Abstract
This paper discusses the so-called multigraph, in which multiple connections (arcs) between two given nodes and loops are possible. The authors present the concept of using Shapley values to analyse both elements (nodes and arcs) of a multigraph. The results obtained allow to evaluate the importance of a given element (node or arc) as an element of a whole structure of the graph. The authors proposed a new cooperative game solution and Shapley value which may determine the evaluation of multigraph elements (which may, among others, determine the cost of its use or the volume of flows).
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Notes
- 1.
There are several claims regarding modified assumptions. E.g., Young (1985) demonstrated that Shapley value is the only value that satisfies effectiveness, symmetry, and strong monotonicity, van den Brink (2001), by modifying contributions to the coalition, Myerson (1977), with balanced contributions to the coalition.
- 2.
In a directed graph such node is source or target node.
- 3.
\( card\left\{ B \right\}\, or\, card\left( B \right) \) represents cardinality of a set B.
- 4.
Note that the graph structure is not explicitly used here, but it can be assumed that this structure affects the probabilities of a given coalition of players (vertices).
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Forlicz, S., Mercik, J., Stach, I., Ramsey, D. (2018). The Shapley Value for Multigraphs. In: Nguyen, N., Pimenidis, E., Khan, Z., Trawiński, B. (eds) Computational Collective Intelligence. ICCCI 2018. Lecture Notes in Computer Science(), vol 11056. Springer, Cham. https://doi.org/10.1007/978-3-319-98446-9_20
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