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

Abstract

Abstract. Newman's measure for (dis)assortativity, the linear degree correlation coefficient ρD, 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 ρD conform our desired objective. Spectral metrics (eigenvalues of graph-related matrices), especially, the largest eigenvalue λ1 of the adjacency matrix and the algebraic connectivity μN−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 λ1 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 λ1 increase with ρD. A new upper bound for the algebraic connectivity μN−1 decreases with ρD. Applying the degree-preserving rewiring algorithm to various real-world networks illustrates that (a) assortative degree-preserving rewiring increases λ1, but decreases μN−1, even leading to disconnectivity of the networks in many disjoint clusters and that (b) disassortative degree-preserving rewiring decreases λ1, but increases the algebraic connectivity, at least in the initial rewirings.

Assortative and disassortative mixing investigated using the spectra of graphs

“…1(a)-(e)). The largest eigenvalue exhibits an increasing trend as already discussed in [28,29]. As network is rewired entailing disassortativity, spectral distribution (ρ(λ)) acquires a very different structure than those of the assortative networks.…”

On the Conduciveness of Random Network Graphs for Maximal Assortative or Maximal Dissortative Matching

“…Motivated by this observation, Winterbach et al (2012) introduced an algorithm to compute a network with maximal or minimal assortativity given a vector of valid node degrees using degree-preserving rewiring (Maslov & Sneppen, 2002) and weighted b-matching (Muller-Hannemann & Schwartz, 1999). Degree-preserving link rewiring is effective in decreasing or increasing the assortativity of a network graph without affecting the degree distribution of the vertices (Van Mieghem et al, 2010). Holme & Zhao (2007) also found that an increase in the assortativity of a graph (accomplished through degree-preserving rewiring) also contributes to an increase in the maximum modularity, average hop count, effective graph resistance as well as a decrease in the number of clusters.…”

Generating graphs that approach a prescribed modularity

“…Some weaknesses in modularity optimization have also been determined, such as the incapability to detect communities smaller than a resolution limit [5] or the breaking up of large random sub-graphs into separate communities [6]. A spectral analysis of the modularity as well as correlation with other metrics, such as assortativity [23,24], has been conducted in [25].…”