Local Homophily on Bicolored Graphs is $\mathbf{P}$-complete

TL;DR

This paper introduces a local transformation called local homophily on bicolored graphs, demonstrating its computational complexity by showing it can simulate Boolean circuits and is P-complete.

Contribution

It defines local homophily, analyzes its computational complexity, and proves P-completeness for connectivity problems under this transformation.

Findings

Local homophily can simulate Boolean circuits.

Determining vertex connectivity under local homophily is P-complete.

The transformation models adaptive network dynamics.

Abstract

We propose a local transformation on bicolored graphs, which we call local homophily, inspired by adaptive networks and based on majority dynamics and homophily. In this transformation, a vertex updates its color to match the majority of its neighbors, while neighbors of the same color become connected and neighbors of the opposite color become disconnected. We show how to simulate Boolean circuits using local homophily and establish that determining whether a given pair of vertices becomes connected under iterative applications of local homophily is $\mathbf{P}$-complete under logspace reductions.