A Message-Passing Algorithm for Counting Short Cycles in a Graph

TL;DR

This paper introduces a message-passing algorithm that efficiently counts short cycles in graphs, especially bipartite ones, with improved complexity and memory usage over existing methods.

Contribution

The paper presents a novel message-passing algorithm capable of counting specific cycle lengths in bipartite and general graphs, with better efficiency for sparse graphs.

Findings

Efficient counting of short cycles in bipartite graphs.

Algorithm outperforms existing methods in complexity and memory for sparse graphs.

Applicable to graphs of various cycle lengths based on girth.

Abstract

A message-passing algorithm for counting short cycles in a graph is presented. For bipartite graphs, which are of particular interest in coding, the algorithm is capable of counting cycles of length g, g +2,..., 2g - 2, where g is the girth of the graph. For a general (non-bipartite) graph, cycles of length g; g + 1, ..., 2g - 1 can be counted. The algorithm is based on performing integer additions and subtractions in the nodes of the graph and passing extrinsic messages to adjacent nodes. The complexity of the proposed algorithm grows as $O(g|E|^2)$, where $|E|$ is the number of edges in the graph. For sparse graphs, the proposed algorithm significantly outperforms the existing algorithms in terms of computational complexity and memory requirements.