Robust Algorithms for Finding Cliques in Random Intersection Graphs via Sum-of-Squares

We study efficient algorithms for recovering cliques in dense random intersection graphs (RIGs). In this model, $d = n^{\Omega(1)}$ cliques of size approximately $k$ are randomly planted by choosing the vertices to participate in each clique independently with probability $\delta$. While there has been extensive work on recovering one, or multiple disjointly planted cliques in random graphs, the natural extension of this question to recovering overlapping cliques has been, surprisingly, largely unexplored. Moreover, because every vertex can be part of polynomially many cliques, this task is significantly harder than in case of disjointly planted cliques (as recently studied by Kothari, Vempala, Wein and Xu [COLT’23]). In this work we obtain the first efficient algorithms for recovering the community structure of RIGs both from the perspective of exact and approximate recovery. Our algorithms are further robust to noise, monotone adversaries, a certain, optimal number of edge corruptions, and work whenever $k \gg \sqrt{n \log(n)}$. Our techniques follow the proofs-to-algorithms framework utilizing the sum-of-squares hierarchy. An essential component are certificates for the absence of large cliques outside of the ground-truth. Instead of spectral certificates, a central ingredient are modified versions of the biclique certificates, recently used for semi-random planted clique by Buhai, Kothari and Steurer [STOC’23]. To turn these certificates into robust and efficient algorithms that do not produce “false positives”, we rely on an extremely sharp concentration property for pseudo-distributions which might be of independent interest. Our techniques further extend to the related task of efficient \emph{refutation}, and lead to algorithms that can not only recover the ground-truth, but also certify the optimality of this clustering.