Random Algebraic Graphs and Their Convergence to ErdőS–Rényi

Abstract

ABSTRACTA random algebraic graph is defined by a group with a uniform distribution over it and a connection with expectation satisfying . The random graph with vertex set is formed as follows. First, independent variables are sampled uniformly from . Then, vertices are connected with probability . This model captures random geometric graphs over the sphere, torus, and hypercube; certain instances of the stochastic block model; and random subgraphs of Cayley graphs. The main question of interest to the current paper is: when is a random algebraic graph statistically and/or computationally distinguishable from ? Our results fall into two categories. (1) Geometric. We focus on the case and use Fourier‐analytic tools. We match and extend the following results from the prior literature: For hard threshold connections, we match for , and for ‐Lipschitz connections we extend the results of when to the non‐monotone setting. (2) Algebraic. We provide evidence for an exponential statistical‐computational gap. Consider any finite group and let be a set of elements formed by including each set of the form independently with probability Let be the distribution of random graphs formed by taking a uniformly random induced subgraph of size of the Cayley graph . Then, and are statistically indistinguishable with high probability over if and only if . However, low‐degree polynomial tests fail to distinguish and with high probability over when