The $k$-Cap Process on Geometric Random Graphs

TL;DR

This paper studies the $k$-cap process on geometric random graphs, revealing unexpected convergence behaviors that model neural activity and contribute to understanding complex network dynamics.

Contribution

It introduces analysis of the $k$-cap process on geometric random graphs, highlighting novel convergence properties and behaviors in this specific network model.

Findings

Revealed surprising convergence behaviors of the $k$-cap process

Identified unique dynamics in geometric random graphs

Provided insights into neural activity modeling

Abstract

The $k$-cap (or $k$-winners-take-all) process on a graph works as follows: in each iteration, exactly $k$ vertices of the graph are in the cap (i.e., winners); the next round winners are the vertices that have the highest total degree to the current winners, with ties broken randomly. This natural process is a simple model of firing activity in the brain. We study its convergence on geometric random graphs, revealing rather surprising behavior.