Shapley value centrality (Michalak et al. 2013) is the Shapley value of
each node in a coalition game whose worth \(v(C)\) counts the nodes a
coalition \(C\) covers. Each game has an exact closed form. In game 1 a
coalition covers its members and their neighbors, and
$$SV(v) = \sum_{u \in \{v\} \cup N(v)} \frac{1}{1 + k_u}.$$
In game 2 a coalition covers its members and the nodes with at least
\(k\) neighbors in it. In game 3 it covers the nodes within
shapley_cutoff hops of it, and \(N(v)\) is replaced by the set
of nodes within that distance.
Usage
centrality_shapley_game1(x, ...)
centrality_shapley_game2(x, shapley_k = 2, ...)
centrality_shapley_game3(x, shapley_cutoff = 2, ...)Arguments
- x
Network input accepted by
centrality.- ...
Further arguments to
centrality, such asnormalized.- shapley_k
Neighbor threshold \(k\) for game 2 (default 2).
- shapley_cutoff
Hop cutoff for game 3 (default 2).
Details
The values of every game sum to the number of nodes. Game 2 with \(k = 1\) and game 3 with cutoff 1 equal game 1. Degrees exclude self-loops, and edge weights are ignored. On a directed network coverage runs along out-edges and the denominators use in-degrees, the extension the source states. Distances in game 3 are hop counts.
References
Michalak, T. P., Aadithya, K. V., Szczepanski, P. L., Ravindran, B., & Jennings, N. R. (2013). Efficient computation of the Shapley value for game-theoretic network centrality. Journal of Artificial Intelligence Research, 46, 607-650.
Examples
centrality_shapley_game1(regulation_net)
#> Explore Plan Monitor Adapt Reflect Discuss Synthesize
#> 0.6500000 1.6428571 0.6428571 1.2833333 0.5428571 0.9833333 1.1761905
#> Evaluate Create Share
#> 0.9261905 1.1761905 0.9761905
centrality_shapley_game2(regulation_net)
#> Explore Plan Monitor Adapt Reflect Discuss Synthesize
#> 0.7166667 1.4523810 0.6190476 0.8166667 0.6690476 1.1333333 1.4357143
#> Evaluate Create Share
#> 1.1023810 1.1023810 0.9523810
centrality_shapley_game3(regulation_net)
#> Explore Plan Monitor Adapt Reflect Discuss Synthesize
#> 0.7138889 1.1107143 1.0190476 1.1678571 0.6583333 0.8329365 1.3107143
#> Evaluate Create Share
#> 1.0329365 0.9678571 1.1857143