The normalized Laplacian spectrum and eigentime identities of hype-cubes
aa r X i v : . [ m a t h . SP ] S e p THE NORMALIZED LAPLACIAN SPECTRUM ANDEIGENTIME IDENTITIES OF HYPE-CUBES
YANGYANG CHEN AND YI ZHAO
Abstract.
Many popular graph metrics encode average properties of individ-ual network elements. Complementing these conventional graph metrics, theeigenvalue spectrum of the normalized Laplacian describes a network’s struc-ture directly at a systems level, without referring to individual nodes or connec-tions. In this paper, we study the spectrum and their applications of normalizedLaplacian matrices of hype-cubes, a special kind of Cayley graphs. We deter-mine explicitly all the eigenvalues and their corresponding multiplicities by arecursive method. By using the relation between normalized Laplacian spec-trum and eigentime identity, we derive the explicit formula to the eigentimeidentity for random walks on the hype-cubes and show that it grows linearlywith the network size. Moreover, we compute the number of spanning trees ofthe hype-cubes. Introduction
Recently, the theory of complex networks has attracted wide attention and be-comes an area of great interest [1, 2], for its advances in the understanding of manynatural and social systems. A central issue in the study of complex systems is tounderstand the topological structure and to further unveil how various structuralproperties affect the dynamical processes occurring on diverse systems [3]. Froma graph-theoretic perspective, the spectrum of the standard Laplacian matrix ofa network contains tremendous information about the underlying network, whichprovides useful insights into the intrinsic structural features of the network [4] andplays a fundamental role in the dynamical behavior of the network. For example,the resistance distance [5], relaxation dynamic in the framework of generalizedGaussian structure [3, 6, 7], fluorescence depolarization by quasiresonant energytransfer [8, 9, 10], continuous-time quantum walks [11, 12, 13], average trappingtime [14] and so on. Thus it is important to study the spectrum of standard Lapla-cian matrices of complex networks. For the techniques to compute the spectrumof standard Laplacian matrices of complex systems, we refer the reader to earlierworks [3, 15, 16, 17].
Mathematics Subject Classification.
Key words and phrases.
Cayley graph, Hype-cubes, Normalized Laplacian spectrum, Eigen-time identity, Spanning trees.
Compared to standard Laplacian matrices, the spectrum of normalized Lapla-cian matrices have received little attention [18, 19, 20]. However, the eigenvaluesand eigenvectors of normalized Laplacian matrix of a network also contain muchimportant information about its structure and dynamical processes. For example,the number of spanning tress of a connected network is determined by the productof all nonzero eigenvalues[21]; the nonzero eigenvalues and their orthonormalizedeigenvectors can be used to describe the resistor resistance between any pair ofnodes [22]. Many interesting quantities of random walks, like mixing time [23],Kemeny constant [24] and eigentime identity [25], are related to the normalizedspectrum. Moreover, many problems in chemical physics [26, 27] are closely relatedto eigenvalues and eigenvectors of normalized Laplacian matrix.In [28], Julaiti et al. studied the spectrum of normalized Laplacian matrices of afamily of fractal trees and dendrimers modeled by Cayley trees, both of which werebuilt in an iterative way. They used recursive method to determine all the eigen-values and their corresponding multiplicities. As an application, they obtained anexplicit solution to the eigentime identity for random walks. Eigentime identityfor random walks were also studied in [29], for a family of treelike networks andpolymer networks.It should be pointed out that [28] did not give an explicit formula of the normal-ized spectrum. Instead, they gave a recursive relation governing the eigenvalues offractal trees at two successive generations and for the Cayley tress, the eigenvalueswere described as roots of several small-degree polynomials defined recursively.Generally, it should be a difficult problem to determine explicitly the normalizedLaplacian spectrum of networks. However, we believe that for vertex symmetricnetworks [30], the study of normalized Laplacian spectrum will be relatively easy,due to the symmetry of the network structure. Recall that for vertex symmetricnetworks, there exists a good model, the celebrated Cayley graph model [30]. TheCayley graph model has a simple mathematical