Characterization of Simplicial Complexes by Counting Simplets Beyond Four Nodes

TL;DR

This paper introduces a novel sampling algorithm called SC3 for counting simplets in simplicial complexes, enabling detailed local pattern analysis and characterization of complex systems with improved accuracy and scalability.

Contribution

The paper extends color coding algorithms from graphs to simplicial complexes and provides theoretical analysis and extensive experiments demonstrating its effectiveness.

Findings

SC3 is more accurate than baseline methods.

SC3 is faster and more scalable.

SC3 effectively characterizes and clusters real-world simplicial complexes.

Abstract

Simplicial complexes are higher-order combinatorial structures which have been used to represent real-world complex systems. In this paper, we concentrate on the local patterns in simplicial complexes called simplets, a generalization of graphlets. We formulate the problem of counting simplets of a given size in a given simplicial complex. For this problem, we extend a sampling algorithm based on color coding from graphs to simplicial complexes, with essential technical novelty. We theoretically analyze our proposed algorithm named SC3, showing its correctness, unbiasedness, convergence, and time/space complexity. Through the extensive experiments on sixteen real-world datasets, we show the superiority of SC3 in terms of accuracy, speed, and scalability, compared to the baseline methods. Finally, we use the counts given by SC3 for simplicial complex analysis, especially for characterization, which is further used for simplicial complex clustering, where SC3 shows a strong ability of characterization with domain-based similarity.