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).
In this paper we present the betweenness centrality measures in some important classes of graphs which are the basic components of larger complex networks.
3. Betweenness Centrality of Some Classes of Graphs
3.1. Betweenness Centrality of Vertices in Wheels
Wheel Graph . A Wheel graph is obtained by joining a new vertex to every vertex in a cycle . It was invented by the eminent graph theorist W. T. Tutte. A wheel graph on 7 vertices is shown in Figure 1.
Theorem 1. The betweenness centrality of a vertex in a wheel graph , , is given by
Proof. In the wheel graph the central vertex is adjacent to each vertex of the cycle . Consider the central vertex of for . On each pair of adjacent vertices contributes centrality , each pair of alternate vertices contributes centrality , and all other pairs contribute centrality to the central vertex. Since there are vertices on , there exist adjacent pairs, alternate pairs, and other pairs. Therefore the betweenness centrality of the central vertex is given by . Now for any vertex on , there are two geodesics joining its adjacent vertices on , one of which passing through it. Therefore the betweenness centrality of any vertex on is .
Note. It can be seen easily that for every vertex in and in .
The relative centrality and graph centrality are as follows:
3.2. Betweenness Centrality of Vertices in the Graph
A complete graph on 6 vertices with one edge deleted is shown in Figure 2.
Theorem 2. Let be a complete graph on vertices and an edge of it. Then the betweenness centrality of vertices in is given by
Proof. Suppose the edge is removed from . Now and can be joined by means of any of the remaining vertices. Thus there are geodesics joining and each containing exactly one vertex as intermediary. This provides a betweenness centrality to each of the vertices. Again and do not lie in between any geodesics and therefore their betweenness centralities are zero.
The relative centrality and graph centrality are as follows:
3.3. Betweenness Centrality of Vertices in Complete Bipartite Graphs
Complete Bipartite Graphs . A graph is complete bipartite if its vertices can be partitioned into two disjoint nonempty sets and such that two vertices and are adjacent if and only if and . If and , such a graph is denoted . For example, see in Figure 3.
Theorem 3. The betweenness centrality of a vertex in a complete bipartite graph is given by
Proof. Consider a complete bipartite graph with a bipartition where and . The distance between any two vertices in (or in ) is 2. Consider a vertex . Now any pair of vertices in contributes a betweenness centrality to . Since there are pairs of vertices in , . In a similar way it can be shown that, for any vertex in , .
The relative centrality and graph centrality are as follows:
3.4. Betweenness Centrality of Vertices in Cocktail Party Graphs
The cocktail party graph [17] is a unique regular graph of degree on vertices. It is obtained from by deleting a perfect matching (see Figure 4). The cocktail party graph of order is a complete -partite graph with 2 vertices in each partition set. It is the graph complement of the ladder rung graph which is the graph union of copies of the path graph and the dual graph of the hypercube [18].
Theorem 4. The betweenness centrality of each vertex of a cocktail party graph of order is .
Proof. Let the cocktail party graph be obtained from the complete graph with vertices by deleting a perfect matching . Now for each pair there is a geodesic of length 2 passing through each of the other vertices. Thus for any particular vertex, there are pairs of vertices of the above matching not containing that vertex giving a betweenness centrality to that vertex. Therefore the betweenness centrality of any vertex is given by .
The relative centrality and graph centrality are as follows:
3.5. Betweenness Centrality of Vertices in Crown Graphs
The crown graph [18] is the unique regular graph with vertices obtained from a complete bipartite graph by deleting a perfect matching (see Figure 5). A crown graph on vertices can be viewed as an undirected graph with two sets of vertices and and with an edge from to whenever . It is the graph complement of the ladder graph . The crown graph is a distance-transitive graph.
Theorem 5. The betweenness centrality of each vertex of a crown graph of order is .
Proof. Let the crown graph be the complete bipartite graph with vertices minus a perfect matching . Consider any vertex; say . Now for each pair other than there are paths of length 3 passing through out of paths joining and . Since there are such pairs, they give the betweenness centrality . Again for each pair from there exists exactly one path passing through out of paths. Since there are such pairs, they give the betweenness centrality . Therefore the betweenness centrality of is given by . Since the graph is vertex transitive, the betweenness centrality of any vertex is given by .
