Facets of Random Symmetric Edge Polytopes, Degree Sequences, and Clustering

TL;DR

This paper explores the structure of symmetric edge polytopes derived from random graphs, identifying thresholds and relationships between clustering and polytope facets through theoretical analysis and empirical sampling.

Contribution

It provides new insights into the facet structure of symmetric edge polytopes for random graphs, linking clustering coefficients to polytope complexity.

Findings

Threshold probability for facet similarity with complete graphs in Erdős-Rényi models

Higher clustering correlates with more facets in fixed degree sequence graphs

Empirical evidence from MCMC sampling supports the clustering-facet relationship

Abstract

Symmetric edge polytopes are lattice polytopes associated with finite simple graphs that are of interest in both theory and applications. We investigate the facet structure of symmetric edge polytopes for various models of random graphs. For an Erd\H{o}s-Renyi random graph, we identify a threshold probability at which with high probability the symmetric edge polytope shares many facet-supporting hyperplanes with that of a complete graph. We also investigate the relationship between the average local clustering, also known as the Watts-Strogatz clustering coefficient, and the number of facets for graphs with either a fixed number of edges or a fixed degree sequence. We use well-known Markov Chain Monte Carlo sampling methods to generate empirical evidence that for a fixed degree sequence, higher average local clustering in a connected graph corresponds to higher facet numbers in the associated symmetric edge polytope.