Dynamics and self-similarity in min-driven clustering

TL;DR

This paper analyzes a mean-field clustering model driven by the smallest cluster merging with randomly chosen clusters, establishing conditions for self-similarity, characterizing eternal solutions, and deriving an explicit solution formula.

Contribution

It introduces a continuous dynamical system framework for the clustering process and characterizes self-similar and eternal solutions using a Levy-Khintchine formula.

Findings

Conditions for approach to self-similar form are established.

Eternal solutions are characterized via a Levy-Khintchine formula.

An explicit solution formula for the model is derived and extended.

Abstract

We study a mean-field model for a clustering process that may be described informally as follows. At each step a random integer $k$ is chosen with probability $p_k$, and the smallest cluster merges with $k$ randomly chosen clusters. We prove that the model determines a continuous dynamical system on the space of probability measures supported in $(0,\infty)$, and we establish necessary and sufficient conditions for approach to self-similar form. We also characterize eternal solutions for this model via a Levy-Khintchine formula. The analysis is based on an explicit solution formula discovered by Gallay and Mielke, extended using a careful choice of time scale.