Cmjv02i06p0790

J. Comp. & Math. Sci. Vol.2 (6), 790-797 (2011)

Betweeness-Centrality of Grid Networks INDRA RAJASINGH, BHARATI RAJAN and FLORENCE ISIDO. D. Department of Mathematics, Loyola College, Chennai 600 034, India ABSTRACT Estimating the importance or centrality of the nodes in large networks has recently attracted increased interest. Betweenness is one of the most important centrality indices, which basically counts the number of shortest paths going through a node.

Betweenness has been used in diverse applications such as social network analysis or route planning. In this paper we find a formula to obtain Betweenness-Centrality for grids.

Keywords: Social networks, betweeness-centrality, grid.

1. INTRODUCTION In social network analysis, graphs are used to model relationships between actors or participants in a social setting.

Each node or vertex in the graph represents a participant or actor. Each link or edge represents a connection or relationship between two participants. A variety of graph1. algorithms have been developed to analyze the structure of social networks and to assess the roles or importance of the individual actors. Since the September 11 bombing of the World Trade Center, social network analysis has emerged as a potential vehicle for modeling and analyzing the structure of terrorist networks11, 21. There are a variety of measures to assess the “importance” or centrality of each actor in a social network22.

The most popular of these centrality measures requires the computation or

enumeration of shortest paths between all pairs of nodes in the graph. Such computations can be time consuming in large graphs. Moreover, they may become problematic even in more moderately-sized networks when changing data or “what-if” scenario analysis warrants frequent recomputation.

2. AN OVERVIEW OF THE PAPER An essential tool for the analysis of social networks is centrality indices defined on the vertices of the graph2, 19, 10. They are designed to rank the actors according to their position in the network and are interpreted as the prominence of actors embedded in a social structure. Many centrality indices are based on shortest paths linking pairs of actors, measuring, the average distance from other actors, or the ratio of shortest paths an

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

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Indra Rajasingh, et al., J. Comp. & Math. Sci. Vol.2 (6), 790-797 (2011)

actor lies on. Many network-analytic studies rely at least in part on an evaluation of these indices.

With the increasing practicality of electronic data collection and, of course, the advent of the Web, there is a likewise increasing demand for the computation of centrality indices on networks with thousands of actors. Several notions of centrality originating from social network analysis are in use to determine the structural prominence of Web pages15, 4, 3.

However, there is a bottleneck in existing implementations, due to the particularly important betweeness-centrality which makes comparative index10,1, centrality analyses of networks with more than a few hundred actors prohibitive. As a remedy, network analysts are now suggesting simpler indices, for instance based only on linkages between the neighbors of each actor9, to at least obtain rough approximations of Betweennesscentrality.

A shortest path between two vertices is called a geodesic. In this paper we derive a formula to find the number of geodesics between every pair of vertices of a grid. The centrality indices relevant here are defined in Section III. An elegant formula for finding betweenness-centrality of grid is derived in Section IV. 3. PRELIMINARIES Consider a graph , , where V is the set of vertices representing actors or nodes in the social network, and E, the set of edges representing the relationships between the actors. The number of vertices and edges are denoted by n and m, respectively. The

graphs can be directed or undirected. A path from vertex s to t is defined as a sequence of edges , , 0 1, where and . We use d(s, t) to denote the shortest distance between vertices s and t in G. Let us denote the total number of geodesics between vertices s and t by , and the number of geodesics between s and t passing through vertex v by . In this paper, we extend the definition to any subset of .

We begin with the definitions of certain centrality parameters10, 20.

Definition 1: The degree centrality DC of a vertex v is simply the degree of v in G. This is a simple local measure, based on the notion of neighborhood. This index is useful in case of static graphs, for situations when we are interested in finding vertices that have the most direct connections to other vertices.

Definition 2: Stress centrality20 is a metric based on shortest path counts. It is defined as

Intuitively, this metric deals with the work done by each vertex in a communication network. The number of geodesics that contain an element v will give an estimate of the amount of stress a vertex v is under, assuming that communications will be carried out through geodesics all the time.

This index can be calculated using a variant of the all-pair shortest-paths algorithm that calculates and stores all geodesics between any pair of vertices.

