Graph Spectra for Complex Networks by Piet van Mieghem

By Piet van Mieghem

A concise and self-contained advent to the idea of graph spectra and its functions to the learn of advanced networks.

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3), Var [] = Q 1 X 2 2O = H [G] n = Q Q n=1 Especially in order to understand the density function of the adjacency eigenvalues (for example, in Section 7), this stochastic interpretation is helpful. 28. 1 General properties 31 } = 1. That form of P3×3 corresponds with a subgraph of three nodes that are fully connected. Hence, fQ 3 = 2× the number of triangles NJ in J. From art. 3) n=1 29. 4) G5Jn where Jn is the set of all subgraphs of J with exactly n nodes and f|fohv (G) is the number of cycles in a subgraph G 5 Jn .

Since \ x = x (art. 5). 2 The number of walks 33. 8) l=1 m=1 For example, Q0 = Q> Q1 = 2O. 9) n=1 The number of walks {Q0 > Q1 > = = = > Q10 } in the graph of Fig. 1 ignoring directions and up to n = 10 is {6> 18> 56> 174> 542> 1688> 5258> 16378> 51016> 158910> 494990}. Since the adjacency matrix D is symmetric, art. 11) q=1 PQ where {Wq x = m=1 ({q )m is the sum of the components of the eigenvector {q . When the (normalized) eigenvector {1 = sxQ as in regular graphs (art. 41) where 1 = u, the number of all walks with n hops equals Qn = Q un .

15. Quotient matrix. 11) D = V W V ³ ´ ¡ ¢ 1 where V W V = diag |F11 | > |F12 | > = = = > |F1n | . The quotient matrix of the matrix D of the example in art. 14 is 6 5 1 2 0 D = 7 4 2 1 8 3 3 0 3 0 We can verify that (D )lm denotes the average row sum of the block matrix ( D)l>m . 11. If the row sum of each block matrix Dl>m is constant, then the partition  is called equitable (or regular). In that case, Dl>m x = ( D)l>m x or  DV = VD . Also, a partition  is equitable if, for any l and m, the number of neighbors that a node in Fl has in the cell Fm does not depend on the choice of a node in Fl .

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