Asymptotic modularity of some graph classes

Authors:
Fabien de Montgolfier;Mauricio Soto;Laurent Viennot
Affiliations:
LIAFA, UMR 7089 CNRS - Université Paris Diderot, France;LIAFA, UMR 7089 CNRS - Université Paris Diderot, France;INRIA and Université Paris Diderot, France
Venue:
ISAAC'11 Proceedings of the 22nd international conference on Algorithms and Computation
Year:
2011

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Abstract

Modularity has been introduced as a quality measure for graph partitioning. It has received considerable attention in several disciplines, especially complex systems. In order to better understand this measure from a graph theoretical point of view, we study the modularity of a variety of graph classes. We first consider simple graph classes such as tori and hypercubes. We show that these regular graph families have asymptotic modularity 1 (that is the maximum possible). We extend this result to the general class of unit ball graphs of bounded growth metrics. Our most striking result concerns trees with bounded degree which also appear to have asymptotic modularity 1. This last result can be extended to graphs with constant average degree and to some power-law graphs.