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

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Indra Rajasingh, et al., J. Comp. & Math. Sci. Vol.2 (6), 790-797 (2011)

Betweenness-Centrality is another shortest path enumeration-based metric introduced by Freeman in10.

Betweeness-Centrality Betweenness-Centrality of a vertex v is defined as

where

ŕłžŕł&#x; . ŕłžŕł&#x;

This metric can be thought of as normalized stress centrality. BetweennessCentrality of a vertex measures the control a vertex has over communication in the network, and can be used to identify key actors in the network. High centrality indices indicate that a vertex can reach other vertices on relatively short paths, or that a vertex lies on a considerable fraction of shortest paths connecting pairs of other vertices.

This index has been extensively used in recent years for analysis of social as well as other large scale complex networks.

Some applications include biological networks13, 18, 8, study of sexual networks and AIDS17, identifying key actors in terrorist networks16,6,organizational behavior5, supply chain management7 and transportation networks12.

multigrid network is perfectly suitable for solving discretized elliptic differential equations by assigning each grid point to its counterpart in the array, as the iterative methods for solving the resulting linear systems require only nearest neighbor grid point iteration14.

In this paper we determine the betweenness-centrality of any vertex in a grid. We also extend the definition of betweenness-centrality of a vertex to a set S of vertices in the grid and determine the same when S induces an edge and also a cycle of length 4.

Definition 3: Betweenness-Centrality of a set S‍ Řżâ€Źáˆş áˆť is defined as

∑ , \ where ŕłžŕł&#x;áˆşŕł„áˆť and

ŕłžŕł&#x;

to be the number of geodesics from s to t, passing through at least one vertex of S. Two Dimensional Grid

Let denote a path on n vertices. For, m , n ≼ 2, is defined as the two dimensional grid with m rows and n columns. It is denoted by .

4. MAIN RESULTS

Remark: The vertex of ! in the ith row and jth column is denoted by , " , 1 #, 1 " . For convenience we denote the number of geodesics in ! between vertices , " and %, & as , and that between , " and %, & passing through ', ( as , ', (

The grid and multigrid structures of interconnection networks provide a very useful communication pattern to implement a lot of algorithms in many parallel computing systems. In particular, the

Theorem 1: The number of geodesics between the vertices , " and %, & in the grid ! is given by & * " + % * , ) ,. % *

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

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Indra Rajasingh, et al., J. Comp. & Math. Sci. Vol.2 (6), 790-797 (2011)

Proof: A geodesic between the vertices , " and %, & is of length % * + & * " and is composed of % * vertical moves and & * " horizontal moves in ! . The !

number of such geodesics is ! ! . & * " + % * Thus , ) ,â Ť % * Theorem 2: The number of geodesics between vertices , " and %, & of ! passing through a vertex ', ( is given by , ', ( ( * " + ' * & * ( + % * ' ) ,) , % * ' ' * Proof: We have , ', ( = ,"# "#, = ( * " + ' * & * ( + % * ' ) ,) ,â Ť % * ' ' * We need the following notations to determine the Betweenness-Centrality measures of a grid ! .

Notation 1: For - . ! , let !/-0, the subgrid induced by - , be denoted by -.

Let 12, 3, 4, 5 . ! where 2 ', ( , , 1 , 1, 1 , 1, .

Let 6, , , 7, , $ , , % be eight subgrids of ! as shown in Figure 1. In other words, , : 1

1, 1

1 , , : 1

1, 2

, , : 2

, 1

1 ,

, : 2

, 2

, ଵ , : 1

1,

1 , ଶ , : 2 , 1 , ଷ , : 1, 1 1 , ସ , :

1, 1

.

A

a,b

E3

a,b+1

Z a+1,b

C

B

E1

E4 a+1,b+1

E2

D

Figure 1

Lemma 1: Let 8 = 234 2 be a cycle in the grid ! where 2 ', ( , 3 ', ( + 1 ,4 ' + 1, ( + 1 , ' + 1, ( , for some a, b, 1 ' #, 1 ( . Then 8 3 + , where s and t are any pair of vertices in the grid.

Proof: The geodesics passing through 2 are partitioned into paths that pass through 3 and those that pass through . Thus the geodesics passing through 3 and together contain all the paths passing through 2.

