Exotic phase transitions of k-cores in clustered networks

TL;DR

This paper investigates how clustering influences the emergence of k-cores in networks, revealing exotic phase transitions including both discontinuous and continuous types, unlike traditional models.

Contribution

It introduces a new k-core calculation method accounting for clustering, showing novel phase transition behaviors in clustered networks.

Findings

3-cores can emerge via both discontinuous and continuous phase transitions

Clustering leads to exotic phase transition phenomena in k-cores

A new calculation method for k-cores in clustered networks

Abstract

The giant $k$-core --- maximal connected subgraph of a network where each node has at least $k$ neighbors --- is important in the study of phase transitions and in applications of network theory. Unlike Erd\H{o}s-R\'enyi graphs and other random networks where $k$-cores emerge discontinuously for $k\ge 3$, we show that transitive linking (or triadic closure) leads to 3-cores emerging through single or double phase transitions of both discontinuous and continuous nature. We also develop a $k$-core calculation that includes clustering and provides insights into how high-level connectivity emerges.