Simple and efficient four-cycle counting on sparse graphs

We consider the problem of counting 4-cycles ($C_4$) in a general undirected graph $G$ of $n$ vertices and $m$ edges (in bipartite graphs, 4-cycles are also often referred to as $\textit{butterflies}$). There have been a number of previous algorithms for this problem; some of these are based on fast matrix multiplication, which is attractive theoretically but not practical, and some of these are based on randomized hash tables. We develop a new simpler algorithm for counting $C_4$, which has several practical improvements over previous algorithms; for example, it is fully deterministic and avoids any expensive arithmetic in its inner loops. The algorithm can also be adapted to count 4-cycles incident to each vertex and edge. Our algorithm runs in $O(m\bar\delta(G))$ time and $O(n)$ space, where $\bar \delta(G) \leq O(\sqrt{m})$ is the $\textit{average degeneracy}$ parameter introduced by Burkhardt, Faber & Harris (2020).