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Exponential generalised network descriptors
1. | Faculty of Civil Engineering, Architecture and Geodesy, Matice hrvatske 15, University of Split, Croatia |
2. | Faculty of Science, Bijenička cesta 30, University of Zagreb, Croatia |
3. | Faculty of Science, Rudera Boškovića 33, University of Split, Croatia |
In communication networks theory the concepts of networkness and network surplus have recently been defined. Together with transmission and betweenness centrality, they were based on the assumption of equal communication between vertices. Generalised versions of these four descriptors were presented, taking into account that communication between vertices $ u $ and $ v $ is decreasing as the distance between them is increasing. Therefore, we weight the quantity of communication by $ \lambda^{d(u,v)} $ where $ \lambda \in \left\langle0,1 \right\rangle $. Extremal values of these descriptors are analysed.
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doi: 10.1016/j.dam.2013.04.005. |
show all references
References:
[1] |
S. Antunovic, T. Kokan, T. Vojkovic and D. Vukicevic,
Generalised network descriptors, Glasnik Matematicki, 48 (2013), 211-230.
doi: 10.3336/gm.48.2.01. |
[2] |
A. L. Barabasi, Linked: How Everything is Connected to Everything Else and What It Means, Persus Publishing, Cambridge, 2002. Google Scholar |
[3] |
B. Bollobas, Modern Graph Theory, Springer, New York, 1998.
doi: 10.1007/978-1-4612-0619-4. |
[4] |
S. P. Borgatti and M. G. Everett,
A graph-theoretic perspective on centrality, Social Networks, 28 (2006), 466-484.
doi: 10.1016/j.socnet.2005.11.005. |
[5] |
U. Brandes,
A faster algorithm for betweenness centrality, J. Math. Sociol., 25 (2001), 163-177.
doi: 10.1080/0022250X.2001.9990249. |
[6] |
G. Caporossi, M. Paiva, D. Vukicevic and M. Segatto,
Centrality and betweenness: Vertex and edge decomposition of the Wiener index, MATCH Commun. Math. Comput. Chem, 68 (2012), 293-302.
|
[7] |
C. Dangalchev,
Residual closeness and generalized closeness, International Journal of Foundations of Computer Science, 22 (2011), 1939-1948.
doi: 10.1142/S0129054111009136. |
[8] |
L. Freeman,
A set of measures of centrality based on betweenness, Sociometry, 40 (1977), 35-41.
doi: 10.2307/3033543. |
[9] |
L. Freeman,
Centrality in social networks: Conceptual clarification, Social Networks, 1 (1978), 215-239.
doi: 10.1016/0378-8733(78)90021-7. |
[10] |
S. Gago, J. Coroničová Hurajová and T. Mandaras,
On decay centrality in graphs, Mathematica Scandinavica, 123 (2018), 39-50.
doi: 10.7146/math.scand.a-106210. |
[11] |
M. O. Jackson and A. Wolinsky,
A strategic model of social and economic networks, Journal of Economic Theory, 71 (1996), 44-74.
doi: 10.1006/jeth.1996.0108. |
[12] |
M. E. J. Newman, Networks: An Introduction, Oxford University Press, Oxford, 2010.
doi: 10.1093/acprof:oso/9780199206650.001.0001.![]() ![]() |
[13] |
D. Vukicevic and G. Caporossi,
Network descriptors based on betweenness centrality and transmission and their extremal values, Discrete Applied Mathematics, 161 (2013), 2678-2686.
doi: 10.1016/j.dam.2013.04.005. |

Descriptor | ||
Lower bound | Upper bound | |
broom (starting vertex) | complete graph * | |
open problem | broom (starting vertex) | |
path (end vertices) | complete graph * | |
open problem | star (center) | |
broom (starting vertex) | vertex-transitive graph | |
vertex-transitive graph | star (center) | |
broom (starting vertex) | vertex-transitive graph | |
vertex-transitive graph | star (center) | |
Descriptor | ||
Lower bound | Upper bound | |
broom (starting vertex) | complete graph * | |
open problem | broom (starting vertex) | |
path (end vertices) | complete graph * | |
open problem | star (center) | |
broom (starting vertex) | vertex-transitive graph | |
vertex-transitive graph | star (center) | |
broom (starting vertex) | vertex-transitive graph | |
vertex-transitive graph | star (center) | |
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