Extensions Of Closeness Centrality?

Rong YangDepartment of Mathematics

and Computer ScienceWestern Kentucky University

Bowling Green, KY 42101Rong.Yang@wku.edu

Leyla ZhuhadarDepartment of Computer Engineering

and Computer ScienceUniversity of LouisvilleLouisville, KY 40292

l0zhuh01@louisville.edu

ABSTRACTThe concept of vertex centrality has long been studied in or-der to help understand the structure and dynamics of com-plex networks. It has found wide applicability in practicalas well as theoretical areas. Closeness centrality is one of thefundamental approaches to centrality, but one difficulty withusing it is that it degenerates for disconnected graphs. Somealternate versions of closeness centrality have been proposedwhich rectify this problem. This paper points out that theseare not true extensions of closeness centrality in the sensethat they do not rank the vertices of connected graphs inthe same way that closeness centrality does.

Keywordsgraph structure, closeness, centrality

1. INTRODUCTIONNetwork analysts have long been concerned with ideas of

centrality as a mechanism for understanding graph structureand how it influences systems modeled with graphs. Anytype of centrality attempts to measure how central (in somesense) a vertex is in its containing graph, and many notionsof centrality have been proposed for various purposes. Threeof the most important centrality measures were introducedby Freeman in a seminal paper [5], and all three can berelated to concerns about communication. Degree central-ity, which essentially just measures the degree of a vertex,takes the point of view that the degree of a vertex servesas a proxy for the amount of communication activity mightbe involved in. Betweenness centrality uses the geodesics(shortest paths) passing through a vertex to rank its impor-tance in being able to control the communication of others.Closeness centrality is concerned with the distance from avertex to the other vertices, and gives an indication of howwell a vertex can protect its communications from controlby others. Other significant approaches to centrality includeBonacich’s eigenvector centrality [1, 2], which refines degree

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centrality, and Newman’s random walk attack [8] which re-laxes the geodesic restraint in betweenness centrality.Applications of these ideas are quite numerous. Among

the most important is the PageRank algorithm [3], basedon eigenvector centrality, which is the basis for the popularGoogle technology and the detection of community structurein networks [6].A long-standing disadvantage of closeness centrality has

been the fact that it is only applicable to connected graphs.This is because the classical definition of closeness centralityis based on the sum of all the distances between a particularvertex and all the other vertices. Since in a disconnectedgraph some vertices will be at infinite distance from thatparticular vertex, the resulting sum will be infinite for allvertices, which leads to a totally uninformative centralityindex. Some recent work [7, 4] has proposed some differ-ent approaches to closeness centrality which can be appliedto disconnected graphs. It is the purpose of this paper todemonstrate that these newer approaches yield fundamen-tally different centrality results than the classical definitionand so cannot be considered to be extensions of it.

2. TERMINOLOGYAll graphs considered are finite and simple (no loops, no

multiple edges). A graph is represented as an ordered pairG = (V, E) where V is the set of vertices of G and E isthe set of edges of G. The number of vertices of G is theorder of G and the number of edges of G is its size. Agraph with order p and size q is referred to as a (p, q)-graph.The shortest path between two vertices is called a geodesicand the length of such a path is called the geodesic distancebetween the two vertices, or simply the distance betweenthe vertices. The distance matrix of G is the n × n matrixD = D(G) = DG whose (i, j)-entry is the distance betweenvertex i and vertex j. (Of course, the distance from a vertexto itself is 0.) If v and w are two vertices of G, the distancebetween v and w is denoted by d(v, w).

3. THE ALTERNATE DEFINITIONSThe traditional definition of the closeness centrality of a

vertex v from [5] is

CC(v) =1∑

w∈V −{v} d(v, w). (1)

If G is disconnected, at least one of the terms in the sum-mation is ∞, so the summation is ∞ and the centrality ofevery point is 0, which is not very informative. The usual

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work around for this dilemma is to compute the closenesscentrality for each vertex with respect to the connected com-ponent which contains it.In [7], Latora and Marchiori introduce a notion of effi-

ciency for networks. In the process of this development,they are led to the sum

C1(v) =∑

w∈V −{v}

1d(v, w) . (2)

Although Latora and Marchiori never use the term “close-ness” in their paper and certainly do not claim that C1 isa centrality measure, in [4], Dangalchev refers to it as “anew definition for point closeness”. During his work (whichis really focused on measuring network vulnerability), heproposes another related definition of closeness:

C2(v) =∑

w∈V −{v}

12d(v,w) . (3)

C2 is advocated primarily because of the ease of compu-tation compared to C1. Both C1 and C2 have the advan-tage of elegantly yielding reasonable results for disconnectedgraphs. That is to say, they yield measures that are not 0on all the vertices because, in both cases, any infinite dis-tances simply make a contribution of 0 while finite distancescontribute nonzero amounts.

4. A NEGATIVE RESULTThe two alternate definitions of closeness centrality given

should not be considered as “extensions of the definition ofcloseness centrality to disconnected graphs” because theydo not always rank the vertices of connected graphs in thesame order as the classical definition of closeness centrality(and note that their authors do not claim that they do). Forexample, in the complete bipartite graph shown in Figure 1,we have Cc(1) = 1

10 > Cc(2) = Cc(3) = 111 , while C1(2) =

C1(3) = 256 > C1(1) = 4 and C2(1) = C2(2) = C2(3) = 2.

Figure 1: A Complete Bipartite Graph

Both C1 and C2 are defined using the same general ap-proach. In each case we have a function F : Z+ → R+ andthe associated “centrality measure” which we will denote byCF is given by

CF (v) =∑

wεV −{v}F (d (v, w)) (4)

The above example shows that both C1 and C2 fail to al-ways rank vertices in the same order as closeness centrality.The question then arises as to whether a different choice forthe function F would allow us to construct a genuine ex-tension of closeness centrality to all finite graphs, includingdisconnected ones.

5. CONCLUSIONIt has been shown that two proposed alternatives to close-

ness centrality do not, in fact provide extensions of closenesscentrality to disconnected graphs. In future work we willestablish that no extension of the type considered here ispossible.

6. REFERENCES[1] Bonacich, P. Technique for analyzing overlappingmemberships. Sociological Methodology 4 (1972),176–185.

[2] Bonacich, P. Power and centrality: A family ofmeasures. American Journal of Sociology 92, 5 (1987),1170–1182.

[3] Brin, S., and Page, L. The anatomy of a large scalehyper-textual web search engine, in proc. of 7th WWWConference, 1998.

[4] Dangalchev, C. Residual closeness in networks.Physica A: Statistical Mechanics and its Applications365, 2 (2006), 556–564.

[5] Freeman, L. Centrality in social networks conceptualclarification. Social networks 1, 3 (1979), 215–239.

[6] Girvan, M., and Newman, M. Community structurein social and biological networks. Proceedings of theNational Academy of Sciences of the United States ofAmerica 99, 12 (2002), 7821.

[7] Latora, V., and Marchiori, M. Efficient behavior ofsmall-world networks. Physical Review Letters 87, 19(2001), 198701.

[8] Newman, M. A measure of betweenness centralitybased on random walks. Social networks 27, 1 (2005),39–54.

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