Some remarks on modularity density

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 propose a modification of the original formulation to overcome this issue. Moreover, we show that a clique can be divided into two or more parts when maximizing the modularity density. We also compare the solution found by maximizing the modularity density with that obtained by maximizing the modularity on the Zachary karate club network.