Upper and lower bounds of various centrality measures on planar and sparse graphs

Abstract

In this thesis we address the problem of computing closeness centrality, Harmonic centrality and a few related centrality measures that operate on the shortest paths in a graph. We consider sparse graphs, especially planar graphs and this makes our results widely applicable to real-world networks such as social, geographical, citation, biological, communication, etc. on which centrality values are often evaluated in practice. We introduce a generalisation of Harmonic centrality and two simplifications of betweenness centrality, a more well-known but more complicated notion of centrality. We show that closeness, Harmonic and number-farness centrality values of all nodes of a planar graph can be computed in o(n2). On the other hand for sparse graphs we show that the optimal algorithms for computing these values of all nodes cannot be truly subquadratic. These problems are, therefore, computationally no different from betweenness centrality. We also show that one centrality measures that involves shortest paths passing through a particular node can be computed inO(n2) in planar graphs and no faster, making it a harder problem compared to the others but probably slightly easier compared to betweenness centrality which, as of now, requires O(n2 log n) for planar graphs. One of the centralities that we introduce, between number-farness centrality, has a tight bound of O(n2) for one node and all nodes in the case of sparse graphs, putting it into a league of its own. Based on these results, we conjecture that for planar graphs, computing betweenness centrality of only a single node can possibly be done in subquadratic time but not of all nodes.