Betweenness Centrality of Cartesian Product of Graphs

# Betweenness Centrality of Cartesian Product of Graphs

Sunil Kumar R sunilstands@gmail.com Kannan Balakrishnan mullayilkannan@gmail.com
###### Abstract

Betweenness centrality is a widely-used measure in the analysis of large complex networks. It measures the potential or power of a vertex to control the communication over the network under the assumption that information primarily flows over the shortest paths between them. In this paper we prove several results on betweenness centrality of Cartesian product of graphs.
Keywords: Betweenness centrality, Pairwise dependency, Cartesian product, Geodetic graph

## 1 Introduction

Several Centrality measures have so far been studied and their importance is increasing day by day. Betweenness centrality has a vital role in the analysis of networks.[1, 2, 3, 4] It has many applications in a variety of domains such as biological networks [5, 6, 7, 8, 9], study of sexual networks and AIDS, identifying key actors in terrorist networks, transportation networks, supply chain management, bio-informatics-protein interaction networks[14, 15], food webs etc. Betweennes centrality [17, 18] indicates the betweenness of a vertex (or an edge) in a network and it measures the extent to which a vertex (or an edge) lies on the shortest paths between pairs of other vertices. It is quite difficult to find out the betweenness centrality of a vertex in a large graph. The computation of this index based on direct application of definition becomes impractical as the number of nodes increases and has complexity in the order of . The fastest exact algorithm due to Brandes requires space and time where is the number of nodes and the number of edges in the graph. Exact computations of betweenness centrality can take a lot of time even for Brandes algorithm. But a large network can be thought of as it is made by joining smaller networks together. There are several graph operations which results in a larger graph and many of the properties of larger graphs can be derived from its constituent graphs. Graph operations are used for constructing new classes of graphs. Cartesian product is an important graph operation.

It is assumed that the graphs taken here are simple undirected connected graphs. Graph-reference may be given for making the context clear.

## 2 Background

The concept of betweenness centrality of a vertex was first introduced by Bavelas in 1948 . The importance of the concept of vertex centrality is that how a vertex acts as a bridge among all the pairs of vertices in joining them by shortest paths. It gives the potential of a vertex for control of information flow in the network[21, 22]. The order of a graph is the number of vertices in ; it is denoted by . The same notation is used for the number of elements (cardinality) of a set. Thus, . The distance between two vertices , denoted by , is the length of the shortest path in between and . A shortest path joining vertices and is called a geodesic between and . A graph is a geodetic graph if every pair of vertices of is connected by a unique shortest path. The diameter, , of a graph is given by . Two vertices and of with are diametrical vertices. The interval consists of all vertices on geodesics joining and in .

A graph is vertex-transitive if every vertex in can be mapped to any other vertex by some automorphism. Similarly a graph is edge-transitive if its automorphism group acts transitively on the set of edges.

A definition to betweenness centrality of a vertex in a graph , given by Freeman  is as follows

###### Definition 2.1 (Betweenness Centrality).

If , the betweenness centrality for is defined as

 B(x)=∑u≠v≠xδ(u,v|x)

provided where is the number of shortest - paths and is the number of shortest - paths containing . The ratio is called the pair-dependancy of the pair of vertices on .

Observe that lies on the shortest path between two vertices , iff . The number of shortest - paths passing through is given by

 σ(u,v|x)=σ(u,x)×σ(x,v) (1)
###### Definition 2.2 (Cartesian Product, ).

The Cartesian product of two graphs and , denoted by , is a graph with vertex set , where two vertices and are adjacent if and , or and . The graphs and are called factors of the product .

For any , the subgraph of induced by is called as -fiber or -layer, denoted by . Similarly, we can define -fiber or -layer. They are isomorphic to and , respectively. contains copies of and copies of . Projections are the maps from a product graph to its factors. They are weak homomorphisms in the sense that they respect adjacency. The two projections on namely and defined by and refer to the corresponding -, - coordinates. Thus an edge in is mapped into a single vertex by one of the projections or and into an edge by the other. If and are connected, then is also connected. Assuming isomorphic graphs are equal, Cartesian product is commutative as well as associative.

For a graph and , the degree of a vertex is denoted by , or simply . Furthermore, we denote by the minimum degree of a graph . The minimum degree is additive under Cartesian products, i.e. . Recall that the symbol denotes the set of neighbours of a vertex in a graph . Thus .

###### Definition 2.3 (Cartesian product of several graphs,).

The Cartesian product of the graphs is defined on the -tuples , where in such a way that two -tuples and are adjacent if there exists an index such that and for . The -tuples are called coordinate vectors, and the are the coordinates.

