Abstract
Betweenness centrality-measuring how many shortest paths pass through a vertex-is one of the most important network analysis concepts for assessing the relative importance of a vertex. The well-known algorithm of Brandes [J. Math. Sociol. '01] computes, on an $n$-vertex and $m$-edge graph, the betweenness centrality of all vertices in $O(nm)$ worst-case time. In later work, significant empirical speedups were achieved by preprocessing degree-one vertices and by graph partitioning based on cut vertices. We contribute an algorithmic treatment of degree-two vertices, which turns out to be much richer in mathematical structure than the case of degree-one vertices. Based on these three algorithmic ingredients, we provide a strengthened worst-case running time analysis for betweenness centrality algorithms. More specifically, we prove an adaptive running time bound $O(kn)$, where $k < m$ is the size of a minimum feedback edge set of the input graph.
Highlights
One of the most important building blocks in network analysis is to determine a vertex’s relative importance in the network
We focus on shortest paths between pairs of maximal induced paths P1max and P2max, and how to efficiently determine how these paths affect the betweenness centrality of each vertex
In the proof of Proposition 12, we consider two cases for every pair P1max = P2max ∈ Pmax of maximal induced paths: First, we look at how the shortest paths between vertices in P1max and P2max affect the betweenness centrality of those vertices that are not contained in the two maximal induced paths, and second, how they affect the betweenness centrality of those vertices that are contained in the two maximal induced paths
Summary
One of the most important building blocks in network analysis is to determine a vertex’s relative importance in the network. More empirically oriented work of Baglioni et al [2], Puzis et al [12], and Sariyüce et al [13] (see Section 2 for a description of their approaches), our main result is the mathematically rigorous analysis of an algorithm for Betweenness Centrality that runs in O(kn) time, where k denotes the feedback edge number of the input graph G. Our most profound contribution is to analyze the worst-case running time of the proposed betweenness centrality algorithm based on degreeone-vertex processing [2], usage of cut vertices [12, 13], and our degree-two-vertex processing. For j ≤ k we set [j, k] := {j, j + 1, . . . , k}
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