  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.
