Modularity Resolution Limit

Consider a network consisting of a ring of nc cliques, each clique having Nc nodes and m$($m$1)/2$ links. The neighboring cliques are connected by a single link $($Image $9\mathrm{.}34).$ The network has an obvious community structure, each community corresponding to a clique.

Determine the modularity Msingle of this natural partition, and the modularity Mpairs of the partition in which pairs of neighboring cliques are merged into a single community, as indicated by the dotted lines in Image $9\mathrm{.}34.$

Show that only for nc $<2$L will the modularity maximum predict the intuitively correct community partition, where L$=$ncm$($m$1)/2+$nc

Discuss the consequences of violating the above inequality.