## Graduate Theses and Dissertations

2010

Thesis

M.S.C.S.

Computer Science

#### Major Professor

Rahul Tripathi, Ph.D.

#### Keywords

Betweenness centrality, Social networks, Randomized algorithms, Experimental algorithmics, Graphs

#### Abstract

In network analysis, it is useful to identify important vertices in a network. Based on the varying notions of importance of vertices, a number of centrality measures are defined and studied in the literature. Some popular centrality measures, such as betweenness centrality, are computationally prohibitive for large-scale networks. In this thesis, we propose a new centrality measure called k-path centrality and experimentally compare this measure with betweenness centrality. We present a polynomial-time randomized algorithm for distinguishing high k-path centrality vertices from low k-path centrality vertices in any given (unweighted or weighted) graph.

Specifically, for any graph G = (V,E) with n vertices and for every choice of parameters alpha between (0,1), epsilon between (0,1/2), and integer k between [1,n], with probability at least 1-1/n 2 our randomized algorithm distinguishes all vertices v in V that have k-path centrality Ck(v) more than n (alpha)*(1+2*epsilon) from all vertices v in V that have k-path centrality Ck(v) less than n (alpha)*(1-2*epsilon). The running time of the algorithm is O(k (2)*epsilon (-2)*n (1-alpha)*ln(n)). Next, we present a polynomial-time randomized approximation algorithm for computing the k-path centrality values of all vertices in any given (unweighted or weighted) graph. Specifically, for any graph and for every choice of parameters alpha between (0,1/2) and integer k between [1,n], with probability at least 1-1/n 2 our randomized approximation algorithm computes the k-path centrality value of every vertex within an additive error of at most n (1/2+alpha).

The running time of the algorithm is O(k (3)*n (1-2*alpha)*ln(n)). Theoretically and experimentally, our algorithms are (for suitable choices of parameters) significantly faster than the best known deterministic algorithm for computing exact betweenness centrality values (Brandes' algorithm). Through experimentations on both real and randomly generated networks, we demonstrate that vertices that have high betweenness centrality values also have high k-path centrality values.

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