title: Spectral graph analysis of modularity and assortativity.
abstract: Expressions and bounds for Newman's modularity are presented. These results reveal conditions for or properties of the maximum modularity of a network. The influence of the spectrum of the modularity matrix on the maximum modularity is discussed. The second part of the paper investigates how the maximum modularity, the number of clusters, and the hop count of the shortest paths vary when the assortativity of the graph is changed via degree-preserving rewiring. Via simulations, we show that the maximum modularity increases, the number of clusters decreases, and the average hop count and the effective graph resistance increase with increasing assortativity.
authors.affiliation: Faculty of Electrical Engineering, Mathematics, and Computer Science, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands. p.f.a.vanmieghem@tudelft.nl
authors.forename: P
authors.initials: P
authors.lastname: Van Mieghem
authors.forename: X
authors.initials: X
authors.lastname: Ge
authors.forename: P
authors.initials: P
authors.lastname: Schumm
authors.forename: S
authors.initials: S
authors.lastname: Trajanovski
authors.forename: H
authors.initials: H
authors.lastname: Wang