Shapley Meets Tutte

2026-07-25

AI Summary

This study addresses the problem of quantifying the contribution of pre-aligned local connections—such as road segments or database associations—to the overall connectivity of a network, with applications in adversarial defense, resilience planning, and benefit allocation. To this end, it introduces the Shapley value from cooperative game theory into networks featuring pairwise pre-aligned agents, establishing a novel framework that measures each local connection’s contribution based on its failure probability. The key theoretical innovation lies in forging rigorous connections between the Shapley value and fundamental graph-theoretic constructs—namely, the Tutte polynomial and chromatic polynomial—as well as the partition function of the Potts model from statistical physics. This linkage elucidates the structural influence of individual links on global connectivity and offers new mathematical tools and perspectives for network reliability analysis.

Abstract

We initiate the study of cooperative games where there exist groups of pre-aligned agent. For example, in a road network, the agents are cross-roads and pre-aligned groups are road-segments. For a set of datasets, the agents are attributes and each database which connects two (or more) attributes is pre-aligned. In this paper, each pre-aligned group has size two. Our goal is to find a way to determine the contribution of each individual local connection (pre-aligned couple) to the connectivity of the network, having an adversarial attack in mind or planning a defense of the network, or wanting to split the profits of the network between various owners of the individual local connections of the network. We model these contributions by Shapley values of the connectivity augmented 'local values' which are characteristic functions determined, e.g., by failure probabilities of local connections. We link the concepts of potential and Shapley values of these connectivity augmented local values to the world of chromatic and Tutte polynomials and the partition function of the Potts model from statistical physics.

Problem

• cooperative games
• Shapley values
• network connectivity
• pre-aligned groups
• Tutte polynomials

Innovation

• Shapley value
• Tutte polynomial
• Potts model
• cooperative game
• network connectivity