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

• Published in: The European physical journal. B, Condensed matter physics Vol. 76; no. 4; pp. 643 - 652
• Main Authors: Van Mieghem, P., Wang, H., Ge, X., Tang, S., Kuipers, F. A.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01-08-2010; EDP Sciences; Springer
• Subjects: Algorithms; Applied sciences; Complex Systems; Computer science; control theory; systems; Computer systems and distributed systems. User interface; Condensed Matter Physics; Exact sciences and technology; Fluid- and Aerodynamics; Information retrieval. Graph; Interdisciplinary Physics; Physics; Physics and Astronomy; Software; Solid State Physics; Theoretical computing; Regular Graph; Algebraic Connectivity; Complete Bipartite Graph; Adjacency Matrix; Degree Correlation; Correlation coefficient; Process dynamics; Lower bound; Adjacency matrix; Graph theory; Distributed system; Synchronization; Modeling; Laplacian; Computer virus; Upper bound; Network topology; Metric; Eigenvalue problem; Bipartite graph; Regular graph
• ISSN: 1434-6028; 1434-6036
• DOI: 10.1140/epjb/e2010-00219-x

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