Recovery of clique hyperedges in random intersection graphs

Determine when the cliques corresponding to the sets C_j can be recovered from the observed random intersection graph, where each vertex i independently selects a random subset S_i of [m], each element j belongs to S_i independently with probability p, and C_j denotes the set of vertices whose selected subset contains j.

Background

The paper identifies random intersection graphs as a specific non-uniform hypergraph recovery problem. Each element j of the ground set [m] induces a clique C_j consisting of vertices i for which j belongs to S_i, and the observed graph is the union of these cliques. The unresolved question is whether, and in which parameter regimes, the underlying clique family can be reconstructed from the graph alone. The paper notes that prior work establishes a phase transition for total-variation convergence of random intersection graphs to Erdős–Rényi graphs, motivating the expectation of a corresponding phase transition for recoverability.