Community Detection in the Multi-View Stochastic Block Model
Yexin Zhang Zhongtian Ma Qiaosheng Zhang Zhen Wang Xuelong Li
This paper considers the problem of community detection on multiple
potentially correlated graphs from an information-theoretical perspective. We
first put forth a random graph model, called the multi-view stochastic block
model (MVSBM), designed to generate correlated graphs on the same set of nodes
(with cardinality $n$). The $n$ nodes are partitioned into two disjoint
communities of equal size. The presence or absence of edges in the graphs for
each pair of nodes depends on whether the two nodes belong to the same
community or not. The objective for the learner is to recover the hidden
communities with observed graphs. Our technical contributions are two-fold: (i)
We establish an information-theoretic upper bound (Theorem~1) showing that
exact recovery of community is achievable when the model parameters of MVSBM
exceed a certain threshold. (ii) Conversely, we derive an information-theoretic
lower bound (Theorem~2) showing that when the model parameters of MVSBM fall
below the aforementioned threshold, then for any estimator, the expected number
of misclassified nodes will always be greater than one. Our results for the
MVSBM recover several prior results for community detection in the standard SBM
as well as in multiple independent SBMs as special cases.