The European Physical Journal B

, Volume 76, Issue 4, pp 643–652

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

Authors

  • P. Van Mieghem
    • Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science
    • Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science
  • X. Ge
    • Northeastern University, College of Information Science and Engineering
  • S. Tang
    • Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science
  • F. A. Kuipers
    • Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science
Interdisciplinary Physics

DOI: 10.1140/epjb/e2010-00219-x

Cite this article as:
Van Mieghem, P., Wang, H., Ge, X. et al. Eur. Phys. J. B (2010) 76: 643. doi:10.1140/epjb/e2010-00219-x

Abstract

Newman’s measure for (dis)assortativity, the linear degree correlation coefficient \(\rho _{D}\), is reformulated in terms of the total number Nk 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.

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2010