Similarly all the geodesics passing through 4 are nothing but paths passing through 3 or . So 8 9 2 + 3 + 4 + : * 2 * 4 .

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

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Indra Rajasingh, et al., J. Comp. & Math. Sci. Vol.2 (6), 790-797 (2011)

Hence 8 3 + â Ť

A

B

E1

We now state the key result of this paper.

Theorem 3: Let ! be a grid and 8 a cycle as in Lemma 1.Then the BetweennessCentrality of Z is given by ఙೞŕł&#x; áˆşŕŻ“áˆť ఙೞŕł&#x; áˆşŕŻ“áˆť

∑ ௌ‍×?â€ŹŕŽť ௌ‍×?â€ŹŕŽşâ€Ť×Ťâ€ŹŕŽžŕ°­ ఙೞŕł&#x; ఙೞŕł&#x; ௧‍×?â€ŹŕŽž â€Ť×Ťâ€ŹŕŽźâ€Ť×Ťâ€ŹŕŽž ௧‍×?â€ŹŕŽžŕ°° â€Ť×Ťâ€ŹŕŽ˝â€Ť×Ťâ€ŹŕŽžŕ°Ž ŕ°Ž ŕ°Ż ఙೞŕł&#x; áˆşŕŻ“áˆť ఙೞŕł&#x; áˆşŕŻ“áˆť ఙೞŕł&#x; áˆşŕŻ“áˆť âˆ‘ŕŻŚâ€Ť×?â€ŹŕŽžŕ°­â€Ť×Ťâ€ŹŕŽžŕ°°

âˆ‘ŕŻŚâ€Ť×?â€ŹŕŽžŕ°°

âˆ‘ŕŻŚâ€Ť×?â€ŹŕŽ˝â€Ť×Ťâ€ŹŕŽžŕ°Ž ఙೞŕł&#x; ఙೞŕł&#x; ఙೞŕł&#x; ௧‍×?â€ŹŕŽźâ€Ť×Ťâ€ŹŕŽžŕ°Ż ௧‍×?â€ŹŕŽžŕ°Ž ௧‍×?â€ŹŕŽžŕ°Ż

∑

where 6, , , 7, , $ , and % are as defined in Notation 1.

Proof: It is clear that 8 0 if , ଵ , , , , ଷ , , , ଵ , , , ସ , , , ସ , , , ଶ , , , ଶ , or , ଷ .

On the other hand 8 contributes to 8 whenever

; 6 < , ; % < 7 < $ , ; , ; $ < < , ; < % , ; < , ; % , ; $ and ; 7 < $ , ; .

Thus ŕ°&#x2122;ŕł&#x17E;ŕł&#x; á&#x2C6;şŕŻ&#x201C;á&#x2C6;ť ŕ°&#x2122;ŕł&#x17E;ŕł&#x; á&#x2C6;şŕŻ&#x201C;á&#x2C6;ť â&#x2C6;&#x2018; ௌâ&#x20AC;Ť×?â&#x20AC;ŹŕŽť

ௌâ&#x20AC;Ť×?â&#x20AC;ŹŕŽşâ&#x20AC;Ť×Ťâ&#x20AC;ŹŕŽžŕ°­ ŕ°&#x2122;ŕł&#x17E;ŕł&#x; ௧â&#x20AC;Ť×?â&#x20AC;ŹŕŽžŕ°Ž â&#x20AC;Ť×Ťâ&#x20AC;ŹŕŽźâ&#x20AC;Ť×Ťâ&#x20AC;ŹŕŽžŕ°Ż ŕ°&#x2122;ŕł&#x17E;ŕł&#x; ௧â&#x20AC;Ť×?â&#x20AC;ŹŕŽžŕ°° â&#x20AC;Ť×Ťâ&#x20AC;ŹŕŽ˝â&#x20AC;Ť×Ťâ&#x20AC;ŹŕŽžŕ°Ž ŕ°&#x2122; á&#x2C6;şŕŻ&#x201C;á&#x2C6;ť ŕ°&#x2122; á&#x2C6;şŕŻ&#x201C;á&#x2C6;ť ŕ°&#x2122; á&#x2C6;şŕŻ&#x201C;á&#x2C6;ť â&#x2C6;&#x2018;ௌâ&#x20AC;Ť×?â&#x20AC;ŹŕŽžŕ°­â&#x20AC;Ť×Ťâ&#x20AC;ŹŕŽžŕ°° ŕł&#x17E;ŕł&#x; â&#x2C6;&#x2018;ௌâ&#x20AC;Ť×?â&#x20AC;ŹŕŽžŕ°° ŕł&#x17E;ŕł&#x; â&#x2C6;&#x2018;ௌâ&#x20AC;Ť×?â&#x20AC;ŹŕŽ˝â&#x20AC;Ť×Ťâ&#x20AC;ŹŕŽžŕ°Ž ŕł&#x17E;ŕł&#x; â Ť ŕ°&#x2122;ŕł&#x17E;ŕł&#x; ŕ°&#x2122;ŕł&#x17E;ŕł&#x; ŕ°&#x2122;ŕł&#x17E;ŕł&#x; ௧â&#x20AC;Ť×?â&#x20AC;ŹŕŽžŕ°Ž ௧â&#x20AC;Ť×?â&#x20AC;ŹŕŽžŕ°Ż ௧â&#x20AC;Ť×?â&#x20AC;ŹŕŽźâ&#x20AC;Ť×Ťâ&#x20AC;ŹŕŽžŕ°Ż

