Four-Cycle Counting in Low-Degeneracy Graph Streams
低退化度图流中的四环计数
Sebastian Lüderssen, Stefan Neumann, Pan Peng
AI总结 提出两种基于子图采样的算法,分别使用两遍和一遍流式扫描,在低退化度图上实现四环数量的(1+ε)近似,空间复杂度达到理论最优或接近最优。
Comments KDD 2026
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我们研究了在任意顺序边流给出的图中,对四环数量进行$(1+\varepsilon)$近似的问题。我们提出了两种基于采样诱导子图的新算法。第一个贡献是一个两遍算法,使用$\widetilde{O}(\kappa m / \sqrt{T})$空间,其中$m$是边数,$T$是四环数,$\kappa$是图的退化度。该算法改进了现有的理论界限,并且在常数退化度图上被证明是最优的,匹配已知的$\Omega(m/\sqrt{T})$下界(忽略低阶因子)。第二个贡献是一个一遍算法,当四环不是高度集中在单个节点、边或楔形周围时,该算法保持准确;这种结构性质在稀疏社交和协作网络中很常见。我们在各种真实世界图流上评估了这两种算法。两遍算法始终优于最先进的方法,使用更少的空间达到所需的精度。一遍算法在四环均匀分布时具有竞争力,与我们的理论分析一致。与最近的几项工作不同,我们的算法即使在非二分图(如社交网络)上也表现良好。
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.