Clustering in Block Markov Chains

Jaron Sanders (Corresponding author), Alexandre Proutière, Se-Young Yun

Research output: Contribution to journalArticleAcademicpeer-review

5 Citations (Scopus)
51 Downloads (Pure)

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.
Original languageEnglish
Pages (from-to)3488-3512
Number of pages25
JournalThe Annals of Statistics
Volume48
Issue number6
DOIs
Publication statusPublished - 1 Dec 2020

Keywords

  • math.PR
  • cs.IT
  • math.IT
  • math.ST
  • stat.TH
  • Mixing times
  • Community detection
  • Markov chains
  • Asymptotic analysis
  • Clustering
  • Change of measure
  • Information theory

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