Hindawi Publishing Corporation International Journal of Combinatorics Volume 2014, Article ID 241723, 12 pages http://dx.doi.org/10.1155/2014/241723 Research Article Betweenness Centrality in Some Classes of Graphs Sunil Kumar Raghavan Unnithan, Balakrishnan Kannan, and Madambi Jathavedan Department of Computer Applications, Cochin University of Science and Technology, Cochin 682022, India Correspondence should be addressed to Sunil Kumar Raghavan Unnithan; [email protected] 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. There are several centrality measures that have been introduced and studied for real-world networks. They account for thedifferent 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 vertex centrality is in the potential of a vertex for control of information flow in the network. Positions are viewed Betweenness centrality plays an important role in analysis of as structurally central to the degree to which they stand social networks [1, 2], computer networks [3], and many other between others and can therefore facilitate, impede, or bias types of network data models [4–9]. the transmission of messages. Freeman in his papers [5, 16] In the case of communication networks the distance from classified betweenness centrality into three measures. The other units is not the only important property of a unit. three measures include two indexes of vertex centrality—one What is more important is which units lie on the shortest based on counts and one on proportions—and one index of paths (geodesics) among pairs of other units. Such units overall network or graph centralization. have control over the flow of information in the network. Betweenness centrality is useful as a measure of the potential 2.1. Betweenness Centrality of a Vertex. Betweenness central- of a vertex for control of communication. Betweenness ity (V) for a vertex V is defined as centrality [10–14] indicates the betweenness of a vertex in a network and it measures the extent to which a vertex lies on (V) (V) = ∑ , the shortest paths between pairs of other vertices. In many (1) real-world situations it has quite a significant role. ≠V≠ Determining betweenness is simple and straightforward where isthenumberofshortestpathswithvertices when only one geodesic connects each pair of vertices, where and as their end vertices, while (V) is the number of the intermediate vertices can completely control communi- those shortest paths that include vertex V [16]. High centrality cation between pairs of others. But when there are several scores indicate that a vertex lies on a considerable fraction of geodesics connecting a pair of vertices, the situation becomes shortest paths connecting pairs of vertices. morecomplicatedandthecontroloftheintermediatevertices gets fractionated. (i) Every pair of vertices in a connected graph provides avaluelyingin[0, 1] to the betweenness centrality of all other vertices. 2. Background (ii) If there is only one geodesic joining a particular pair The concept of betweenness centrality was first introduced of vertices, then that pair provides a betweenness by Bavelas in 1948 [15]. The importance of the concept of centrality 1 to each of its intermediate vertices and 2 International Journal of Combinatorics zero to all other vertices. For example, in a path graph, 1 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. 2 6 (iii) If there are geodesics of length 2 joining a pair of vertices, then that pair of vertices provides a betweenness centrality 1/ to each of the intermediate vertices. 0 Freeman [16]provedthatthemaximumvaluetakenby (V) is achieved only by the central vertex in a star as the central vertex lies on the geodesic (which is unique) joining 3 5 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 ( −1 ). The betweenness 2 centrality of each pendant vertex is zero since no pendant 4 vertex lies in between any geodesic. Again it can be seen that Figure 1: Wheel graph 7. 