A “quantitative function” for community detection called modularity density has been proposed by Li, Zhang, Wang, Zhang, and Chen in [Phys. Rev. E 77, 036109 (2008)]. We study the modularity density maximization problem and we discuss some features of the optimal solution. More precisely, we show that in the optimal solution there can be communities having negative modularity density, and we pr...

We investigate axioms that intuitively ought to be satisfied by graph clustering objective functions. Two tailored for graph clustering objectives are introduced, and the four axioms introduced in previous work on distance based clustering are reformulated and generalized for the graph setting. We show that modularity, a standard objective for graph clustering, does not satisfy all these axioms...

Modularity proposed by Newman and Girvan is the most commonly used measure when the nodes of a graph are grouped into communities consisting of tightly connected nodes. However, some authors pointed out drawbacks of the modularity, the main issue of which is resolution limit. Resolution limit refers to the sensitivity of the modularity to the total number of edges in the graph, which leaves sma...

Two penalized-balanced and normalized-versions of the Newman-Girvan modularity are introduced and estimated by the non-negative eigenvalues of the modularity and normalized modularity matrix, respectively. In this way, the partition of the vertices that maximizes the modularity can be obtained by applying the k-means algorithm for the representatives of the vertices based on the eigenvectors be...