Fully Dynamic Betweenness Centrality

Abstract

We present fully dynamic algorithms for maintaining betweenness centrality (BC) of vertices in a directed graph $G=(V,E)$ with positive edge weights. BC is a widely used parameter in the analysis of large complex networks. We achieve an amortized $O({\nu ^*}^2 \cdot \log ^3 n)$ time per update with our basic algorithm, and $O({\nu ^*}^2 \cdot \log ^2 n)$ time with a more complex algorithm, where $n = |V|$, and ${\nu ^*}$ bounds the number of distinct edges that lie on shortest paths through any single vertex. For graphs with $\nu ^* = O(n)$, our algorithms match the fully dynamic all pairs shortest paths (APSP) bounds of Demetrescu and Italiano [8] and Thorup [28] for unique shortest paths, where $\nu ^*=n-1$. Our first algorithm also contains within it, a method and analysis for obtaining fully dynamic APSP from a decremental algorithm, that differs from the one in [8].

This work was supported in part by NSF grants CCF-0830737 and CCF-1320675.