characterization. It should be aninteresting and fascinating problem to study the normalized Laplacian spectrumof Cayley graph networks.In this paper, we study the normalized Laplacian spectrum of a special kind ofCayley graphs, the hype-cube or n -cube, which is a network of 2 n vertices, withdegree n and diameter n . We determine explicitly all the characteristic polyno-mials of the normalized Laplacian matrices of the n -cubes by a recursive method.Particularly, these polynomials are factorized into products of monomials and theroots of these polynomials are elegantly distributed between the closed intervalfrom 0 to 2. As an application of these results, we obtain directly the eigentimeidentity for random walks on these n -cubes and the number of spanning trees.The rest of this paper is organized as follows. In the next section, we recallbriefly the Cayley graph models and a special Cayley graph, the hype-cubes or n -cubes. Section 3 contains the main result of this paper, where we compute theeigenvalues of normalized Laplacian matrix of hype-cubes. Then in Section 4, we ORMALIZED LAPLACIAN SPECTRUM 3 give two applications of the normalized Laplacian spectrum, namely, the explicitformulas for eigentime identities for random walks and number of spanning trees.Finally, the last section contains our conclusions.2.
Cayley graph models and hype-cubes
A graph or network is denoted by Γ = Γ(
V, E ), where V is the set of verticesand E ⊂ V × V is the set of edges. We only consider graphs that are finite,undirected, loop-free and devoid of multiple edges in this paper. If ( v , v ) ∈ E , v and v are adjacent. Recall that a network is said to be vertex symmetric, if forany two vertices v and w , there exists an automorphism of the network that maps v into w . Vertex symmetric networks have the property that the network viewedfrom any vertex of the network looks the same. In such a network, congestionproblems are minimized since the load will be distributed through all the vertices.It is well-known that Cayley graph model is an excellent model for vertex sym-metric networks. It was shown in [30] that most vertex symmetric networks canbe represented using this model, and that every vertex symmetric network can berepresented by a simple extension of this model [30, Theorem 3].We recall the construction of this model briefly. Let G be a finite group, with agenerating subset S , namely, all group elements of G can be expressed as a finiteproduct of the powers of the elements in S . The Cayley graph of the group G withrespect to the subset S , denoted by Cay( G, S ), has vertices that are elements of G and edges that are ordered pairs ( g, gs ) for g ∈ G, s ∈ S . We always requirethat e S and S = S − , where e is the identity element of G . Then Cay( G, S )can be taken as a simple undirected graph. For more definitions and basic resultson graphs and groups we refer the reader to [31].For example, let G n = Z n and S n = { ( x , ..., x n ) ∈ G n : only one x i is 1 } , where Z = Z / Z denotes the group with only two elements and Z n denotes the n thdirect product of Z . Then Γ n = Cay( G n , S n ) is the well-known hype-cube or n -cube, which is a network of 2 n vertices, with degree n and diameter n [30].3. Normalized Laplacian spectrum of hype-cubes
Normalized Laplacian matrix.
Let Γ be a network. Denote by A its ad-jacency matrix, the entry A ( i, j ) of which is 1 (or 0) if nodes i and j are (not)adjacent in Γ. Then the standard Laplacian matrix of Γ is defined as L = D − A ,where D is the diagonal degree matrix of Γ with its i th diagonal entry being thedegree of node i in Γ. Since L is real symmetric, all its eigenvalues are real num-bers. Actually, L is positive semi-definite and thus has nonnegative eigenvalues.Moreover, 0 is always an eigenvalue of L and the multiplicity of the eigenvalue 0is equal to the number of connected components of Γ. For these facts, see, forexample [32]. In this paper, we only consider connected networks.The normalized Laplacian matrix of Γ is defined as L = I − D − / AD − / ,where I denotes the identity matrix with the same order as that of A . If a is a Y. Y. CHEN AND Y. ZHAO eigenvalue of L , then 0 ≤ a ≤
2, see for example [21]. By the normalized Laplacianspectrum of a network, we mean all the eigenvalues of the normalized Laplacianmatrix. Recently, it was pointed out by [28] that one has to treat the standardLaplacian matrix and normalized Laplacian matrix separately, since they behavequite differently. The main goal of this paper is to determine the normalizedLaplacian spectrum of hype-cubes.3.2.
