Approximating Spanning Centrality with Random Bouquets

Spanning Centrality is a measure used in network analysis to determine the importance of an edge in a graph based on its contribution to the connectivity of the entire network. Specifically, it quantifies how critical an edge is in terms of the number of spanning trees that include that edge. The current state-of-the-art for All Edges Spanning Centrality~(AESC), which computes the exact centrality values for all the edges, has a time complexity of $\mathcal{O}(mn^{3/2})$ for $n$ vertices and $m$ edges. This makes the computation infeasible even for moderately sized graphs. Instead, there exist approximation algorithms which process a large number of random walks to estimate edge centralities. However, even the approximation algorithms can be computationally overwhelming, especially if the approximation error bound is small. In this work, we propose a novel, hash-based sampling method and a vectorized algorithm which greatly improves the execution time by clustering random walks into {\it Bouquets}. On synthetic random walk benchmarks, {\it Bouquets} performs $7.8\times$ faster compared to naive, traditional random-walk generation. We also show that the proposed technique is scalable by employing it within a state-of-the-art AESC approximation algorithm, {\sc TGT+}. The experiments show that using Bouquets yields more than $100\times$ speed-up via parallelization with 16 threads.