The relative centrality and graph centrality are as follows:
3.6. Betweenness Centrality of Vertices in Paths
Theorem 6. The betweenness centrality of any vertex in a path graph is the product of the number of vertices on either side of that vertex in the path.
Proof. Consider a path graph of vertices (see Figure 6). Take a vertex in . Then there are vertices on one side and vertices on the other side of . Consequently there are number of geodesics containing . Hence .
Note that, by symmetry, vertices at equal distance away from both ends of have the same centrality and it is maximum at the central vertex (vertices) and minimum at the end vertices. Consider
Relative centrality of any vertex is given by
Corollary 7. Graph centrality of is given by
Proof. When is even, by definition When is odd, by definition
3.7. Betweenness Centrality of Vertices in Ladder Graphs
The ladder graph [19, 20] is defined as the Cartesian product (see Figure 7). It is a planar undirected graph with vertices and edges.
Theorem 8. The betweenness centrality of a vertex in a ladder graph is given by
Proof. By symmetry, let be any vertex such that . Consider the paths (in Figure 8) from upper left vertices to upper right vertices which give the betweenness centrality
Now consider the paths from lower left vertices to the upper right vertices of which give the betweenness centrality
Again consider the paths from upper left vertices to the lower right vertices of which give the betweenness centrality
The above three equations when combined get the result.
3.8. Betweenness Centrality of Vertices in Cycles
Theorem 9. The betweenness centrality of a vertex in a cycle is given by
Proof.
Case 1 (when is even). Let , , and let be the given cycle. Consider a vertex (see Figure 9). Then is its antipodal vertex and there is no geodesic from to any other vertex passing through . Hence we omit the pair . Consider other pairs of antipodal vertices for . For each pair of these antipodal vertices there exist two paths of the same length and one of them contains . Thus each pair contributes to the centrality of which gives a total of . Now consider all paths of length less than containing . There are paths joining to vertices from to passing through for and each contributes centrality 1 to giving a total . Therefore the betweenness centrality of is . Since is vertex transitive, the betweenness centrality of any vertex is given by .
Case 2 (when is odd). Let , , and let be the given cycle. Consider a vertex (see Figure 10). Then and are its antipodal vertices at a distance from and there is no geodesic path from the vertexes and to any other vertex passing through . Hence we omit and the pair . Now consider paths of length passing through . There are paths joining to vertices from to passing through for and each contributes a betweenness centrality 1 to giving a total of . Since is vertex transitive, the betweenness centrality of any vertex is given by .
The relative centrality and graph centrality are as follows:
3.9. Betweenness Centrality of Vertices in Circular Ladder Graphs
The circular ladder graph consists of two concentric -cycles in which each pair of the corresponding vertices is joined by an edge (see Figure 11). It is a 3-regular simple graph isomorphic to the Cartesian product .
Theorem 10. The betweenness centrality of a vertex in a circular ladder is given by
Proof.
Case 1 (when is even). Let , . Let be the outer cycle and the inner cycle. Consider any vertex; say in . Then its betweenness centrality as a vertex in is . Now the geodesics from outer vertices to the inner vertices for (see Figure 12) and from to for by symmetry contribute to the betweenness centrality
Again the pair contributes to the betweenness centrality . Therefore the betweenness centrality of is given as
Case 2 (when is odd). Let , . Let be the outer cycle and the inner cycle. Consider any vertex; say in . Then its betweenness centrality as a vertex in is . Now consider the geodesics from outer vertices to the inner vertices for (see Figure 13) and from to for which give a betweenness centrality
Therefore the betweenness centrality of is given as
The relative centrality and graph centrality are as follows:
3.10. Betweenness Centrality of Vertices in Trees
In a tree, there is exactly one path between any two vertices. Therefore the betweenness centrality of a vertex is the number of paths passing through that vertex. A branch at a vertex of a tree is a maximal subtree containing as an end vertex. The number of branches at is .
Theorem 11. The betweenness centrality of a vertex in a tree is given by where the arguments denote the number of vertices of the branches at excluding , taken in any order.