Normalized Laplacian spectrum of hype-cubes.
Recall that Γ n = Cay( G n , S n ) denote the n -cube, where G n = Z n and S n = { ( x , ..., x n ) ∈ G n : only one x i is 1 } . We array the vertices in Γ n in lexicographicalorder. Denote by A n the corresponding adjacency matrix and D n the diagonaldegree matrix of Γ n . Obviously, D n = nI n , where I n denotes the identity matrixof order 2 n . The normalized Laplacian matrix of Γ n is L n = I n − D − / n A n D − / n ,which is equal to I n − D − n A n , since the degree matrix D n is a scalar matrix. Let g n ( λ ) = det( λI − L n ) = det(( λ − I + D − A ) be the characteristic polynomial ofthe normalized Laplacian matrix L n of Γ n . We sometimes omit the subscripts ifit causes no confusions. The main goal of this subsection is to find all the roots of g n ( λ ).Let f n ( λ ) = det(( λ − D n + A n ). It is clear that g n ( λ ) = n − n f n ( λ ). Thus itsuffices to find all the roots of f n ( λ ). Denote by A n − the corresponding adjacencymatrix of the ( n − n − . The following observation makes it possible forus to derive all the roots of g n ( λ ) in a recursive way. Proposition 3.1.
For every n ≥ , we have A n = (cid:18) A n − II A n − (cid:19) . Proof.
This follows directly from the lexicographical order of the vertices of the n -cube. (cid:3) By Proposition 3.1, we have(1) ( λ − D n + A n = (cid:18) ( λ − nI + A n − II ( λ − nI + A n − (cid:19) . We shall need the following elementary lemma from linear algebra.
Lemma 3.2.
Let B = (cid:18) A II A (cid:19) , where A is a matrix of order m and I is the identity matrix of the same order.Then det( B ) = det( A + I )det( A − I ) . ORMALIZED LAPLACIAN SPECTRUM 5
Proof.
By adding the second row of B to the first row and then subtracting thefirst column from the second column, we havedet( B ) = det (cid:18) A + I I A − I (cid:19) = det( A + I )det( A − I ) . (cid:3) Recall that f n ( λ ) = det(( λ − D n + A n ). By the block decomposition (1) andLemma 3.2, we have the following recursive relation between f n ( λ ) and f n − ( λ ). Proposition 3.3.
For every n ≥ , we have (2) f n ( λ ) = f n − ( nn − λ ) f n − ( nλ − n − . Proof.
By definition, f n ( λ ) = det(( λ − D n + A n )= det(( n ( λ −
1) + 1) I + A n − )det(( n ( λ − − I + A n − ) , the right hand side of the above equation is easily checked to be f n − ( nn − λ ) f n − ( nλ − n − , finishing the proof of the proposition. (cid:3) It follows directly that f ( λ ) = λ ( λ − f n ( λ ) explicitly. Theorem 3.4.
For n ≥ , (3) f n ( λ ) = n n n Y k =0 ( λ − kn )( nk ) . Proof.
