Betweenness centrality is an important index widely used in different domains such as social networks, traffic networks and the world wide web. However, even for mid-size networks that have only a few hundreds thousands vertices, it is computationally expensive to compute exact betweenness scores. Therefore in recent years, several approximate algorithms have been developed. In this paper, first given a network @math and a vertex @math , we propose a Metropolis-Hastings MCMC algorithm that samples from the space @math and estimates betweenness score of @math . The stationary distribution of our MCMC sampler is the optimal sampling proposed for betweenness centrality estimation. We show that our MCMC sampler provides an @math -approximation, where the number of required samples depends on the position of @math in @math and in many cases, it is a constant. Then, given a network @math and a set @math , we present a Metropolis-Hastings MCMC sampler that samples from the joint space @math and @math and estimates relative betweenness scores of the vertices in @math . We show that for any pair @math , the ratio of the expected values of the estimated relative betweenness scores of @math and @math respect to each other is equal to the ratio of their betweenness scores. We also show that our joint-space MCMC sampler provides an @math -approximation of the relative betweenness score of @math respect to @math , where the number of required samples depends on the position of @math in @math and in many cases, it is a constant.
Lee et. al. @cite_19 proposed an algorithm to efficiently update betweenness centralities of vertices when the graph obtains a new edge. They reduced the search space by finding a candidate set of vertices whose betweenness centralities can be updated. Bergamini et. al. @cite_24 presented approximate algorithms that update betweenness scores of all vertices when an edge is inserted or the weight of an edge decreases. They used the algorithm of @cite_26 as the building block. Hayashi et. al. @cite_30 proposed a fully dynamic algorithm for estimating betweenness centrality of all vertices in a large dynamic network. Their algorithm is based on two data structures: hypergraph sketch that keeps track of SPDs, and two-ball index that helps to identify the parts of hypergraph sketches that require updates.
@cite_24 @cite_19 @cite_26 @cite_30
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The betweenness centrality of a vertex in a graph is a measure for the participation of the vertex in the shortest paths in the graph. The Betweenness centrality is widely used in network analyses. Especially in a social network, the recursive computation of the betweenness centralities of vertices is performed for the community detection and finding the influential user in the network. Since a social network graph is frequently updated, it is necessary to update the betweenness centrality efficiently. When a graph is changed, the betweenness centralities of all the vertices should be recomputed from scratch using all the vertices in the graph. To the best of our knowledge, this is the first work that proposes an efficient algorithm which handles the update of the betweenness centralities of vertices in a graph. In this paper, we propose a method that efficiently reduces the search space by finding a candidate set of vertices whose betweenness centralities can be updated and computes their betweenness centeralities using candidate vertices only. As the cost of calculating the betweenness centrality mainly depends on the number of vertices to be considered, the proposed algorithm significantly reduces the cost of calculation. The proposed algorithm allows the transformation of an existing algorithm which does not consider the graph update. Experimental results on large real datasets show that the proposed algorithm speeds up the existing algorithm 2 to 2418 times depending on the dataset. Measuring the relative importance of each vertex in a network is one of the most fundamental building blocks in network analysis. Among several importance measures, betweenness centrality, in particular, plays key roles in many real applications. Considerable effort has been made for developing algorithms for static settings. However, real networks today are highly dynamic and are evolving rapidly, and scalable dynamic methods that can instantly reflect graph changes into centrality values are required. In this paper, we present the first fully dynamic method for managing betweenness centrality of all vertices in a large dynamic network. Its main data structure is the weighted hyperedge representation of shortest paths called hypergraph sketch. We carefully design dynamic update procedure with theoretical accuracy guarantee. To accelerate updates, we further propose two auxiliary data structures called two-ball index and special-purpose reachability index. Experimental results using real networks demonstrate its high scalability and efficiency. In particular, it can reflect a graph change in less than a millisecond on average for a large-scale web graph with 106M vertices and 3.7B edges, which is several orders of magnitude larger than the limits of previous dynamic methods. We consider the incremental computation of the betweenness centrality of all vertices in a large complex network modeled as a graph G = (V, E), directed or undirected, with positive real edge-weights. The current widely used algorithm to compute the betweenness centrality of all vertices in G is the Brandes algorithm that runs in O(mn + n^2 log n) time, where n = |V| and m = |E|. We present an incremental algorithm that updates the betweenness centrality score of all vertices in G when a new edge is added to G, or the weight of an existing edge is reduced. Our incremental algorithm runs in O(m' n + n^2) time, where m' is the size of a certain subset of E*, the set of edges in G that lie on a shortest path. We achieve the same bound for the more general incremental update of a vertex v, where the edge update can be performed on any subset of edges incident to v. Our incremental algorithm is the first algorithm that is asymptotically faster on sparse graphs than recomputing with the Brandes algorithm. Our algorithm is also likely to be much faster than the Brandes algorithm on dense graphs since m*, the size of E*, is often close to linear in n. Our incremental algorithm is very simple and the only data structures it uses are arrays, lists, and stack. We give an efficient cache-oblivious implementation that incurs O(scan(n^2) + n sort(m')) cache misses, where scan and sort are well-known measures for efficient caching. We also give a static algorithm for computing betweenness centrality of all vertices that runs in time O(m* n + n^2 log n), which is faster than the Brandes algorithm on any graph with n log n = o(m) and m* = o(m). Betweenness centrality ranks the importance of nodes by their participation in all shortest paths of the network. Therefore computing exact betweenness values is impractical in large networks. For static networks, approximation based on randomly sampled paths has been shown to be significantly faster in practice. However, for dynamic networks, no approximation algorithm for betweenness centrality is known that improves on static recomputation. We address this deficit by proposing two incremental approximation algorithms (for weighted and unweighted connected graphs) which provide a provable guarantee on the absolute approximation error. Processing batches of edge insertions, our algorithms yield significant speedups up to a factor of 104 compared to restarting the approximation. This is enabled by investing memory to store and efficiently update shortest paths. As a building block, we also propose an asymptotically faster algorithm for updating the SSSP problem in unweighted graphs. Our experimental study shows that our algorithms are the first to make in-memory computation of a betweenness ranking practical for million-edge semi-dynamic networks. Moreover, our results show that the accuracy is even better than the theoretical guarantees in terms of absolute errors and the rank of nodes is well preserved, in particular for those with high betweenness.