Glasnik Matematicki, Vol. 48, No. 2 (2013), 211230.
GENERALISED NETWORK DESCRIPTORS
Suzana Antunović, Tonći Kokan, Tanja Vojković and Damir Vukičević
Faculty of Civil Engineering, Architecture and Geodesy, University of Split, Matice hrvatske 15, 21 000 Split, Croatia
email: santunovic@gradst.hr
Faculty of Economics, University of Split, Cvite Fiskovića 5,
21000 Split , Croatia
email: tkokan@gmail.com
Department of Mathematics, University of Split, Teslina 12, 21 000 Split, Croatia
email: tanja@pmfst.hr
Department of Mathematics, University of Split, Teslina 12, 21 000 Split, Croatia
email: vukicevic@pmfst.hr
Abstract. Transmission and betweenness centrality are key concepts in communication networks theory. Based on this concept, new concepts of networkness and network surplus have recently been defined. However, all these four concepts
include unrealistic assumption about equal communication between vertices. Here, we propose more realistic assumption that the amount of communication of vertices decreases as their distance increases. We assume that amount of
communication between vertices u and v is proportional to d(u,v)^{Λ } where Λ < 0.
Taking this into account generalised versions of these four descriptors are defined. Extremal values of these descriptors are analysed.
2010 Mathematics Subject Classification.
05C35, 05C12.
Key words and phrases. Betweenness, transmission, networkness, network surplus, complex networks.
Full text (PDF) (access from subscribing institutions only)
DOI: 10.3336/gm.48.2.01
References:

J. M. Anthonisse,
The rush in a directed graph,
Stichting Mathematisch Centrum, Amsterdam, 1971, 110.

A.L. Barabási,
Linked: How everything is connected to everything else and what it means,
Persus Publishing, Cambridge, 2002.

A. Barrat, M. Barthélemy and A. Vespignani,
Dynamical processes on complex networks,
Cambridge University Press, Cambridge, 2008.
MathSciNet
CrossRef

B. Bollobás, Modern graph theory,
SpringerVerlag, New York, 1998.
MathSciNet
CrossRef

S. P. Borgatti and M. G. Everett,
A graphtheoretic perspective on centrality,
Social Networks 28 (2006), 466484.
CrossRef

U. Brandes,
A faster algorithm for betweenness centrality,
J. Math. Sociol. 25 (2001), 163177.
CrossRef

U. Brandes and T. Erlebach (eds.),
Network analysis  methodological foundations,
SpringerVerlag, Berlin, 2005.

G. Caporossi, M. Paiva, D. Vukičević and M. Segatto,
Centrality and betweenness: vertex and edge decomposition of the Wiener index, MATCH Commun. Math. Comput. Chem 68 (2012), 293302.
MathSciNet

L. Freeman,
A set of measures of centrality based on betweenness,
Sociometry 40 (1977), 3541.
CrossRef

L. Freeman,
Centrality in social networks: conceptual clarification,
Social Networks 1 (1978), 215239.
CrossRef

S. Gago, J. Hurajová and T. Madaras,
Nodes on the betweenness centrality of a graph,
Math. Slovaca 62 (2012), 112.
MathSciNet
CrossRef

M. Girvan, M. E. J. Newman,
Community structure in social and biological networks,
Proc. Natl. Acad. Sci. USA 99 (2002), 78217826.
MathSciNet
CrossRef

I. Gutman,
A property of the Wiener number and its modifications,
Indian J. Chem. 36 (1997), 128132.

I. Gutman, D. Vidović and Lj. Popović,
On graph representation of organic molecules  Cayley's plerograms vs. his kenograms,
J. Chem. Soc. Faraday Trans. 94 (1998), 857860.
CrossRef

M. E. J. Newman,
Networks. An introduction,
Oxford University Press, Oxford, 2010.
MathSciNet
CrossRef

D. Vukičević and G. Caporossi,
Network descriptors based on betweenness centrality and transmission and their extremal values,
submitted to Discrete Appl. Math.

H. Wiener,
Structural determination of paraffin boiling points,
J. Amer. Chem. Soc. 69 (1947), 1720.
CrossRef
Glasnik Matematicki Home Page