â&#x2C6;&#x2018;

Illustration

E3

Z

E4

E2 C

D

& &

Betweenness-centrality of Z in !% % is 8 =% + + $ + % + + + * +

'

(

'

$

)

$ '

'

(

' + + + % + 4?+= + % + $ + % + ? $

'

$ $ ) '

+=% + + $ + + + * + + % + )

$ $

'

$ $ 4?+=* + $ + + ?+=% + + $ + + + )

?= . *

Notation 2: If 12, 35, where 2 ', ( , 3 ', ( + 1 for some a, b,1 ' #, 1 ( then the subgrids 6, , , 7, and % are as in Notation 1, whereas

1 , ( , 1 ' * 15 and

$ 1 , ( , ' + 2 #5.

Notation 3: If 125, where 2 ', ( for some a, b, 1 ' #, 1 ( , then the subgrids 6, , , 7, and $ are as in Notation 2, whereas

1 ', " , 1 " ( * 15 and

% 1 ', " , ( + 1 " 5.

See Figure 2.

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

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Indra Rajasingh, et al., J. Comp. & Math. Sci. Vol.2 (6), 790-797 (2011)

A

E1

E4

E3

C

B

E2

D

Figure 2

Lemma 2: Let M mxn be a grid and A 2, 3 be an edge where 2 ', ( , 3 ', ( + 1 , for some a, b, 1 ' #, 1 ( . The total number of geodesics passing through a and b is

3. Similarly 3 contains the total number of geodesics passing through only 3 and 2.

Therefore Ď&#x192;+, Îą + Ď&#x192;+, β contains the total number of geodesics passing through only 2, only 3 and twice that of 2 and 3. Hence. Ď&#x192;+, e Ď&#x192;+, Îą + Ď&#x192;+, β * Ď&#x192;+, ιβ â Ť We use the above two lemmas in the following theorem.

Theorem 4: Let ! be a grid and e = (2, 3) be an edge where 2 ', ( , 3 ', ( + 1 for some a, b,1 ' #, 1 ( .Then the Betweenness-Centrality of e is given by

ௌ௧

1

=â&#x2C6;&#x2018;

where , " and %, & . â Ť

â&#x2C6;&#x2018;ௌâ&#x20AC;Ť×?â&#x20AC;ŹŕŽžŕ°­â&#x20AC;Ť×Ťâ&#x20AC;ŹŕŽžŕ°°

Lemma 3:Let M mxn be a grid and A 2, 3 be an edge where 2 ', ( , 3 ', ( + 1 , for some a, b, 1 ' #, 1 ( . Then the total number of geodesics passing through the edge e between any pair of vertices s and t in the grid is given by

where 6, , , 7, , $ , and % are as defined in Notation 2.

Ď&#x192;+, e Ď&#x192;+, Îą + Ď&#x192;+, β * Ď&#x192;+, ιβ .

Proof: By definition, Ď&#x192; st (e) is the total number of geodesics passing through atleast one of the vertices of e. In other words it is the total number of geodesics passing through either only 2, only 3, or 2 and 3.

