Emergence in graphs with near-extreme constraints

TL;DR

This paper studies entropy-maximizing graphons under near-extreme constraints, revealing unique, multipodal solutions and phase transitions, thus expanding understanding of graph structure phases.

Contribution

It introduces the analysis of near-extreme constraints, showing the existence of multiple phases and characterizing their structures and transitions.

Findings

Near-extreme constraints lead to unique, multipodal graphons.

Multiple phases exist with distinct podal structures.

Phase transitions occur between different structural phases.

Abstract

We consider entropy-optimal graphons associated with extreme and near-extreme constraints on the densities of edges and triangles. We prove that the optimizers for near-extreme constraints are unique and multipodal and are perturbations of the previously known unique optimzers for extreme constraints. This proves the existence of infinitely many phases. We determine the podal structures in these phases and prove the existence of phase transitions between them.