingraph G q ( g ) and the q new vertices it generates at iteration g + 1 . Plugging Eq. (30) into the right-hand side of Eq. (31)leads to d u ( g + 1) X j ≤ N q ( g ) j ∼ u y j + X j>N q ( g ) j ∼ u y j = 1 d u ( g + 1) X j ≤ N q ( g ) j ∼ u y j + X j ≤ N q ( g ) j ∼ u − ( y u + y j ) = 1 d u ( g + 1) X j ≤ N q ( g ) j ∼ u − y u = − q + 1) y u . (32)Therefore, the constraint on y in Eq. (28) is equivalent tothe constraint provided by M q ( g ) equations in Eq. (30). Thematrix form of these M q ( g ) equations can be written as · · · · · · − R ⊤ g . . . · · · · · · y = , (33)where R ⊤ g is the transpose of R g , and the unmarked entriesare vanishing. It is straightforward that the right partitionof the matrix in Eq. (33) is an M q ( g ) × qM q ( g ) matrix, witheach row corresponding to an edge e x , x = 1 , , . . . , M q ( g ) ,in graph G q ( g ) . Moreover, in each row associated with e x , repeats q times, corresponding to the q vertices newlycreated by edge e x . Since the row vectors of the matrix in Eq. (33) are linearlyindependent, the dimension of the solution space of Eq. (33)is N q ( g + 1) − M q ( g ) = ( q − M q ( g ) + N q ( g ) . Therefore,the multiplicity of eigenvalue − q +1 for matrix P g +1 is ( q − M q ( g ) + N q ( g ) . Theorem 1.
Let Λ g , g ≥ , be the set of the N q ( g ) eigenvalues λ ( g ) , λ ( g ) , . . . , λ N q ( g ) ( g ) for matrix P g , satisfying λ ( g ) ≥ λ ( g ) ≥ . . . ≥ λ N q ( g ) ( g ) ≥ − . Then the N q ( g + 1) eigenvalues for P g +1 forming the set Λ g +1 canbe listed in a descending order as Λ g +1 = (cid:26) λ ( g ) + qq + 1 , λ ( g ) + qq + 1 , . . . , λ N q ( g ) ( g ) + qq + 1 ,q − q + 1 , q − q + 1 , . . . , q − q + 1 | {z } M q ( g ) − N q ( g ) , − q + 1 , − q + 1 , . . . , − q + 1 | {z } ( q − M q ( g )+ N q ( g ) (cid:27) . (34) Proof:
We prove this theorem by induction on g . First,for g = 0 , it is easy to verify that the statement holds. Forgraph G q ( g ) , g ≥ , assume that the relation between Λ g − and Λ g is valid. We now prove that the result is true forgraph G q ( g + 1) .For each eigenvalue λ i ( g ) ∈ Λ g , i = 1 , , . . . , N q ( g ) , wehave λ i ( g ) > − by the assumption. Therefore, for i =1 , , . . . , N q ( g ) , λ i ( g ) + qq + 1 > q − q + 1 , (35)which implies λ i ( g )+ qq +1 = q − q +1 and λ i ( g )+ qq +1 = − q +1 . ByLemma 1, λ i ( g )+ qq +1 is an eigenvalue of P g +1 with the samemultiplicity of λ i ( g ) as an eigenvalue of P g , namely, m g +1 (cid:18) λ i ( g ) + qq + 1 (cid:19) = m g ( λ i ( g )) . (36)Moreover, by Lemma 1, for each eigenvalue λ of P g +1 satisfying λ = − q +1 and λ = q − q +1 , ( q + 1) λ − q must bean eigenvalue of P g , which means λ can be expressed by as λ = λ i ( g )+ qq +1 with i ∈ { , , . . . , N q ( g ) } . Therefore, the sumof multiplicity of all eigenvalues of P g +1 excluding − q +1 and q − q +1 is N q ( g ) , that is, m g +1 (cid:18) λ / ∈ (cid:26) − q + 1 , q − q + 1 (cid:27)(cid:19) = N q ( g ) . (37)We proceed to compute the multiplicity m g +1 (cid:16) q − q +1 (cid:17) ofeigenvalue q − q +1 for matrix P g +1 , which obeys m g +1 (cid:18) − q + 1 (cid:19) + m g +1 (cid:18) q − q + 1 (cid:19) + m g +1 (cid:18) λ / ∈ (cid:26) − q + 1 , q − q + 1 (cid:27)(cid:19) = N q ( g + 1) . (38)Using Eq. (37) and Lemma 2, one obtains m g +1 (cid:18) q − q + 1 (cid:19) = M q ( g ) − N q ( g ) . (39)Combining Eqs. (35) (36) (39) and Lemma 2 yields (34). For g = 0 , G q (0) is a complete graph with q + 2 vertices. The set of the eigenvalues of matrix P is Λ = n , − q +1 , − q +1 , . . . , − q +1 o . By recursively applying Theo-rem 1, we can obtain all the eigenvalues matrix P g for g ≥ .Using Theorem 1 and the one-to-one correspondencebetween matrices ˜ L g and ˜ P g , we can also obtain relationfor the set of eigenvalues for ˜ L g and ˜ L g +1 . Theorem 2.