Now 2 contains the total number of geodesics passing through only 2 and 2 and

ŕ°&#x2122;ŕł&#x17E;ŕł&#x; á&#x2C6;şŕŻ&#x2DC;á&#x2C6;ť + ௌâ&#x20AC;Ť×?â&#x20AC;ŹŕŽşâ&#x20AC;Ť×Ťâ&#x20AC;ŹŕŽžŕ°­ ŕ°&#x2122;ŕł&#x17E;ŕł&#x; ௧â&#x20AC;Ť×?â&#x20AC;ŹŕŽžŕ°° â&#x20AC;Ť×Ťâ&#x20AC;ŹŕŽ˝â&#x20AC;Ť×Ťâ&#x20AC;ŹŕŽžŕ°Ž

ŕ°&#x2122;ŕł&#x17E;ŕł&#x; á&#x2C6;şŕŻ&#x2DC;á&#x2C6;ť ŕ°&#x2122;ŕł&#x17E;ŕł&#x;

௧â&#x20AC;Ť×?â&#x20AC;ŹŕŽźâ&#x20AC;Ť×Ťâ&#x20AC;ŹŕŽžŕ°Ż

+ â&#x2C6;&#x2018;ௌâ&#x20AC;Ť×?â&#x20AC;ŹŕŽžŕ°° ௧â&#x20AC;Ť×?â&#x20AC;ŹŕŽžŕ°Ž

â&#x2C6;&#x2018;

ŕ°&#x2122;ŕł&#x17E;ŕł&#x; á&#x2C6;şŕŻ&#x2DC;á&#x2C6;ť ŕ°&#x2122;ŕł&#x17E;ŕł&#x;

ŕ°&#x2122;ŕł&#x17E;ŕł&#x; á&#x2C6;şŕŻ&#x2DC;á&#x2C6;ť ௌâ&#x20AC;Ť×?â&#x20AC;ŹŕŽť ௧â&#x20AC;Ť×?â&#x20AC;ŹŕŽžŕ°Ž â&#x20AC;Ť×Ťâ&#x20AC;ŹŕŽźâ&#x20AC;Ť×Ťâ&#x20AC;ŹŕŽžŕ°Ż ŕ°&#x2122;ŕł&#x17E;ŕł&#x;

+ â&#x2C6;&#x2018;ௌâ&#x20AC;Ť×?â&#x20AC;ŹŕŽ˝â&#x20AC;Ť×Ťâ&#x20AC;ŹŕŽžŕ°Ž

+

ŕ°&#x2122;ŕł&#x17E;ŕł&#x; á&#x2C6;şŕŻ&#x2DC;á&#x2C6;ť

௧â&#x20AC;Ť×?â&#x20AC;ŹŕŽžŕ°Ż

ŕ°&#x2122;ŕł&#x17E;ŕł&#x;

,

Theorem 5: The Betweenness-Centrality of a vertex ', ( in ! is given by =â&#x2C6;&#x2018;

ŕł&#x17E;ŕł&#x; -./ŕ°­ ŕł&#x17E;ŕł&#x; /ŕ°° .0./ŕ°Ž

+â&#x2C6;&#x2018;

ŕł&#x17E;ŕł&#x; 1 /ŕ°Ž .2./ŕ°Ż ŕł&#x17E;ŕł&#x;

+ â&#x2C6;&#x2018; /ŕ°° /ŕ°Ž

ŕł&#x17E;ŕł&#x; + ŕł&#x17E;ŕł&#x;

+ â&#x2C6;&#x2018; /ŕ°­ ./ŕ°° 2./ŕ°Ż

â&#x2C6;&#x2018; 0./ŕ°Ž /ŕ°Ż

ŕł&#x17E;ŕł&#x; ŕł&#x17E;ŕł&#x;

ŕł&#x17E;ŕł&#x; , ŕł&#x17E;ŕł&#x;

where 6, , , 7, , $ , and % are as defined in Notation 3.

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

Indra Rajasingh, et al., J. Comp. & Math. Sci. Vol.2 (6), 790-797 (2011)

5. CONCLUSION In this paper the concept of Betweenness-Centrality of a vertex has been extended to a set of vertices of . The problem becomes interesting when the set S is the centre of the graph.

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Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)