for “ Identifying network communities with a high resolution ” Jianhua

where Qi = eii M − ( ai M ) is the contribution made by community ci. It can be seen easily that after moving vertex v from ci to cj , the only terms that will be changed are Qi and Qj . Therefore, the difference between the new modualarity Q and the original modularity can be computed by: Q − Q = Q′i + Q ′ j − Qi − Qj . After the move, the total degrees within community i and j can be given by: e′ii = eii − 2d v i , and e′jj = ejj + 2d v j , where di is the number of connections that v has in community i. Similarly, the total degrees for the vertices in community i and j can be given by: a′i = ai − d , and a′j = aj + d , where d is the degree of v. Therefore,