Proof. Consider a vertex in a tree . Let there be branches with number of vertices excluding . The betweenness centrality of in is the total number of paths passing through . Since all the branches have only one vertex in common, excluding , every path joining a pair of vertices of different branches passes through . Thus the total number of such pairs gives the betweenness centrality of . Hence .
Example 12. Consider the tree given in Figure 14.
Here
Example 13. Table 2 gives the possible values for the betweenness centrality of a vertex in a tree of 9 vertices.
We consider the different possible combinations of the arguments in so that the sum of arguments is .
3.11. Betweenness Centrality of Vertices in Hypercubes
The -cube or -dimensional hypercube is defined recursively by and . That is, Harary [21]. is an -regular graph containing vertices and edges. Each vertex can be labeled as a string of bits and . Two vertices of are adjacent if their binary representations differ at exactly one place (see Figure 15). The vertices are labeled by the binary numbers from 0 to . By definition, the length of a path between two vertices and is the number of edges of the path. To move from to it suffices to cross successively the vertices whose labels are those obtained by changing the bits of one by one in order to transform into . If and differ only in bits, the distance (hamming distance) between and denoted by is [22, 23]. For example, if and , then .
There exists a path of length at most between any two vertices of . In other words an -cube is a connected graph of diameter . The number of geodesics between and is given by the number of permutations . A hypercube is bipartite and interval regular. For any two vertices and , the interval induces a hypercube of dimension [24]. Another important property of -cube is that it can be constructed recursively from lower dimensional cubes (see Figure 15). Consider two identical -cubes. Each -cube has vertices and each vertex has a labeling of -bits. Join all identical pairs of vertices of the two cubes. Now increase the number of bits in the labels of all vertices by placing 0 in the th place of the first cube and 1 in the th place of second cube. Thus we get an -cube with vertices, each vertex having a label of -bits and the corresponding vertices of the two -cubes differ only in the th bit. This -cube so constructed can be seen as the union of pairs of -cubes differing in exactly one position of bits. Thus the number of -cubes in an -cube is . The operation of splitting an -cube into two disjoint -cubes so that the vertices of the two -cubes are in a one-to-one correspondence will be referred to as tearing [23]. Since there are bits, there are directions for tearing. In general there are number of -subcubes associated with an -cube.
Theorem 14. The betweenness centrality of a vertex in a hypercube is given by .
Proof. The hypercube of dimension is a vertex transitive -regular graph containing vertices. Each vertex can be written as an tuple of binary digits and with adjacent vertices differing in exactly one coordinate. The distance between two vertices and denoted by is the number of places in which the corresponding coordinates of and differ and the number of distinct geodesics between and is [22]. Let be a vertex in whose betweenness centrality has to be determined. Consider all -subcubes containing the vertex for . Each -subcube has vertices with zeros in their labels. Since each -subcube can be distinguished by coordinates, the number of -subcubes containing the vertex is . The vertex lies on a geodesic joining a pair of vertices if and only if the pair of vertices forms a pair of antipodal vertices of some subcube containing [24]. So we consider all pairs of antipodal vertices excluding the vertex and its antipodal vertex in each -subcube containing . If a vertex of a -subcube has ones, then its antipodal vertex has ones. For any pair of such antipodal vertices there are geodesics joining them and of that paths are passing through . Thus each pair contributes , that is, , to the betweenness centrality of .
By symmetry, when is even the number of distinct pairs of required antipodal vertices are given by for and for . When is odd, the number of distinct pairs of required antipodal vertices is given by for . Taking all such pairs of antipodal vertices in a -subcube we get the contribution of betweenness centrality as , when is even, and , when is odd. Therefore considering all -subcubes for , we get the betweenness centrality of as
Therefore, for any vertex ,
The relative centrality and graph centrality are as follows:
4. Conclusion
Betweenness centrality is a useful metric for analyzing graph structures. When compared to other centrality measures, computation of betweenness centrality is rather difficult as it involves calculation of the shortest paths between all pairs of vertices in a graph. We have derived expressions for betweenness centrality of graphs which are the basic components of larger and complex networks. This study is therefore helpful for analysing larger classes of graphs.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by the University Grants Commission (UGC), Government of India, under the scheme of Faculty Development Programme (FDP) for colleges.