Four-Cycle Counting in Low-Degeneracy Graph Streams

2026-06-16

AI Summary

This work addresses the problem of efficiently approximating the count of 4-cycles in graph edge streams presented in arbitrary order. To this end, the authors propose two induced subgraph sampling–based algorithms: a two-pass algorithm that achieves theoretically optimal space complexity on graphs with bounded degeneracy, marking the first optimal solution in this streaming model; and a single-pass algorithm tailored for sparse networks where 4-cycles are uniformly distributed, demonstrating robust performance on non-bipartite graphs such as social networks. Experimental evaluation shows that the two-pass algorithm significantly outperforms existing methods on real-world graph streams, while the single-pass variant also exhibits strong performance in its intended application scenarios.

Abstract

We study the problem of $(1+\varepsilon)$-approximating the number of four-cycles in graphs given as arbitrary order edge streams. We propose two new algorithms based on sampling induced subgraphs. Our first contribution is a two-pass algorithm that uses $\widetilde{O}(κm / \sqrt{T})$ space, where $m$ is the number of edges, $T$ is the number of four-cycles, and $κ$ is the graph's degeneracy. This algorithm improves upon existing theoretical bounds and is provably optimal for constant-degeneracy graphs, matching the known $Ω(m/\sqrt{T})$ lower bound up to lower-order factors. Our second contribution is a one-pass algorithm that remains accurate when four-cycles are not highly concentrated around individual nodes, edges, or wedges; this structural property is common in sparse social and collaboration networks. We evaluate both algorithms on a variety of real-world graph streams. The two-pass algorithm consistently outperforms state-of-the-art methods, using substantially less space to achieve a desired accuracy. The one-pass algorithm is competitive when four-cycles are evenly distributed, matching our theoretical analysis. Unlike several recent works, our algorithms perform well even on non-bipartite graphs such as social networks.

Problem

• four-cycle counting
• graph streams
• degeneracy
• subgraph approximation
• edge streaming

Innovation

• four-cycle counting
• graph streaming
• degeneracy
• subgraph sampling
• space-efficient algorithm