Faster and Generalized Temporal Triangle Counting, via Degeneracy Ordering

TL;DR

This paper introduces DOTTT, a fast and general algorithm for counting directed temporal triangles in large, directed, timestamped graphs, improving efficiency and flexibility over previous methods.

Contribution

The paper presents a novel degeneracy-based algorithm, DOTTT, that exactly counts all variants of temporal triangles with separate time constraints, achieving better theoretical and practical performance.

Findings

Runs in $O(mrac{ ext{degeneracy}}{ ext{log }m})$ time, matching static algorithms.

Runs twice as fast as existing temporal triangle counters.

Successfully processes large Bitcoin network data in under an hour.

Abstract

Triangle counting is a fundamental technique in network analysis, that has received much attention in various input models. The vast majority of triangle counting algorithms are targeted to static graphs. Yet, many real-world graphs are directed and temporal, where edges come with timestamps. Temporal triangles yield much more information, since they account for both the graph topology and the timestamps. Temporal triangle counting has seen a few recent results, but there are varying definitions of temporal triangles. In all cases, temporal triangle patterns enforce constraints on the time interval between edges (in the triangle). We define a general notion $(\delta_{1,3}, \delta_{1,2}, \delta_{2,3})$-temporal triangles that allows for separate time constraints for all pairs of edges. Our main result is a new algorithm, DOTTT (Degeneracy Oriented Temporal Triangle Totaler), that exactly counts all directed variants of $(\delta_{1,3}, \delta_{1,2}, \delta_{2,3})$-temporal triangles. Using the classic idea of degeneracy ordering with careful combinatorial arguments, we can prove that DOTTT runs in $O(m\kappa\log m)$ time, where $m$ is the number of (temporal) edges and $\kappa$ is the graph degeneracy (max core number). Up to log factors, this matches the running time of the best static triangle counters. Moreover, this running time is better than existing. DOTTT has excellent practical behavior and runs twice as fast as existing state-of-the-art temporal triangle counters (and is also more general). For example, DOTTT computes all types of temporal queries in Bitcoin temporal network with half a billion edges in less than an hour on a commodity machine.