图上粘合运算的Shapley指数

doi:10.15960/j.cnki.issn.1007-6093.2024.04.012

Abstract

Abstract:

How to allocate profit reasonablely is an important issue in cooperative game research. The distribution rule based on Shapley value, proposed by Shapley, the winner of the Nobel Prize in Economics, is the commonly used one in cooperative games. The cooperative game theory on graphs greatly enriches the research methods of game theory, and Shapley value on graphs has been widely applied in node influence, community detection and link prediction of social networks. Shapley distance on graphs is suggested based on Shapley value and it can be used to measure the cost of one vertex to access another one. Analogous to Wiener index and Kirchhoff index in graph theory, a new graph parameter, namely Shapley index, is proposed. In this paper, we establish the analytical expressions of Shapley distance and Shapley index of three kinds of conglutinate graphs, such as friendship graphs, book graphs and generalized rose graphs. These empirical examples provide methodological guidance for the computation of Shapley index of other complex topological structures.

Key words: graph theory, game theory, bonding operation, Shapley distance, Shapley index

CLC Number:

O157.5

Qingfeng DONG, Zhendong GU, Qianru ZHOU, Shuming ZHOU. Shapley index of the bonding operation on graphs[J]. Operations Research Transactions, 2024, 28(4): 123-134.

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URL: https://www.ort.shu.edu.cn/EN/10.15960/j.cnki.issn.1007-6093.2024.04.012

https://www.ort.shu.edu.cn/EN/Y2024/V28/I4/123

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Shapley距离	$ d_{FR_{n}}^{Sh}=1 $	$ d_{FR_{n}}^{Sh}=2 $
顶点对个数	$ 3n $	$ 4\binom{n}{2} $

Shapley距离	$ d_{B_n}^{Sh}=1 $	$ d_{B_n}^{Sh}=\frac{3}{2} $	$ d_{B_n}^{Sh}=\frac{20}{11} $	$ d_{B_n}^{Sh}=\frac{15}{7} $
顶点对个数	$ 3n+1 $	$ 2n $	$ n^2-n $	$ n^2-n $

Shapley距离	$ 1 $	$ 1+\frac{1}{m} $	$ \frac{3}{2} $	$ 2 $	$ 2+\frac{4}{m^{2}+3m} $	$ 2+\frac{14m+4}{2m^{3}+7m^{2}} $
顶点对个数	$ 2mn $	$ n $	$ n\binom{m}{2} $	$ \binom{n}{2}\binom{m}{1}\binom{m}{1} $	$ \binom{n}{2}\binom{1}{1}\binom{m}{1} $	$ \binom{n}{2} $

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Shapley index of the bonding operation on graphs

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