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.
@article{arxiv.1701.04241,
title = {Modularity-like objective function in annotated networks},
author = {Jia-Rong Xie and Bing-Hong Wang},
journal= {arXiv preprint arXiv:1701.04241},
year = {2017}
}