Betweenness Centrality : Algorithms and Lower Bounds

One of the most fundamental problems in large scale network analysis is to determine the importance of a particular node in a network. Betweenness centrality is the most widely used metric to measure the importance of a node in a network. Currently the fastest known algorithm [5], to compute betweenness of all nodes, requires O(nm) time for unweighted graphs and O(nm + n logn) time for weighted graphs, where n is the number of nodes and m is the number of edges in the network. In this paper, we present structural properties and lower bounds for computing betweenness. We prove that any path comparison based algorithm cannot compute betweenness of all nodes in less than O(nm) time. We resort to algebraic techniques and present an algebraic method for computing betweenness centrality of all nodes in a network. For unweighted graphs, our algorithm runs in time O(nωDiam(G)), where ω < 2.376 is the exponent of matrix multiplication and Diam(G) is the diameter of the graph. For weighted graphs with integer weights taken from the range {1, 2, . . . ,M}, we present an algorithm that runs in time O(MnωDiam(G)). Hence, our algorithms perform better for dense graphs (m ≫ n) with small diameter and small integer weights. We present a randomized parallel algorithm for computing betweenness centrality of all nodes in a network. We compute the betweenness in two stages (which we call the forward pass and the backward pass). Our algorithm for forward pass runs in O(n) time using O(m log n) processors for unweighted graphs and O(n log n logM) time using O(m) processors for weighted graphs with integer weights taken from the range {1, 2, . . . ,M}. Our backward pass algorithm runs in O(n) time using O(n) processors for both weighted and unweighted graphs. For bounded-degree graphs, we present an improved backward pass algorithm that runs in O(n logm) time using O(m) processors for unweighted graphs and O(Mn logm) time using O(m) processors for weighted graphs. We also briefly analyze some of the future directions by relating the complexity of betweenness to the current techniques of All-Pairs Shortest Paths.