Clustering in Block Markov Chains

TL;DR

This paper introduces optimal clustering algorithms for Block Markov Chains that can accurately recover cluster structures from minimal trajectory data, matching the fundamental information-theoretic limits.

Contribution

It develops two algorithms that achieve the theoretical detection limit for clustering in BMCs, providing provable guarantees and efficiency.

Findings

Algorithms reach the fundamental detectability limit.

Clustering accuracy is proven under specified conditions.

Methods are efficient and optimal in trajectory length.

Abstract

This paper considers cluster detection in Block Markov Chains (BMCs). These Markov chains are characterized by a block structure in their transition matrix. More precisely, the $n$ possible states are divided into a finite number of $K$ groups or clusters, such that states in the same cluster exhibit the same transition rates to other states. One observes a trajectory of the Markov chain, and the objective is to recover, from this observation only, the (initially unknown) clusters. In this paper we devise a clustering procedure that accurately, efficiently, and provably detects the clusters. We first derive a fundamental information-theoretical lower bound on the detection error rate satisfied under any clustering algorithm. This bound identifies the parameters of the BMC, and trajectory lengths, for which it is possible to accurately detect the clusters. We next develop two clustering algorithms that can together accurately recover the cluster structure from the shortest possible trajectories, whenever the parameters allow detection. These algorithms thus reach the fundamental detectability limit, and are optimal in that sense.