Let Σ g , g ≥ , be the set of the N q ( g ) eigenvalues σ ( g ) , σ ( g ) , . . . , σ N q ( g ) ( g ) for matrix ˜ L g , satisfying σ ( g ) ≤ σ ( g ) ≤ . . . ≤ σ N q ( g ) ( g ) ≤ . Then the N q ( g +1) eigenvalues for ˜ L g +1 forming the set Σ g +1 can be listedin an increasing order as Σ g +1 = (cid:26) σ ( g ) q + 1 , σ ( g ) q + 1 , . . . , σ N q ( g ) ( g ) q + 1 , q + 1 , q + 1 , . . . , q + 1 | {z } M q ( g ) − N q ( g ) , q + 2 q + 1 , q + 2 q + 1 , . . . , q + 2 q + 1 | {z } ( q − M q ( g )+ N q ( g ) (cid:27) . (40) Proof:
The proof is easily obtained by combining therelation λ i ( g ) = 1 − σ i ( g ) and Theorem 1.The set Σ of eigenvalues for matrix ˜ L is Σ = n , q +2 q +1 , q +2 q +1 , . . . , q +2 q +1 o . For g ≥ , by recursively applyingTheorem 2, we can obtain the exact expressions for alleigenvalues for matrix ˜ L g for any q and g , given by Σ g = (cid:26) , q + 2( q + 1) g +1 , q + 2( q + 1) g +1 , . . . , q + 2( q + 1) g +1 | {z } q +1 , q + 1) g , q + 1) g , . . . , q + 1) g | {z } M q (0) − N q (0) ,q + 2( q + 1) g , q + 2( q + 1) g , . . . , q + 2( q + 1) g | {z } ( q − M q (0)+ N q (0) , q + 1) g − , q + 1) g − , . . . , q + 1) g − | {z } M q (1) − N q (1) ,q + 2( q + 1) g − , q + 2( q + 1) g − , . . . , q + 2( q + 1) g − | {z } ( q − M q (1)+ N q (1) , · · · · · · , q + 1 , q + 1 , . . . , q + 1 | {z } M q ( g − − N q ( g − , q + 2 q + 1 , q + 2 q + 1 , . . . , q + 2 q + 1 | {z } ( q − M q ( g − N q ( g − (cid:27) . (41) PPLICATIONS OF THE SPECTRA
In this section, we apply the above-obtained eigenvaluesand their multiplicities of related matrices to evaluate somerelevant quantities for graph G q ( g ) , including mixing time,mean hitting time also called Kemeny constant, and thenumber of spanning trees. As is well-known, the probability transition matrix P ( G ) ofa graph G characterizes the process of random walks on thegraph. As a classical Markov chain, random walks describevarious phenomena or other dynamical processes in graphs.Many interesting quantities about random walks can beextracted from the eigenvalues of the probability transitionmatrix. In this paper, we only consider mixing time andmean hitting time.For an ergodic random walk on an un-bipartite graph G with N vertices, it has a unique stationary distribu-tion π = ( π , π , . . . , π N ) ⊤ with P Ni =1 π i = 1 , where π i represents the probability that the walker is at vertex i when the random walk converges to equilibrium state [60].The mixing time is defined as the expected time that thewalker needs to approach the stationary distribution. Let λ > λ ≥ λ ≥ · · · ≥ λ N > − be the N eigenvalues for matrix P ( G ) . Then the speed of convergenceto the stationary distribution [61] approximately equals thereciprocal of − λ max , where λ max is the second largesteigenvalue modulus defined by λ max = max( λ , | λ N | ) .Mixing time has found numerous applications in man dif-ferent aspects [47].As our first application of eigenvalues for matrix P g , weuse them to evaluate the mixing time for random walks on G q ( g ) , for which the component of stationary distribution π corresponding to vertex i is π i = d i ( g ) / (2 M q ( g )) . Accord-ing to the above arguments, the second largest eigenvaluemodulus λ max ( g ) of P g is λ max ( g ) = 1 − q +2( q +1) g +1 . Sincethe mixing time is characterized by a parameter, it cannotbe exactly determined [61], but one can evaluate it by usingthe reciprocal of λ max ( g ) . Then, the dominating term of themixing time for random walks on G q ( g ) is ( q +1) g +1 / ( q +2) ,which scales sublinearly with the vertex number N q ( g ) as ( N q ( g )) /θ ( q ) , where θ ( q ) = 2 / log ( q +1)( q +2) / ( q + 1) is the spectral dimension [44] of graph G q ( g ) that is afunction of q . Note that for q = 1 , the spectral dimension θ (2) = 2 ln 3 / ln 2 reduces to the result obtained in [62].Note that it is believed that real-world networks areoften fast mixing with their mixing time at most O (log N ) ,where N is the number of vertices. However, it was exper-imentally reported that the mixing time of some real-worldsocial networks is much higher than anticipated [63]. Ourobtained sublinear scaling of mixing time on graph G sup-ports this recent study, and sheds lights on understandingthe scalings of mixing time. Our second application for our obtained eigenvalues is themean hitting time. For a random walk on graph G , thehitting time H ij , also called first-passage time [64], [65], [66],from vertex i to vertex j , is defined as the expected timetaken by a walker starting from vertex i to reach vertex j for the first time. The mean hitting time H , also known asthe Kemeny constant, is defined as the expected time for arandom walker going from a vertex i to another vertex j that is chosen randomly from all vertices in G according tothe stationary distribution [67], [68]: H = n X j =2 π j H ij . (42) Interestingly, the quantity H is independent of the startingvertex i , and can be expressed in terms of the N − nonzeroeigenvalues σ i , i = 2 , , · · · , N , of the normalized Laplacianmatrix ˜ L ( G ) for graph G , given by [67], [68] H = N X i =2 σ i . (43)Mean hitting time can be applied to measure the efficiencyof user navigation through the World Wide Web [69] andthe efficiency of robotic surveillance in network environ-ments [70]. We refer to the reader to [71] for many otherapplications of mean hitting time.In this subsection, we use the eigenvalues of the normal-ized Laplacian matrix for graph G q ( g ) to compute the meanhitting time of G q ( g ) . Theorem 3.
Let H q ( g ) be the mean hitting time for randomwalk in G q ( g ) . Then, for all g ≥ , H q ( g ) = " ( q + 1) q + 2 − q + 1)2 ( q + 1) g + ( q + 1)(3 q + 7)2( q + 3) (cid:20) ( q + 1)( q + 2)2 (cid:21) g + q + 1 q + 3 . (44) Proof:
By Theorem 2 and Eq. (43), we have H q ( g + 1)= q + 12 ( M q ( g ) − N q ( g ))+ q + 1 q + 2 (( q − M q ( g ) + N q ( g )) + N q ( g ) X i =2 q + 1 σ i ( g )= 3 q ( q + 1)2( q + 2) M q ( g ) − q ( q + 1)2( q + 2) N q ( g ) + ( q + 1) H q ( g ) , (45)which can be rewritten as H q ( g + 1) − ( q + 1)(3 q + 7)2( q + 3) (cid:20) ( q + 1)( q + 2)2 (cid:21) g +1 − q + 1 q + 3= ( q + 1) (cid:26) H q ( g ) − ( q + 1)(3 q + 7)2( q + 3) (cid:20) ( q + 1)( q + 2)2 (cid:21) g − q + 1 q + 3 (cid:27) . (46)With the initial condition H q (0) = ( q +1) ( q +2) , Eq. (46) is solvedto obtain (44).Theorem 3 shows that for g → ∞ , the dependence ofmean hitting time H q ( g ) on the number N q ( g ) of vertices ingraph G g ( g ) is H q ( g ) ∼ N q ( g ) , which implies that the H q ( g ) behaves linearly with N q ( g ) . A spanning tree of an undirected graph G = ( V , E ) with N vertices is a subgraph of G , which is a tree including allthe N vertices. Let τ ( G ) denote the number of spanningtrees in graph G . It has been shown [72], [73] that τ ( G ) canbe expressed in terms of the N − non-zero eigenvalues for normalized Laplacian matrix of G and the degrees of allvertices in G : τ ( G ) = Q i ∈V d i Q Ni =2 σ i ( G ) P i ∈V d i . (47)The number of spanning trees is an important graphinvariant. In the sequel, we will use the above-obtainedeigenvalues to determine this invariant for graph G q ( g ) . Theorem 4.
