Dynamics of k-core percolation in a random graph

TL;DR

This paper investigates the critical behavior of k-core percolation in random graphs, revealing divergence in edge fluctuations near the percolation point and linking it to saddle-node bifurcation theory.

Contribution

It provides a theoretical framework connecting k-core percolation to saddle-node bifurcation, with universal exponents describing fluctuation divergence.

Findings

Edge fluctuations diverge critically near the percolation point

The percolation point corresponds to a saddle-node bifurcation

Universal exponents characterize the divergence behavior

Abstract

We study the edge deletion process of random graphs near a k-core percolation point. We find that the time-dependent number of edges in the process exhibits critically divergent fluctuations. We first show theoretically that the k-core percolation point is exactly given as the saddle-node bifurcation point in a dynamical system. We then determine all the exponents for the divergence based on a universal description of fluctuations near the saddle-node bifurcation.