  We present new algorithms for detecting the emergence of a community in large
networks from sequential observations. The networks are modeled using
Erdos-Renyi random graphs with edges forming between nodes in the community
with higher probability. Based on statistical changepoint detection
methodology, we develop three algorithms: the Exhaustive Search (ES), the
mixture, and the Hierarchical Mixture (H-Mix) methods. Performance of these
methods is evaluated by the average run length (ARL), which captures the
frequency of false alarms, and the detection delay. Numerical comparisons show
that the ES method performs the best; however, it is exponentially complex. The
mixture method is polynomially complex by exploiting the fact that the size of
the community is typically small in a large network. However, it may react to a
group of active edges that do not form a community. This issue is resolved by
the H-Mix method, which is based on a dendrogram decomposition of the network.
We present an asymptotic analytical expression for ARL of the mixture method
when the threshold is large. Numerical simulation verifies that our
approximation is accurate even in the non-asymptotic regime. Hence, it can be
used to determine a desired threshold efficiently. Finally, numerical examples
show that the mixture and the H-Mix methods can both detect a community quickly
with a lower complexity than the ES method.
