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

Benjamin Braun, Kaitlin Bruegge, Matthew Kahle

TL;DR

This work uses well-known Markov Chain Monte Carlo sampling methods to generate empirical evidence that for a fixed degree sequence, higher WattsStrogatz clustering in a connected graph corresponds to higher facet numbers in the associated symmetric edge polytope.

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 Erdős-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.

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

TL;DR

This work uses well-known Markov Chain Monte Carlo sampling methods to generate empirical evidence that for a fixed degree sequence, higher WattsStrogatz clustering in a connected graph corresponds to higher facet numbers in the associated symmetric edge polytope.

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 Erdős-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.
Paper Structure (15 sections, 8 theorems, 7 equations, 17 figures)

This paper contains 15 sections, 8 theorems, 7 equations, 17 figures.

Table of Contents

  1. Introduction
  2. The Erdős-Renyi Model
  3. Facets of $G(n,p)$
  4. Data and Observations
  5. Graph Ensembles via Markov Chain Monte Carlo Sampling
  6. MCMC Sampling Methods
  7. Fixed Number of Edges
  8. Fixed Degree Sequence
  9. Data and Observations
  10. Fixed Number of Edges
  11. Graphs with Hubs
  12. $k$-Regular Graphs
  13. Discussion
  14. Case: $n$ Vertices, $n$ Edges, Fixed Degree Sequence
  15. Final Discussion

Key Result

Theorem 2.1

Let $G=(V,E)$ be a finite simple connected graph. Then $f:V\rightarrow\mathbb{Z}$ is facet-defining if and only if both of the following hold.

Figures (17)

  • Figure 1: Above are the complete graph $K_4$ (left), and the complete bipartite graph $K_{3,2}$ (right).
  • Figure 2: Average local clustering against number of facets for every connected graph $G$ with 8 vertices. The line of best fit for this data is also included.
  • Figure 3: A path graph $P_3$ (top) and the complete graph $K_3$ (bottom) with their symmetric edge polytopes. Facets arising from the same pair of support hyperplanes are bold.
  • Figure 4: The graph $G$ is formed by removing the edge $vw$ from the complete graph $K_4$. The graph $G_f$ is the facet subgraph of $G$ associated to the partition $(\{t,\, v\}, \{u,\,w\})$ of the vertices.
  • Figure 5: Data from an ensemble of 4975 connected graphs from $G(14,0.45)$.
  • ...and 12 more figures

Theorems & Definitions (17)

  • Conjecture 1.1: Braun and Bruegge braunbruegge2022facets
  • Theorem 2.1: Higashitani, Jochemko, Michalek higashitanijochemkomateusz
  • Definition 2.2
  • Lemma 2.3: Chen, Davis, Korchevskaia chen2021facets
  • Definition 2.4
  • Theorem 2.5
  • Lemma 2.6
  • proof : of Theorem \ref{['thm:ERbound']}
  • Corollary 2.7
  • Example 2.8
  • ...and 7 more