  We demonstrate an exact equivalence between two widely used methods of
community detection in networks, the method of modularity maximization in its
generalized form which incorporates a resolution parameter controlling the size
of the communities discovered, and the method of maximum likelihood applied to
the special case of the stochastic block model known as the planted partition
model, in which all communities in a network are assumed to have statistically
similar properties. Among other things, this equivalence provides a
mathematically principled derivation of the modularity function, clarifies the
conditions and assumptions of its use, and gives an explicit formula for the
optimal value of the resolution parameter.