The Cartesian product of -factors is briefly denoted as . The Cartesian product of a graph is donoted as . It is to be noted that the product is connected if and only if each of its factor is connected and the diameter of the product is given by, .

The following proposition shows that the distance between two vertices in the product graph is the sum of the distance between their projections in the factor graphs.

###### Lemma 2.1.

 If and are vertices of a Cartesian product , then

 dG□H[(g,h),(g′,h′)]=dG(g,g′)+dH(h,h′) (2)

This can be generalized to the following lemma.

###### Lemma 2.2 (Distance lemma).

 Let be the Cartesian product of connected graphs, and let and be vertices of . Then

 dG(g,g′)=k∑i=1dGi(gi,g′i)

Lemma 2.1 implies that . In other words, restricted to is . It means that every shortest path in a -fiber ia also a shortest path in . Subgraphs with this property are called isometric. That is, a subgraph of a graph is isometric in if for all . It can be easily seen that -fibers (-fibers) are isometric subgraphs of . Every shortest -path between two vertices of one and the same fiber or is already in that fiber. Such subgraphs are called convex. A subgraph of a graph is convex in if every shortest -path between vertices of is already in .

###### Lemma 2.3.

 Let and be connected graphs. Then all -fibers and -fibers are convex subgraphs of .

For a connected graph and , the interval between and is defined as the set of vertices that lie on shortest - paths; that is,

 IG(u,v)={w∈G:d(u,v)=d(u,w)+d(w,v)}
###### Proposition 2.1.

 Let and be two vertices of , then the vertex lies in if and only if and .

It can be generalized as follows.

###### Proposition 2.2.

Let . Let , and are any three vertices in . Then lies in the shortest path of and if and only if .

### 2.1 The betweenness centrality of vertices in Cartesian product of two graphs

The following proposition shows how the number of geodesics between two vertices and in a product graph is related to the the number of geodesics between their projections in the factor graphs.

###### Proposition 2.3.

If and are vertices of , then the number of shortest paths, , between them in is given by

 σG□H[(g,h),(g′,h′)]=σG(g,g′)×σH(h,h′)×(dG(g,g′)+dH(h,h′)dG(g,g′)) (3)
###### Proof.

Consider the vertices and in . Let denote the distance between and in . Suppose there exists unique shortest paths between and . Every shortest path from to is a sequence of edges and the image of each edge is an edge lying between and or and under the projections and . Let a sequence of edges in the - path in the product makes a sequence of edges in and a sequence of edges in so that . Since and are the same for any - path, the number of shortest paths between and is the number of ways of selecting edges from edges, which is . If there exists shortest paths between and in and shortest paths between and in , then corresponding to each pair there exists shortest paths between and in .
Therefore,
For brevity, we may write

###### Corollary 2.1.

If and are geodetic graphs, then the number of shortest paths between and in is given by

 σG□H[(g,h),(g′,h′)]=(dG(g,g′)+dH(h,h′)dG(g,g′))

By the associativity of , equation 3 can be generalized as

###### Proposition 2.4.

Let . If , are two vertices in such that , and , then

 σG(u,v)=σ1σ2…σn(dd1)(d−d1d2)(d−d1−d2d3)…1
###### Proposition 2.5.

Let , and are any three vertices in . Then

 σG□H(v1,v2|v3)=σG□H(v1,v3)×σG□H(v3,v2) (4)
###### Theorem 2.1.

If , be any distinct vertices in then the betweenness centrality of in is given by

 BG□H(x)=∑u≠v≠xδG□H(u,v|x)

where

 δG□H(u,v|x)=δG(g,g′|g0)×δH(h,h′|h0))×d1×d2d

where

 d1=(dG(g,g0)+dH(h,h0)dG(g,g0)),d2=(dG(g0,g′)+dH(h0,h′)dG(g0,g′)),d=(dG(g,g′)+dH(h,h′)dG(g,g′))

###### Proof.

The result follows from the definition of betweenness centrality and from equations 1-6
Hence

 δG□H(u,v|x)=σG□H(u,v|x)σG□H(u,v)=σG(g,g′|g0)σG(g,g′)×σH(h,h′|h0)σH(h,h′)×d1×d2d

## 3 Wiener index of a graph

The Wiener index  of a graph , denoted by is the sum of the distances between all (unordered) pairs of vertices of . That is,

 W(G)=∑i

or,

 W(G)=12∑u,v∈V(G)d(u,v)

Wiener index is also named total status or total distance of a graph. The Wiener index of Cartesian product [30, 31, 32] of two graphs and is given by

 W(G□H)=|G|2W(H)+|H|2W(G) (5)

It can be extended to

 W(□ni=1Gi)=n∑i=1(W(Gi)∏j≠i|Gj|2) (6)

The betweenness centrality of is given by

 ∑v∈V(G)B(v)=W(G)−(|G|2) (7)

The average distance of a graph , denoted by is given by

 μ(G)=Total distanceNo. of distinct pairs=W(G)(|G|2)

#### 3.0.1 Grid graphs

Grid graphs are the cartesian product of path graphs. represents a rectangular grid . If and are any two vertices of , then and where or .

