Community Detection with Side Information: Exact Recovery under the Stochastic Block Model

TL;DR

This paper investigates how different types of side information can improve the phase transition for exact community detection in stochastic block models, providing tight conditions and proposing an efficient algorithm.

Contribution

It characterizes the conditions under which side information enhances exact recovery in SBM and introduces a new algorithm leveraging partial recovery and local improvements.

Findings

Side information with error probability lpha improves phase transition if lpha is sufficiently small.

Revealing a large fraction of labels (psilon) enhances recovery when psilon decreases with n.

The paper provides tight necessary and sufficient conditions for the benefit of side information.

Abstract

The community detection problem involves making inferences about node labels in a graph, based on observing the graph edges. This paper studies the effect of additional, non-graphical side information on the phase transition of exact recovery in the binary stochastic block model (SBM) with $n$ nodes. When side information consists of noisy labels with error probability $\alpha$, it is shown that phase transition is improved if and only if $\log(\frac{1-\alpha}{\alpha})=\Omega(\log(n))$. When side information consists of revealing a fraction $1-\epsilon$ of the labels, it is shown that phase transition is improved if and only if $\log(1/\epsilon)=\Omega(\log(n))$. For a more general side information consisting of $K$ features, two scenarios are studied: (1)~$K$ is fixed while the likelihood of each feature with respect to corresponding node label evolves with $n$, and (2)~The number of features $K$ varies with $n$ but the likelihood of each feature is fixed. In each case, we find when side information improves the exact recovery phase transition and by how much. The calculated necessary and sufficient conditions for exact recovery are tight except for one special case. In the process of deriving inner bounds, a variation of an efficient algorithm is proposed for community detection with side information that uses a partial recovery algorithm combined with a local improvement procedure.