Sphere of Influence Centrality via Shapley Values: Empirical Approximation and Network Coverage Analysis

Node centrality is a fundamental problem in network analysis, yet classical metrics fail to capture the collective, coalitional nature of influence. We present a systematic empirical evaluation of the Shapley-value-based framework for the sphere of influence problem – selecting $m$ nodes to maximize network coverage under three reachability criteria: single-hop, $k$-hop, and multi-path connectivity – using exact polynomial-time algorithms due to Michalak et al. Evaluation across three diverse real-world networks (Euroroad, Facebook TV Shows, and Cora) demonstrates that practical approximation ratios consistently approach 0.9, substantially exceeding the theoretical $(1-1\mathrm{/}e)$ lower bound, and that the Shapley-based approach dramatically outperforms a degree-based baseline, particularly in hub-and-spoke topologies. In the most striking case, Shapley-based selection identifies just 26 nodes (under 1% of the Cora network) sufficient to influence half the graph under 3-hop reachability, compared to substantially larger sets required by the naive baseline.