We prove (3) by induction on n . Obviously the case n = 1 holds. Assumethat (3) holds for n −
1, where n >
1. We shall show that then it also holds for n . Y. Y. CHEN AND Y. ZHAO
By (2) and the induction hypothesis on n −
1, we have f n ( λ ) = f n − ( nn − λ ) f n − ( nλ − n − n − n − n − Y k =0 ( nn − λ − kn − n − k )( n − n − n − Y k =0 ( nλ − n − − kn − n − k )= ( n − n ( nn − P n − k =0 ( n − k ) n − Y k =0 ( λ − kn )( n − k )( λ − k + 2 n )( n − k )= n n n − Y k =0 ( λ − kn )( n − k ) n Y k =1 ( λ − kn )( n − k − )= n n n Y k =0 ( λ − kn )( nk ) , which completes the proof of n case. (cid:3) Recall that g n ( λ ) = det( λI − L n ), where L n denotes the normalized Laplacianmatrix of the n -cube Γ n and we have g n ( λ ) = n − n f n ( λ ). By Theorem 3.4,(4) g n ( λ ) = n Y k =0 ( λ − kn )( nk ) . From this formula, we know all the eigenvalues of normalized Laplacian matrix ofthe n -cube. The eigenvalues are 2 k/n , with multiplicity (cid:0) nk (cid:1) , for each 0 ≤ k ≤ n .These eigenvalues are evenly distributed in the closed interval from 0 to 2. As wementioned in the Introduction, this is partly due to the symmetry of the networkstructure. We believe that similar nice results hold for other vertex symmetricnetworks, and specially for Cayley graph networks.4. Applications of normalized Laplacian spectrum
As described in the Introduction, the normalized Laplacian spectrum of a net-work contains much important information about its structure and dynamicalprocesses. With the normalized Laplacian spectrum of hype-cubes obtained, nowwe can give explicit formulas to the eigentime identity for random walks on the n -cube Γ n and the number of spanning trees of Γ n .4.1. Eigentime identity for random walks.
Firstly we recall the eigentimeidentity for random walks in a general network Γ. Let H ij be the mean-first passagetime from node i to node j in Γ, which is the expected time for a particle startingoff from node i to arrive at node j for the first time, see [28]. The stationarydistribution for random walks on Γ is π = ( π , ..., π N ), where π i = d i / | E (Γ) | , N = | V (Γ) | and d i denotes the degree of node i . Let H represent the eigentimeidentity for random walks in Γ, which is defined as the expected time for a walker ORMALIZED LAPLACIAN SPECTRUM 7 going from a node i to another node j , chosen randomly from all nodes accordinglyto the stationary distribution. That is,(5) H = N X j =1 π j H ij . Note that H is independent of the starting node. It is a global characteristic ofthe network and reflects the architecture of the whole network. By [23, 33], H canbe expressed as(6) H = X λ =0 λ , where the sum is taken over all the nonzero eigenvalues of the normalized Laplacianmatrix of Γ. For recent work on eigentime identities of flower networks withmultiple branches and weighted scale-free triangulation networks, we refer thereader to [34, 35]. We shall give an explicit formula of (6) for hype-cubes. Proposition 4.1.
The eigentime identity for random walks on the n -cube is givenby (7) H (Γ n ) = n X k =1 n (cid:0) nk (cid:1) k . Proof.
This follows directly from Theorem 3.4. (cid:3)
For the asymptotical behavior of the eigentime identity for random walks on Γ n as n → ∞ , we have the following characterization. Proposition 4.2. lim n →∞ H (Γ n )2 n = 1 . Proof.
