Abstract
This paper presents the idea of measuring the formal impact of elements of a communication graph structure consisting of nodes and arcs on its entirety or subparts. Arcs and nodes, depending on the context, can be assigned different interpretations. E.g. in game theory its nodes may represent the players, often referred to as policy makers and arcs symbolize the relationships between them. In another context, however, nodes and arcs of the graph represent elements of technical infrastructure, e.g. a computer. The graph representing the tested relationships is called the communication graph and the influence of the elements on the entire graph (or its subpart) is referred to as power of the element. Taking into account the power of nodes and connections creates so-called incidence-power matrix more completely than the one formerly describing the communication graph.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
It could be for example: traditional mail, e-mail, phone calls, personal meeting, etc. In such a case connections' weights can be modified by adding for example different values for to differentiate connections (and different weights of nodes).
- 2.
As we will see later, this condition is not necessary for games with graph representing communication between players.
- 3.
In Turnovec et al. (2008) one can find introduction of power indices without games theory but based on concept of permutations and their probability.
- 4.
We use the | . | operator to denote the cardinality of a finite set.
References
Alonso-Meijide, J.M., Alvares-Mozos, M., Fiestras-Janeiro, M.G.: Values of games with graph restricted communication and a priori unions. Math. Soc. Sci. 58, 202–213 (2009)
Aumann, R.J., Dreze, J.: Cooperative games with coalitional structures. Int. J. Game Theory 3, 217–237 (1974)
Banzhaf III, J.F.: Weighted voting doesn’t work: and mathematical analysis. Rutgers Law Rev. 19, 317–343 (1965)
Hamiache, G.: A value with incomplete communication. Games Econ. Behav. 26, 59–78 (1999)
Hamiache, G.: A matrix approach to TU games with coalition and communication structures. Soc. Choice Welfares 38, 85–100 (2012)
Mercik, J.: On a priori evaluation of power of veto. In: Herrera-Viedma, E., García-Lapresta, J.L., Kacprzyk, J., Fedrizzi, M., Nurmi, H., Zadrożny, S. (eds.) Consensual Processes. STUDFUZZ, vol. 267, pp. 145–156. Springer, Heidelberg (2011)
Mercik, J.: Classification of committees with vetoes and conditions for the stability of power indices. Neurocomputing 149(Part C), 1143–1148 (2015)
Myerson, R.B.: Graphs and cooperation in games. Math. Oper. Res. 2, 225–229 (1977)
Owen, G.: Values of games with a priori unions. In: Henn, R., Moeschlin, O. (eds.) Mathematical Economics and Game Theory. LNEMS, vol. 141, pp. 76–88. Springer, Heidelberg (1977)
Rosenthal, E.C.: Communication and its costs in graph-restricted games. Soc. Netw. 10, 275–286 (1988)
Shapley, L.S., Shubik, M.: A method of evaluating the distribution of power in a committee system. Am. Polit. Sci. Rev. 48(3), 787–792 (1954)
Shapley, L.S.: A value for n-person games. In: Kuhn, H.W., Tucker, A.W. (eds.) Contributions to the Theory of Games, vol. II. Annals of Mathematical Studies, vol. 28, pp. 307–317 (1953)
Turnovec, F., Mercik, J., Mazurkiewicz, M.: Power indices methodology: decisiveness, pivots, and swings. In: Braham, M., Steffen, F. (eds.) Power, Freedom, and Voting, pp. 23–37. Springer, Berlin (2008)
Vasques-Brage, M., Garcia-Jurado, I., Carreras, F.: The Owen value applied to games with graph-restricted communication. Games Econ. Behav. 12, 42–53 (1996)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mercik, J. (2016). Formal a Priori Power Analysis of Elements of a Communication Graph. In: Nguyen, N.T., Trawiński, B., Fujita, H., Hong, TP. (eds) Intelligent Information and Database Systems. ACIIDS 2016. Lecture Notes in Computer Science(), vol 9621. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49381-6_39
Download citation
DOI: https://doi.org/10.1007/978-3-662-49381-6_39
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-49380-9
Online ISBN: 978-3-662-49381-6
eBook Packages: Computer ScienceComputer Science (R0)