Graph Partitioning Induced Phase Transitions

TL;DR

This paper investigates the percolation phase transitions in graph partitioning on random regular graphs, revealing critical thresholds and cluster scaling behaviors relevant for network robustness and immunization strategies.

Contribution

It characterizes the percolation transition points and cluster properties during graph partitioning, including optimal and non-optimal processes, on random regular graphs.

Findings

Percolation transition at f_c=1-2/k for equal-sized partitions

Cluster size scales as N^{0.4} at the transition

Multiple non-percolation transitions occur for f<f_c

Abstract

We study the percolation properties of graph partitioning on random regular graphs with N vertices of degree $k$. Optimal graph partitioning is directly related to optimal attack and immunization of complex networks. We find that for any partitioning process (even if non-optimal) that partitions the graph into equal sized connected components (clusters), the system undergoes a percolation phase transition at $f=f_c=1-2/k$ where $f$ is the fraction of edges removed to partition the graph. For optimal partitioning, at the percolation threshold, we find $S \sim N^{0.4}$ where $S$ is the size of the clusters and $\ell\sim N^{0.25}$ where $\ell$ is their diameter. Additionally, we find that $S$ undergoes multiple non-percolation transitions for $f<f_c$.