FreSCo: Mining Frequent Patterns in Simplicial Complexes

Abstract: Simplicial complexes are a generalization of graphs that model higher-order relations. In this paper, we introduce simplicial patterns -- that we call simplets -- and generalize the task of frequent pattern mining from the realm of graphs to that of simplicial complexes. Our task is particularly challenging due to the enormous search space and the need for higher-order isomorphism. We show that finding the occurrences of simplets in a complex can be reduced to a bipartite graph isomorphism problem, in linear time and at most quadratic space. We then propose an anti-monotonic frequency measure that allows us to start the exploration from small simplets and stop expanding a simplet as soon as its frequency falls below the minimum frequency threshold. Equipped with these ideas and a clever data structure, we develop a memory-conscious algorithm that, by carefully exploiting the relationships among the simplices in the complex and among the simplets, achieves efficiency and scalability for our complex mining task. Our algorithm, FreSCo, comes in two flavors: it can compute the exact frequency of the simplets or, more quickly, it can determine whether a simplet is frequent, without having to compute the exact frequency. Experimental results prove the ability of FreSCo to mine frequent simplets in complexes of various size and dimension, and the significance of the simplets with respect to the traditional graph patterns.