We have H (Γ n ) = n X k =1 n (cid:0) nk (cid:1) k = n n X k =1 k ( k + 1) (cid:18) nk (cid:19) + n n X k =1 k + 1 (cid:18) nk (cid:19) . Denote by S ( n ) = n n X k =1 k + 1 (cid:18) nk (cid:19) , L ( n ) = n n X k =1 k ( k + 1) (cid:18) nk (cid:19) and T ( n ) = n n X k =1 k + 1)( k + 2) (cid:18) nk (cid:19) . Y. Y. CHEN AND Y. ZHAO
Then(8) S ( n ) = n n + 1) n X k =1 (cid:18) n + 1 k + 1 (cid:19) = n (2 n +1 − n − n + 1) , thus(9) lim n →∞ S ( n )2 n = 1 . On the other hand, T ( n ) = n n + 1)( n + 2) n X k =1 (cid:18) n + 2 k + 2 (cid:19) < n n +1 ( n + 1)( n + 2) , thus lim n →∞ T ( n )2 n = 0 . For any k ≥
1, one has 1( k + 1)( k + 2) < k ( k + 1) ≤ k + 1)( k + 2) , thus T ( n ) < L ( n ) ≤ T ( n ) and therefore(10) lim n →∞ L ( n )2 n = 0 . Note that H (Γ n ) = L ( n )+ S ( n ), thus the proposition follows from (9) and (10). (cid:3) By Proposition 4.2, H (Γ n ) grows linearly with the network size N (Γ n ) of the n -cube as n → ∞ , which is quite different from the fractal trees and Cayley treesas studied in [28]. This indicates that the network structure of hype-cubes areessentially different from that of fractal trees and Cayley trees constructed in [28].4.2. Number of spanning trees.
In addition to eigentime identity, the eigen-values of normalized Laplacian matrix of a connected network also determine thenumber of its spanning trees. Recall that a spanning tree of an undirected graphΓ is a subgraph of Γ that is a tree which includes all the vertices of Γ. In general,a graph may have several spanning trees, but a graph that is not connected willnot contain a spanning tree. By [21, 22], the number of spanning trees N st (Γ) fora connected network Γ is(11) N st (Γ) = Q Ni =1 d i Q λ =0 λ P Ni =1 d i , where λ runs over all the nonzero eigenvalues of the normalized Laplacian matrix.Denote by N st (Γ n ) the number of spanning trees of the n -cube Γ n . ORMALIZED LAPLACIAN SPECTRUM 9
Proposition 4.3. (12) N st (Γ n ) = 2 n − n − n Y k =1 k ( nk ) . Proof.
By (11) and Theorem 3.4, we have N st (Γ n ) = Q d i Q λ =0 λ P d i = n n Q nk =1 ( kn )( nk ) n n = 2 n − n − n Y k =1 k ( nk ) . (cid:3) Conclusions
It is known that numerous structural and dynamical properties of a networkedsystem are encoded in eigenvalues and eigenvectors of its standard Laplacian ma-trix. Compared to standard Laplacian matrices, the spectrum of normalized Lapla-cian matrices have received little attention. Recently, it was pointed out by [28]that it is equally important to compute and analyze the normalized Laplacianspectrum. For example, the normalized Laplacian spectrum of a network is rele-vant in the topological aspects and random walk dynamics that is closely relatedto a large variety of other dynamical processes of the network.Generally, it should be a difficult problem to determine explicitly the normalizedLaplacian spectrum. However, we do believe that for vertex symmetric networks,especially for Cayley graph networks, the study of normalized Laplacian spectrumwill be relatively easy, due to the symmetry of the network structure.In this paper, we have studied the eigenvalue problem of the normalized Lapla-cian matrices of the hype-cubes, a special kind of Cayley graph networks. We de-termined explicitly all the characteristic polynomials of the normalized Laplacianmatrices of the hype-cubes by a recursive method. Particularly, these polynomialswere factorized into products of monomials and the roots of these polynomials areelegantly distributed in the closed interval from 0 to 2. As an application of theseresults, we obtained explicitly the eigentime identity for random walks on thesehype-cubes, which grows linearly with the network size. This is in sharp contrastto fractal trees and Cayley trees as constructed in [28]. Since eigentime identity isan important quantity rooted in the inherent network topology, we conclude thatthe network structure of hype-cubes is essentially different from that constructedin [28]. Moreover, we derived the number of spanning trees of these hype-cubesthrough the normalized Laplacian spectrum.
Acknowledgments
This work was sponsored by the National Natural ScienceFoundation of China (NSFC) under Project No.61573119 and the FundamentalResearch Project of Shenzhen under Project Nos. JCYJ20170307151312215 andKQJSCX20180328165509766.
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