Modified Randomized Modularity Clustering: Adapting the Resolution Limit

Fortunato and Barthélemy (Proc Nat Acad Sci USA 104(1):36–41, 2007) investigated the resolution limit of modularity clustering algorithms. They showed that the goal function of the standard modularity clustering algorithm of Newman and Girvan (Phys Rev E 69(2):026113, 2004) implies that the number of clusters chosen by the algorithm is approximately the square root of the number of edges. The existence of the resolution limit shows that the discovery of the number of clusters is not automatic. In this paper we report on two contributions to solve the problem of automatic cluster detection in graph clustering: We parametrize the goal function of modularity clustering by considering the number of edges as free parameter and we introduce permutation invariance as a general formal diagnostic to recognize good partitions of a graph. The second contribution results from the study of scaling bounds combined with the stability of graph partitions on various types of regular graphs. In this study the connection of the stability of graph partitions with the automorphism group of the graph was discovered.