Consider the rectangular grid . Let denotes the vertex where . Consider the paths and passing through . They divide the rectangular grid into 4 parts namely sharing their common sides. Figure3.1. Now the pairs of vertices in the diagonal regions and contribute to the betweenness centrality of . Hence

where and belongs the diagonal quadrants and then .

###### Note 3.1.

The number of geodesics of length in the grid from to can be obtained from the -ary de Bruijn sequence where ( )

### 3.1 Hamming graphs

Hamming graphs are Cartesian products of complete graphs. If is a Hamming graph, then for some and . The vertices of can be labeled with vector where . Two vertices of are adjacent if the corresponding tuples differ in precisely one coordinate. The distance (named Hamming distance) between two vertices and denoted by is the number of positions in which the two vectors differ.

Hypercubes are Cartesian product of complete graphs . An -dimensional hypercube (or -cube) denoted by is given by, . It can also be defined recursively, . Hypercubes are important classes of graphs having many interesting structural properties. The number of geodesics between is given by . For a connected graph , the condition “ induces a -dimensional hypercube for any two vertices and of implies that ” is a Hamming graph .

###### Lemma 3.1.

 A graph is a nontrivial subgraph of the Cartesisn product of graphs if and only if is a nontrivial subgraph of the Cartesian product of two complete graphs.

###### Proposition 3.1.

Let be the Hamming graph . Then the betweenness centrality of is given by

 B(v)=12r∏i=1ni[r−1−r∑i=11ni]+12 (8)
###### Proof.

Let . Since is vertex transitive, from equations 6 and 7,

 W(H) =r∑i=1W(Kni)r∏j=1,j≠i|Knj|2=r∑i=1(ni2)r∏j=1,j≠in2j =12(r∏i=1ni)2[r−r∑i=11ni] BH(v) =W(H)−(|H|2)|H| =12r∏i=1ni[r−1−r∑i=11ni]+12

###### Corollary 3.1.

If , and are complete graphs, then

for

 B(v)=(p−1)(q−1)2

for

 B(v)=12[2pqr−(pq+pr+qr)+1]
###### Corollary 3.2.

If , then

 B(v)=12[(r−1)nr−rnr−1+1]

when , , the -cube, then

 B(v)=(r−2)2r−2+12

#### 3.1.1 Product of cycles

For the cycle , when is even, and when is odd. Product of cycles is vertex transitive. Therefore from equations 6 and7 we get

for

 W(□ni=1Cni)==18(r∏i=1ni)2r∑i=1ni

and for

 W(□ni=1Cni)==18(r∏i=1ni)2r∑i=1(ni−1ni)
###### Proposition 3.2.

If is the Cartesian product of even cycles.
ie. , , then for

 B(v)=18[r∏i=1nir∑i=1ni−4(r∏i=1ni−1)]

if ,

 B(v)=2r−2r∏i=1ki[r∑i=1ki−2]+12
###### Proposition 3.3.

If is the Cartesian product of odd cycles.
ie. , , then for

 B(v)=18[r∏i=1nir∑i=1(ni−1ni)−4(r∏i=1ni−1)]
###### Corollary 3.3.

Consider two cycles and . Let then

 B(v)=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩18(mn−1)(m+n−4),whenmandnareodd18(mn2+m(m−4)n+4),whenmandnareeven18[mn2+(m2−4m−1)n+4],whenmoddandniseven.

In another form,

 B(v)=⎧⎪ ⎪⎨⎪ ⎪⎩k1k2(k1+k2)+(k12)+(k22)when m=2k1+1,n=2k2+1k1k2(k1+k2−2)+12when m=2k1,n=2k2k1k2(k1+k2−1)+12(k2−1)2when m=2k1+1,n=2k2

## 4 Conclusion

A composite graph can be constructed by applying different graph oparations on smaller graphs and hence many of the structural properties of the composite graphs can be studied from its constituent smaller subgraphs. Here we tried to find the betweenness centrality of Cartesian product of graphs. This can be extended to other products also.

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