Approximate Triangle Counting

Triangle counting is an important problem in graph mining. Clustering coefficients of vertices and the transitivity ratio of the graph are two metrics often used in complex network analysis. Furthermore, triangles have been used successfully in several real-world applications. However, exact triangle counting is an expensive computation. In this paper we present the analysis of a practical sampling algorithm for counting triangles in graphs. Our analysis yields optimal values for the sampling rate, thus resulting in tremendous speedups ranging from \emph{2800}x to \emph{70000}x when applied to real-world networks. At the same time the accuracy of the estimation is excellent. Our contributions include experimentation on graphs with several millions of nodes and edges, where we show how practical our proposed method is. Finally, our algorithm’s implementation is a part of the \pegasus library (Code and datasets are available at (http://www.cs.cmu.edu/~ctsourak/).) a Peta-Graph Mining library implemented in Hadoop, the open source version of Mapreduce.

💡 Research Summary

The paper addresses the fundamental problem of counting triangles in large graphs, a task that underlies many important network analysis metrics such as clustering coefficients, transitivity ratios, and community detection. Exact counting methods based on matrix multiplication have prohibitive time (≈ O(n²·³⁷¹)) and space (O(n²)) requirements, making them unsuitable for real‑world networks that often contain millions of vertices and edges. Simpler exact approaches like node‑iterator or edge‑iterator run in O(m n) time, which still becomes cubic in dense graphs. Consequently, there is strong motivation for fast approximation techniques.

Among approximation methods, streaming‑based algorithms have received much attention, but they typically rely on uniform edge or node sampling and achieve good accuracy only when the graph is relatively dense. The Doulion algorithm, introduced in prior work, proposes a very simple sparsification scheme: each edge is retained independently with probability p and, if kept, is re‑weighted by 1/p. After sparsification, any exact triangle‑counting routine (e.g., node‑iterator) is applied to the resulting graph G′, producing a count t′. The estimator T̂ = t′ / p³ is unbiased (E