Betweenness Ordering Problem : An Efficient Non-Uniform Sampling Technique for Large Graphs.

Centrality measures, erstwhile popular amongst the sociologists and psychologists, has seen wide and increasing applications across several disciplines of late. In conjunction with the big data problems there came the need to analyze big networks and in this connection, centrality measures became of great interest to the community of mathematicians, computer scientists and physicists. While it is an important question to ask how one can rank vertices based on their importance in a network, there hasn’t been a commonly accepted definition, mainly due to the subjectivity of the term “importance”. Amongst a plethora of application specific definitions available in the literature to rank the vertices, closeness centrality, betweenness centrality and eigenvector centrality (page-rank) have been the most important and widely applied ones. In the current paper, we formulate a method to determine the betweenness ordering of two vertices without exactly computing their betweenness indices which is a daunting task for networks of large size. We apply our approach to find the betweenness ordering of two vertices in several synthetic and real world graphs. We compare our method with the available techniques in the literature and show that our method fares several times better than the currently known techniques. We further show that the accuracy of our algorithm gets better with the increase in size and density of the network.