Active learning for community detection in stochastic block models

Abstract

Abstract-The stochastic block model (SBM) is an important generative model for random graphs in network science and machine learning, useful for benchmarking community detection (or clustering) algorithms. The symmetric SBM generates a graph with 2n nodes which cluster into two equally sized communities. Nodes connect with probability p within a community and q across different communities. We consider the case of p = a ln(n)/n and q = b ln(n)/n. In this case, it was recently shown that recovering the community membership (or label) of every node with high probability (w.h.p.) using only the graph is possible if and only if the Chernoff-Hellinger (CH) divergenceIn this work, we study if, and by how much, community detection below the clustering threshold (i.e. D(a, b) < 1) is possible by querying the labels of a limited number of chosen nodes (i.e., active learning). Our main result is to show that, under certain conditions, sampling the labels of a vanishingly small fraction of nodes (a number sub-linear in n) is sufficient for exact community detection even when D(a, b) < 1. Furthermore, we provide an efficient learning algorithm which recovers the community memberships of all nodes w.h.p. as long as the number of sampled points meets the sufficient condition. We also show that recovery is not possible if the number of observed labels is less than n 1−D(a,b) . The validity of our results is demonstrated through numerical experiments.

Active Learning in the Geometric Block Model

“…We propose two active learning algorithms for the GBM that exactly recover the community memberships with high probability using a sub-linear number of queries, even in regimes below the limit of the state-of-the-art unsupervised algorithm in (Galhotra et al 2019). Similar to the result of (Gadde et al 2016) in the SBM, our results offer a smooth trade-off between query complexity and clustering accuracy in the GBM. Both algorithms exploit the idea of motifcounting to remove cross-cluster edges, while combining it with active learning in a different way.…”

“…In the active learning framework, we are allowed to query node labels up to a budget constraint in order to improve overall classification accuracy. The authors of (Gadde et al 2016) showed that a sub-linear number of queries is sufficient to achieve exact recovery below the limit (in terms of difference between p and q) of unsupervised methods in the SBM, and that the number of queries needed for exact recovery depends on how far we are below such limit -hence providing a smooth trade-off between query complexity and clustering accuracy in the SBM.…”

Rényi Entropy Bounds on the Active Learning Cost-Performance Tradeoff

“…In particular, the observable E = g(X N , L N ) can be described by an N × N binary random matrix where element (i, j) is given by E i,j = h(X i , X j , L i , L j ), with h(•) being a properly defined (possibly noisy) function. A renowned example is given by the SBM [9]- [13], a popular random graph model for community detection that generalizes the well known Erdös-Renyi model. In this case, h(X i , X j , L i , L j ) is a noisy binary function defined as…”

“…On the theoretical front, a few recent works have studied the AL problem assuming an underlying statistical model and providing performance guarantees. However, such works focus on specific models, typically graph models such as the stochastic block model (SBM), and their performance analysis is only valid in the asymptotic regime where the number of data items goes to infinity [9]- [13]. As a result, while one would expect a significant performance boost when exploiting the knowledge of the underlying statistical model, their results tend to be pessimistic and with limited practical insight.…”

Chain-referral sampling on stochastic block models

“…In this paper, we consider a particular model with latent community structure: the stochastic block model (SBM) proposed by Holland et al [19]. This model is a useful benchmark for some statistical tasks as recovering community (also called blocks or types in the sequel) structure in network science [14,15,23]. By block structure, we mean that the set of vertices in the graph is partitioned into subsets called blocks and nodes connect to each other with probabilities that depend only on their types, i.e.…”