Triangle Centrality

TL;DR

Triangle centrality is a novel measure for identifying important vertices in graphs based on triangle concentration, offering broad applicability, computational efficiency, and distinct identification of central nodes compared to existing measures.

Contribution

The paper introduces triangle centrality, along with optimal algorithms for its computation, demonstrating its effectiveness and efficiency on sparse graphs and parallel computing models.

Findings

Identifies central vertices differently from existing measures in 30% of cases.

Provides algorithms with near-linear time complexity on sparse graphs.

Achieves near work-optimal parallel computation and optimal MapReduce rounds.

Abstract

Triangle centrality is introduced for finding important vertices in a graph based on the concentration of triangles surrounding each vertex. It has the distinct feature of allowing a vertex to be central if it is in many triangles or none at all. We show experimentally that triangle centrality is broadly applicable to many different types of networks. Our empirical results demonstrate that 30% of the time triangle centrality identified central vertices that differed with those found by five well-known centrality measures, which suggests novelty without being overly specialized. It is also asymptotically faster to compute on sparse graphs than all but the most trivial of these other measures. We introduce optimal algorithms that compute triangle centrality in $O(m\bar\delta)$ time and $O(m+n)$ space, where $\bar\delta\le O(\sqrt{m})$ is the $\textit{average degeneracy}$ introduced by Burkhardt, Faber, and Harris (2020). In practical applications, $\bar\delta$ is much smaller than $\sqrt{m}$ so triangle centrality can be computed in nearly linear time. On a Concurrent Read Exclusive Write (CREW) Parallel Random Access Machine (PRAM), we give a near work-optimal parallel algorithm that takes $O(\log n)$ time using $O(m\sqrt{m})$ CREW PRAM processors. In MapReduce, we show it takes four rounds using $O(m\sqrt{m})$ communication bits and is therefore optimal. We also derive a linear algebraic formulation of triangle centrality which can be computed in $O(m\bar\delta)$ time on sparse graphs.