, Volume 76, Issue 4, pp 643-652
Date: 02 Jul 2010

Influence of assortativity and degree-preserving rewiring on the spectra of networks

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Abstract

Newman’s measure for (dis)assortativity, the linear degree correlation coefficient \(\rho _{D}\) , is reformulated in terms of the total number N k of walks in the graph with k hops. This reformulation allows us to derive a new formula from which a degree-preserving rewiring algorithm is deduced, that, in each rewiring step, either increases or decreases \(\rho _{D}\) conform our desired objective. Spectral metrics (eigenvalues of graph-related matrices), especially, the largest eigenvalue \(\lambda _{1}\) of the adjacency matrix and the algebraic connectivity \(\mu _{N-1}\) (second-smallest eigenvalue of the Laplacian) are powerful characterizers of dynamic processes on networks such as virus spreading and synchronization processes. We present various lower bounds for the largest eigenvalue \(\lambda _{1}\) of the adjacency matrix and we show, apart from some classes of graphs such as regular graphs or bipartite graphs, that the lower bounds for \(\lambda _{1}\) increase with \(\rho _{D}\) . A new upper bound for the algebraic connectivity \(\mu _{N-1}\) decreases with \(\rho _{D}\) . Applying the degree-preserving rewiring algorithm to various real-world networks illustrates that (a) assortative degree-preserving rewiring increases \(\lambda _{1}\) , but decreases \(\mu _{N-1}\) , even leading to disconnectivity of the networks in many disjoint clusters and that (b) disassortative degree-preserving rewiring decreases \(\lambda _{1}\) , but increases the algebraic connectivity, at least in the initial rewirings.