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.

翻译：三角形中心性是一种基于每个顶点周围三角形集中程度来发现图中重要顶点的度量方法。其显著特点是允许顶点在包含大量三角形或完全不包含三角形的情况下均可能成为中心顶点。实验表明，三角形中心性广泛适用于多种不同类型的网络。实证结果表明，在30%的情况下，三角形中心性识别出的中心顶点与五种经典中心性度量方法的结果存在差异，这体现了该方法既具有新颖性又不过度特化的特点。在稀疏图上，其计算渐进复杂度优于除最平凡度量外的所有对比方法。我们提出了在$O(m\bar\delta)$时间和$O(m+n)$空间内计算三角形中心性的最优算法，其中$\bar\delta\le O(\sqrt{m})$为Burkhardt、Faber和Harris（2020）提出的平均退化度。在实际应用中，$\bar\delta$远小于$\sqrt{m}$，因此三角形中心性可在近似线性时间内完成计算。在并发读独占写并行随机存取存储器上，我们提出了近乎工作最优的并行算法，该算法使用$O(m\sqrt{m})$个CREW PRAM处理器，耗时$O(\log n)$。在MapReduce框架中，我们证明该方法仅需四轮计算，通信量为$O(m\sqrt{m})$，因而达到最优。我们还推导了三角形中心性的线性代数形式，该形式在稀疏图上的计算时间复杂度为$O(m\bar\delta)$。