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arXiv · 0807.4473

Dynamics and self-similarity in min-driven clustering

Govind Menon·Barbara Niethammer·Robert L. Pego
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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.

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Govind Menon, Barbara Niethammer, Robert L. Pego. 2008-07-28. Dynamics and self-similarity in min-driven clustering. https://doi.org/10.1090/s0002-9947-2010-05085-8

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