Hypothesis Testing for Automated Community Detection in Networks

The network data has attracted considerable attention in modern statistics. In research on complex network data, one key issue is finding its underlying connection structure given a network sample. The methods that have been proposed in literature usually assume that the underlying structure is a known model. In practice, however, the true model is usually unknown, and network learning procedures based on these methods may suffer from model misspecification. To handle this is...

Many methods have been proposed for community detection in networks. Some of the most promising are methods based on statistical inference, which rest on solid mathematical foundations and return excellent results in practice. In this paper we show that two of the most widely used inference methods can be mapped directly onto versions of the standard minimum-cut graph partitioning problem, which allows us to apply any of the many well-understood partitioning algorithms to the...

Real-world networks usually have community structure, that is, nodes are grouped into densely connected communities. Community detection is one of the most popular and best-studied research topics in network science and has attracted attention in many different fields, including computer science, statistics, social sciences, among others. Numerous approaches for community detection have been proposed in literature, from ad-hoc algorithms to systematic model-based approaches. ...

Network-based clustering methods frequently require the number of communities to be specified \emph{a priori}. Moreover, most of the existing methods for estimating the number of communities assume the number of communities to be fixed and not scale with the network size $n$. The few methods that assume the number of communities to increase with the network size $n$ are only valid when the average degree $d$ of a network grows at least as fast as $O(n)$ (i.e., the dense case)...

Community detection in complex networks has attracted considerable attention, however, most existing methods need the number of communities to be specified beforehand. In this paper, a goodness-of-fit test based on the linear spectral statistic of the centered and rescaled adjacency matrix for the stochastic block model is proposed. We prove that the proposed test statistic converges in distribution to the standard Gaussian distribution under the null hypothesis. The proof us...

Community structure is a commonly observed feature of real networks. The term refers to the presence in a network of groups of nodes (communities) that feature high internal connectivity, but are poorly connected between each other. Whereas the issue of community detection has been addressed in several works, the problem of validating a partition of nodes as a good community structure for a real network has received considerably less attention and remains an open issue. We pr...

We propose and analyze the problems of \textit{community goodness-of-fit and two-sample testing} for stochastic block models (SBM), where changes arise due to modification in community memberships of nodes. Motivated by practical applications, we consider the challenging sparse regime, where expected node degrees are constant, and the inter-community mean degree ($b$) scales proportionally to intra-community mean degree ($a$). Prior work has sharply characterized partial or f...

The problem of community detection in networks is usually formulated as finding a single partition of the network into some "correct" number of communities. We argue that it is more interpretable and in some regimes more accurate to construct a hierarchical tree of communities instead. This can be done with a simple top-down recursive partitioning algorithm, starting with a single community and separating the nodes into two communities by spectral clustering repeatedly, until...

To characterize the community structure in network data, researchers have introduced various block-type models, including the stochastic block model, degree-corrected stochastic block model, mixed membership block model, degree-corrected mixed membership block model, and others. A critical step in applying these models effectively is determining the number of communities in the network. However, to our knowledge, existing methods for estimating the number of network communiti...

The stochastic block model is a popular tool for studying community structures in network data. We develop a goodness-of-fit test for the stochastic block model. The test statistic is based on the largest singular value of a residual matrix obtained by subtracting the estimated block mean effect from the adjacency matrix. Asymptotic null distribution is obtained using recent advances in random matrix theory. The test is proved to have full power against alternative models wit...