International Journal of Combinatorics

Volume 2014, Article ID 241723, 12 pages

http://dx.doi.org/10.1155/2014/241723

## Betweenness Centrality in Some Classes of Graphs

Department of Computer Applications, Cochin University of Science and Technology, Cochin 682022, India

Received 21 May 2014; Accepted 21 October 2014; Published 25 December 2014

Academic Editor: Chris A. Rodger

Copyright © 2014 Sunil Kumar Raghavan Unnithan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

There are several centrality measures that have been introduced and studied for real-world networks. They account for the different vertex characteristics that permit them to be ranked in order of importance in the network. Betweenness centrality is a measure of the influence of a vertex over the flow of information between every pair of vertices under the assumption that information primarily flows over the shortest paths between them. In this paper we present betweenness centrality of some important classes of graphs.

#### 1. Introduction

Betweenness centrality plays an important role in analysis of social networks [1, 2], computer networks [3], and many other types of network data models [4–9].

In the case of communication networks the distance from other units is not the only important property of a unit. What is more important is which units lie on the shortest paths (geodesics) among pairs of other units. Such units have control over the flow of information in the network. Betweenness centrality is useful as a measure of the potential of a vertex for control of communication. Betweenness centrality [10–14] indicates the betweenness of a vertex in a network and it measures the extent to which a vertex lies on the shortest paths between pairs of other vertices. In many real-world situations it has quite a significant role.

Determining betweenness is simple and straightforward when only one geodesic connects each pair of vertices, where the intermediate vertices can completely control communication between pairs of others. But when there are several geodesics connecting a pair of vertices, the situation becomes more complicated and the control of the intermediate vertices gets fractionated.

#### 2. Background

The concept of betweenness centrality was first introduced by Bavelas in 1948 [15]. The importance of the concept of vertex centrality is in the potential of a vertex for control of information flow in the network. Positions are viewed as structurally central to the degree to which they stand between others and can therefore facilitate, impede, or bias the transmission of messages. Freeman in his papers [5, 16] classified betweenness centrality into three measures. The three measures include two indexes of vertex centrality—one based on counts and one on proportions—and one index of overall network or graph centralization.

##### 2.1. Betweenness Centrality of a Vertex

Betweenness centrality for a vertex is defined as where is the number of shortest paths with vertices and as their end vertices, while is the number of those shortest paths that include vertex [16]. High centrality scores indicate that a vertex lies on a considerable fraction of shortest paths connecting pairs of vertices.(i)Every pair of vertices in a connected graph provides a value lying in to the betweenness centrality of all other vertices.(ii)If there is only one geodesic joining a particular pair of vertices, then that pair provides a betweenness centrality 1 to each of its intermediate vertices and zero to all other vertices. For example, in a path graph, a pair of vertices provides a betweenness centrality 1 to each of its interior vertices and zero to the exterior vertices. A pair of adjacent vertices always provides zero to all others.(iii)If there are geodesics of length 2 joining a pair of vertices, then that pair of vertices provides a betweenness centrality to each of the intermediate vertices.Freeman [16] proved that the maximum value taken by is achieved only by the central vertex in a star as the central vertex lies on the geodesic (which is unique) joining every pair of other vertices. In a star with vertices, the betweenness centrality of the central vertex is therefore the number of such geodesics which is . The betweenness centrality of each pendant vertex is zero since no pendant vertex lies in between any geodesic. Again it can be seen that the betweenness centrality of any vertex in a complete graph is zero since no vertex lies in between any geodesic as every geodesic is of length 1.

##### 2.2. Relative Betweenness Centrality

The betweenness centrality increases with the number of vertices in the network, so a normalized version is often considered with the centrality values scaled to between 0 and 1. Betweenness centrality can be normalized by dividing by its maximum value. Among all graphs of vertices the central vertex of a star graph has the maximum value which is . The relative betweenness centrality is therefore defined as

##### 2.3. Betweenness Centrality of a Graph

The betweenness centrality of a graph measures the tendency of a single vertex to be more central than all other vertices in the graph. It is based on differences between the centrality of the most central vertex and that of all others. Freeman [16] defined the betweenness centrality of a graph as the average difference between the measures of centrality of the most central vertex and that of all other vertices.

The betweenness centrality of a graph is defined as where is the largest value of for any vertex in the given graph and is the maximum possible sum of differences in centrality for any graph of vertices which occur in star with the value times of the central vertex, that is, .

Therefore the betweenness centrality of is defined as The index, , determines the degree to which exceeds the centrality of all other vertices in . Since is the ratio of an observed sum of differences to its maximum value, it will vary between 0 and 1. if and only if all are equal, and if and only if one vertex completely dominates the network with respect to centrality. Freeman showed that all of these measures agree in assigning the highest centrality index to the star graph and the lowest to the complete graph (see Table 1).