  We ascertain the modularity-like objective function whose optimization is
equivalent to the maximum likelihood in annotated networks. We demonstrate that
the modularity-like objective function is a linear combination of modularity
and conditional entropy. In contrast with statistical inference methods, in our
method, the influence of the metadata is adjustable; when its influence is
strong enough, the metadata can be recovered. Conversely, when it is weak, the
detection may correspond to another partition. Between the two, there is a
transition. This paper provides a concept for expanding the scope of modularity
methods.