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. ∗ The index, (), determines the degree to which (V ) exceeds the centrality of all other vertices in .Since() 2.2. Relative Betweenness Centrality. The betweenness cen- is the ratio of an observed sum of differences to its maximum trality increases with the number of vertices in the network, value, it will vary between 0 and 1. () = 0 if and only if all so a normalized version is often considered with the centrality ∗ (V) are equal, and () = 1 if and only if one vertex V values scaled to between 0 and 1. Betweenness centrality completely dominates the network with respect to centrality. canbenormalizedbydividing(V) by its maximum value. Freemanshowedthatallofthesemeasuresagreeinassigning Among all graphs of vertices the central vertex of a star −1 the highest centrality index to the star graph and the lowest graph has the maximum value which is ( ).Therelative 2 tothecompletegraph(seeTable 1). betweenness centrality is therefore defined as In this paper we present the betweenness centrality (V) 2 (V) measures in some important classes of graphs which are the (V) = = 0≤ (V) ≤1. Max (V) (−1)(−2) basic components of larger complex networks. (2) 3. Betweenness Centrality of 2.3. Betweenness Centrality of a Graph. The betweenness Some Classes of Graphs centrality of a graph measures the tendency of a single vertex to be more central than all other vertices in the graph. It 3.1. Betweenness Centrality of Vertices in Wheels is based on differences between the centrality of the most centralvertexandthatofallothers.Freeman[16] defined the Wheel Graph .AWheelgraph is obtained by joining betweenness centrality of a graph as the average difference a new vertex to every vertex in a cycle −1.Itwasinvented between the measures of centrality of the most central vertex bytheeminentgraphtheoristW.T.Tutte.Awheelgraphon andthatofallothervertices. 7verticesisshowninFigure 1. The betweenness centrality of a graph is defined as ∗ Theorem 1. The betweenness centrality of a vertex V in a wheel ∑=1 [ (V )− (V)] () = , graph , >5,isgivenby ∗ (3) Max ∑=1 [ (V ) − (V)] ∗ (−1)(−5) where (V ) is the largest value of (V) for any vertex V { ,V ℎ V, ∗ (V) = 2 in the given graph and Max ∑ [(V )−(V)] is the {1 =1 { ,ℎ. maximum possible sum of differences in centrality for any {2 graph of vertices which occur in star with the value −1 −1 (5) times (V) of the central vertex, that is, ( − 1) ( 2 ). Therefore the betweenness centrality of is defined as Proof. In the wheel graph the central vertex is adjacent ∗ to each vertex of the cycle −1. Consider the central vertex 2∑=1 [ (V )− (V)] () = of for >5.On−1 each pair of adjacent vertices con- (−1)2 (−2) tributes centrality 0,eachpairofalternateverticescontributes (4) 1/2 1 ∑ [ (V∗)− (V )] centrality , and all other pairs contribute centrality to =1 −1 or () = . the central vertex. Since there are vertices on −1, (−1) there exist −1adjacent pairs, −1alternate pairs, and International Journal of Combinatorics 3 Table 1: Graphs showing extreme betweenness. (V) (V)() { −1 {( ) for central vertex {1 for central vertex { 2 { 1 { {0 for other vertices {0 for other vertices 000 −1 ( 2 ) − 2( − 1) = ( − 1)( −6)/2 other pairs. Therefore 1 the betweenness centrality of the central vertex is given by (1/2)(−1)+1((−1)(−6)/2) = (−1)(−5)/2.Nowforany vertex on −1, there are two geodesics joining its adjacent vertices on −1, one of which passing through it. Therefore 2 6 the betweenness centrality of any vertex on −1 is 1/2. Note.Itcanbeseeneasilythat(V)=0for every vertex V in 4 and 2 { , V {3 if is the central vertex, 3 5 (V) = {1 (6) { , {3 otherwise 4 in 5. The relative centrality and graph centrality are as follows: Figure 2: Complete graph 6 −(V1, V4). 2 (V) (V) = (−1)(−2) The relative centrality and graph centrality are as follows: (−5) { , if V is the central vertex, 2 (V) −2 (V) = = { 1 { , , (−1)(−2) (−1)(−2) otherwise { 2 { , V ≠ V, V, ∗ 2 2 ∑=1 [ (V )− (V)] −6+4 = {(−1)(−2) ( )= = . (9) (−1) (−1)(−2) {0, otherwise, (7) ∗ ∑ [ (V )− (V )] 4 () = =1 = . (−1) (−1)2 (−2)2 3.2. Betweenness Centrality of Vertices in the Graph −. A complete graph on 6 vertices with one edge deleted is shown in Figure 2. 3.3. Betweenness Centrality of Vertices in Complete Bipartite Graphs Theorem 2. Let be a complete graph on vertices and = (V, V) an edge of it. Then the betweenness centrality of vertices Complete Bipartite Graphs ,.Agraphiscompletebipartite in −is given by if its vertices can be partitioned into two disjoint nonempty sets 1 and 2 such that two vertices and are adjacent if 1 { and only if ∈1 and ∈2.If|1|=and |2|=,such ,V ≠ V, V, (V) = {−2 (8) a graph is denoted ,. For example, see 3,4 in Figure 3. {0, ℎ. Theorem 3. The betweenness centrality of a vertex ina Proof. Suppose the edge (V, V) is removed from .Now complete bipartite graph , is given by V and V can be joined by means of any of the remaining −2vertices.