Let τ q ( g ) = τ ( G q ( g )) be the number of spanningtrees in graph G q ( g ) . Then, for all g ≥ , τ q ( g ) = 2 q +1) q ( q +3)2 [ ( q +1)( q +2)2 ] g +1 − ( q +1 q +3 ) g − ( q +1)2( q +2) q ( q +3)2 · ( q + 2) q q − q ( q +3)2 [ ( q +1)( q +2)2 ] g +1 + ( q +1 q +3 ) g + q q − q +2 q ( q +3)2 . (48) Proof:
First, by Theorem 2, we derive the relation forthe product of all the non-zero eigenvalues for normalizedLaplacian matrix for graph G q ( g + 1) and G q ( g ) : N q ( g +1) Y i =2 σ i ( g + 1)= (cid:18) q + 1 (cid:19) M q ( g ) − N q ( g ) (cid:18) q + 2 q + 1 (cid:19) ( q − M q ( g )+ N q ( g ) N q ( g ) Y i =2 σ i ( g ) q + 1= 2 M q ( g ) − N q ( g ) ( q + 2) ( q − M q ( g )+ N q ( g ) ( q + 1) qM q ( g )+ N q ( g ) − N q ( g ) Y i =2 σ i ( g ) . (49)Second, we derive the relation be between the productof degrees of all vertices in G q ( g + 1) and the product ofdegrees of all vertices in G q ( g ) . For G q ( g + 1) , the degree ofall the new vertices in V ′ g +1 that were generated at iteration g + 1 is q + 1 ; while for each i of those old vertices in V g , wehave d i ( g + 1) = ( q + 1) d i ( g ) . Then, Y i ∈V g +1 d i ( g + 1) = Y i ∈V ′ g +1 d i ( g + 1) Y i ∈V g d i ( g + 1)= ( q + 1) qM q ( g ) Y i ∈V g ( q + 1) d i ( g )= ( q + 1) qM q ( g )+ N q ( g ) Y i ∈V g d i ( g ) . (50)Finally, the sum of degrees of all vertices in G q ( g ) is equalto M q ( g ) . Then, Combining Eqs. (5), (47), (49), and (50),we obtain the following recursive relation for τ q ( g + 1) and τ q ( g ) : τ q ( g + 1) = 2 M q ( g ) − N q ( g )+1 ( q + 2) ( q − M q ( g )+ N q ( g ) − τ q ( g ) . (51)Considering the expressions for M q ( g ) and N q ( g ) in Eqs. (5)and (6), we obtain τ q ( g + 1) = 2 q +1 q +3 [ ( q +1)( q +2)2 ] g +1 − q +1 q +3 × ( q + 2) q q − q +3 [ ( q +1)( q +2)2 ] g +1 + q +1 q +3 τ q ( g ) . (52)With the initial condition τ q (0) = τ ( K q +2 ) = ( q + 2) q ,Eq. (52) is solved to yield (48). ONCLUSION
For many graph products of two graphs, one can analyze thestructural and spectral properties of the resulting graph, ex-pressing them in terms those corresponding the two graphs.Because of this strong advantage, many authors have usedgraph products to generate realistic networks with cyclesat different scales. In this paper, by iteratively using theedge corona product, we proposed a minimal model forcomplex networks called simplicial networks, which cancapture group interactions in real networks, characterizedby a parameter q . We then provided an extensive analysis forrelevant topological properties of the model, most of whichare dependent on q . We show that the resulting networksdisplay some remarkable characteristics of real networks,such as non-trivial higher-order interaction, power-law dis-tribution of vertex degree, small diameter, and high cluster-ing coefficient.Furthermore, we found exact expressions for all theeigenvalues and their multiplicities of the transition prob-ability matrix and normalized Laplacian matrix of ourproposed networks. Using these obtained eigenvalues, wefurther evaluated mixing time, as well as mean hittingtime for random walks on the networks. The former scalessublinearly with the vertex number, while the latter behaveslinearly with the vertex number. The sublinear scaling ofmixing time is contrary to previous knowledge that mixingtime scales at most logarithmically with the vertex number.We also using the obtained eigenvalues to determine thenumber of spanning tree in the networks. Thus, in additionto the advantage of networks generated by other graphproducts, the proposed networks have another obvious ad-vantage that both the eigenvalues and their multiplicities ofrelevant matrix can be analytically and exactly determined,since for previous networks created by graph products,the eigenvalues are only obtained recursively at most. Theexplicit expression for each eigenvalue facilitates to studythose dynamical processes determined by one or severalparticular eigenvalues, such as mixing time considered here.It should be mentioned that many real networks areweighted with variable edge length [74]. For example, inscientific collaboration networks, the collaboration strengthbetween collaborators can be weighted by the number ofpapers they coauthored. It is thus necessary to model theserealistic networks by weighted simplicial complexes [75]. Infuture, as the case of corona product [27], one can also defineextended edge corona product of graphs and use it to buildweighted scale-free networks with rich properties matchingthose of real-world networks [76]. A CKNOWLEDGMENTS
This work was supported in part by the National Nat-ural Science Foundation of China (Nos. 61872093 and61803248), the National Key R & D Program of China (No.2018YFB1305104), Shanghai Municipal Science and Technol-ogy Major Project (No. 2018SHZDZX01) and ZJLab. R EFERENCES [1] M. E. Newman, “The structure and function of complex net-works,”
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