Lack of hyperbolicity in asymptotic erdös–Renyi sparse random graphs

Abstract

In this work, we prove that the giant component of the Erd¨os–Renyi random graph G(n, c/n) for c a constant greater than 1 (sparse regime), is not Gromov δ–hyperbolic for any δ with probability tending to one as n → ∞. We present numerical evidence to show that the “fat” triangles that rule out δ–hyperbolicity are in fact abundant in these graphs. We also present numerical evidence showing that, despite the known fact that these graphs almost surely have zero spectral gap as n→∞, the spectrum is very similar in outline to that of an infinite regular tree with the same (average) degree, which can be calculated analytically.