Efficient Computation of the Shapley Value for Game-Theoretic Network Centrality

Abstract: The Shapley value---probably the most important normative payoff division scheme in coalitional games---has recently been advocated as a useful measure of centrality in networks. However, although this approach has a variety of real-world applications (including social and organisational networks, biological networks and communication networks), its computational properties have not been widely studied. To date, the only practicable approach to compute Shapley value-based centrality has been via Monte Carlo simulations which are computationally expensive and not guaranteed to give an exact answer. Against this background, this paper presents the first study of the computational aspects of the Shapley value for network centralities. Specifically, we develop exact analytical formulae for Shapley value-based centrality in both weighted and unweighted networks and develop efficient (polynomial time) and exact algorithms based on them. We empirically evaluate these algorithms on two real-life examples (an infrastructure network representing the topology of the Western States Power Grid and a collaboration network from the field of astrophysics) and demonstrate that they deliver significant speedups over the Monte Carlo approach. For instance, in the case of unweighted networks our algorithms are able to return the exact solution about 1600 times faster than the Monte Carlo approximation, even if we allow for a generous 10% error margin for the latter method.

• The paper proposes novel polynomial-time algorithms for efficiently and exactly computing Shapley values in game-theoretic network centrality, addressing the inefficiency of prior Monte Carlo methods.
• The research introduces and provides algorithms for five distinct network games that extend classical centrality notions to both weighted and unweighted networks.
• Empirical evaluation shows the new algorithms are significantly faster, achieving speedups up to 1600 times compared to Monte Carlo approximations on tested datasets.

Efficient Computation of the Shapley Value for Game-Theoretic Network Centrality: An Expert Analysis

The paper "Efficient Computation of the Shapley Value for Game-Theoretic Network Centrality" provides an extensive examination of network centrality metrics derived from cooperative game theory, specifically focusing on the calculation of Shapley values in diverse network settings. The authors address a prominent computational challenge within this domain, namely, the inefficiency of existing methods for determining Shapley value-based centrality, predominantly relying on computationally intensive Monte Carlo simulations.

Summary of the Research

The authors propose novel polynomial-time algorithms that offer exact solutions for the computation of the Shapley value in network centrality scenarios, both weighted and unweighted. This is a significant improvement over the prior Monte Carlo methods and is applicable to a range of networks, such as those representing infrastructure or academic collaborations.

The paper introduces several games to extend classical notions of centrality:

• Game 1 (g1): Focuses on unweighted networks where centrality is derived based on nodes directly reachable.
• Game 2 (g2): Expands on g1 by requiring nodes to be reachable through multiple distinct paths, incorporating synergy.
• Game 3 (g3): Applies to weighted networks, evaluating nodes within a specific cutoff distance.
• Game 4 (g4): Further generalizes g3 with an arbitrary function reflecting the influence of distance.
• Game 5 (g5): Considers weighted networks where influence is summed over adjacent nodes with weight exceeding a threshold.

For each game, the authors develop exact and efficient algorithms and verify their accuracy through empirical evaluation on datasets representing the Western States Power Grid and astrophysics collaboration networks.

Key Results

The research demonstrates substantial performance improvements with these algorithms:

• For unweighted networks, the algorithms deliver results up to 1600 times faster than Monte Carlo approximations, with an error margin set at 10%.
• Analytical computation for Shapley values in weighted networks achieves accuracy with polynomial time complexity, outperforming commonly used approximation techniques.

Implications and Future Directions

The implications of this research are multifaceted:

• Practical applications: Efficient computation of Shapley values enables precise identification of key nodes within communication, social, and biological networks, enhancing strategic decision-making in fields such as epidemiology and network optimization.
• Methodological advancement: The development of polynomial-time algorithms represents a methodological breakthrough for computational game theory, influencing how centrality measures could be utilized in broader contexts, including dynamic networks.
• Further exploration: Future research might explore adapting these methodologies to other types of centrality measures or extending them to cooperative games with externalities.

The work effectively bridges theoretical game theory concepts and practical network analysis, offering robust tools for researchers and practitioners seeking advanced insights into network dynamics. Additionally, it opens potential avenues for exploring Shapley value-based metrics for influence maximization and other novel network applications.