Triadic Measures on Graphs: The Power of Wedge Sampling

Graphs are used to model interactions in a variety of contexts, and there is a growing need to quickly assess the structure of a graph. Some of the most useful graph metrics, especially those measuring social cohesion, are based on triangles. Despite the importance of these triadic measures, associated algorithms can be extremely expensive. We propose a new method based on wedge sampling. This versatile technique allows for the fast and accurate approximation of all current variants of clustering coefficients and enables rapid uniform sampling of the triangles of a graph. Our methods come with provable and practical time-approximation tradeoffs for all computations. We provide extensive results that show our methods are orders of magnitude faster than the state-of-the-art, while providing nearly the accuracy of full enumeration. Our results will enable more wide-scale adoption of triadic measures for analysis of extremely large graphs, as demonstrated on several real-world examples

Wedge Sampling for Computing Clustering Coefficients and Triangle Counts on Large Graphs

Graphs are used to model interactions in a variety of contexts, and there is a growing need to quickly assess the structure of such graphs. Some of the most useful graph metrics are based on triangles, such as those measuring social cohesion. Algorithms to compute them can be extremely expensive, even for moderately-sized graphs with only millions of edges. Previous work has considered node and edge sampling; in contrast, we consider wedge sampling, which provides faster and more accurate approximations than competing techniques. Additionally, wedge sampling enables estimation local clustering coefficients, degree-wise clustering coefficients, uniform triangle sampling, and directed triangle counts. Our methods come with provable and practical probabilistic error estimates for all computations. We provide extensive results that show our methods are both more accurate and faster than state-of-the-art alternatives.Comment: Full version of SDM 2013 paper "Triadic Measures on Graphs: The Power of Wedge Sampling" (arxiv:1202.5230

Parallel Algorithms for Small Subgraph Counting

Subgraph counting is a fundamental problem in analyzing massive graphs, often studied in the context of social and complex networks. There is a rich literature on designing efficient, accurate, and scalable algorithms for this problem. In this work, we tackle this challenge and design several new algorithms for subgraph counting in the Massively Parallel Computation (MPC) model: Given a graph $G$ over $n$ vertices, $m$ edges and $T$ triangles, our first main result is an algorithm that, with high probability, outputs a $(1+\varepsilon)$-approximation to $T$, with optimal round and space complexity provided any $S \geq \max{(\sqrt m, n^2/m)}$ space per machine, assuming $T=\Omega(\sqrt{m/n})$. Our second main result is an $\tilde{O}_{\delta}(\log \log n)$-rounds algorithm for exactly counting the number of triangles, parametrized by the arboricity $\alpha$ of the input graph. The space per machine is $O(n^{\delta})$ for any constant $\delta$, and the total space is $O(m\alpha)$, which matches the time complexity of (combinatorial) triangle counting in the sequential model. We also prove that this result can be extended to exactly counting $k$-cliques for any constant $k$, with the same round complexity and total space $O(m\alpha^{k-2})$. Alternatively, allowing $O(\alpha^2)$ space per machine, the total space requirement reduces to $O(n\alpha^2)$. Finally, we prove that a recent result of Bera, Pashanasangi and Seshadhri (ITCS 2020) for exactly counting all subgraphs of size at most $5$, can be implemented in the MPC model in $\tilde{O}_{\delta}(\sqrt{\log n})$ rounds, $O(n^{\delta})$ space per machine and $O(m\alpha^3)$ total space. Therefore, this result also exhibits the phenomenon that a time bound in the sequential model translates to a space bound in the MPC model