On Influence, Stable Behavior, and the Most Influential Individuals in Networks: A Game-Theoretic Approach

We introduce a new approach to the study of influence in strategic settings where the action of an individual depends on that of others in a network-structured way. We propose \emph{influence games} as a \emph{game-theoretic} model of the behavior of a large but finite networked population. Influence games allow \emph{both} positive and negative \emph{influence factors}, permitting reversals in behavioral choices. We embrace \emph{pure-strategy Nash equilibrium (PSNE)}, an important solution concept in non-cooperative game theory, to formally define the \emph{stable outcomes} of an influence game and to predict potential outcomes without explicitly considering intricate dynamics. We address an important problem in network influence, the identification of the \emph{most influential individuals}, and approach it algorithmically using PSNE computation. \emph{Computationally}, we provide (a) complexity characterizations of various problems on influence games; (b) efficient algorithms for several special cases and heuristics for hard cases; and (c) approximation algorithms, with provable guarantees, for the problem of identifying the most influential individuals. \emph{Experimentally}, we evaluate our approach using both synthetic influence games as well as several real-world settings of general interest, each corresponding to a separate branch of the U.S. Government. \emph{Mathematically,} we connect influence games to important game-theoretic models: \emph{potential and polymatrix games}.

💡 Research Summary

The paper introduces “influence games,” a game‑theoretic framework for modeling strategic behavior in networked populations. Each individual is represented by a node in a directed weighted graph and chooses a binary action (−1 or +1). Nodes have heterogeneous thresholds; the net influence on a node is the sum of incoming positive influence from neighbors playing +1 minus the sum from neighbors playing −1. If this net influence exceeds the node’s threshold, the node’s best response is +1, otherwise −1 (ties are indifferent). A configuration where every node plays its best response is a pure‑strategy Nash equilibrium (PSNE), which the authors adopt as the definition of a stable outcome.

The authors formalize a novel “most‑influential‑nodes” (MIN) problem: given a goal (e.g., a desired stable outcome with many +1 actions), find the smallest set S of nodes whose prescribed actions (taken from a target PSNE) force the entire network to converge uniquely to that PSNE. This differs from classic influence‑maximization, which seeks to maximize spread; MIN focuses on guaranteeing a particular equilibrium rather than diffusion dynamics.

Complexity analysis shows that deciding PSNE existence and enumerating all PSNE are NP‑complete and #P‑hard respectively in general graphs. The MIN problem is also NP‑hard, with hardness persisting even under restrictions. However, for special graph families—trees, bipartite graphs, and graphs with only non‑negative weights—the authors present polynomial‑time algorithms based on dynamic programming or reductions to max‑flow/min‑cut.

Algorithmic contributions include:

• Exact algorithms for the special cases mentioned above.
• Greedy and local‑search heuristics for general graphs that iteratively select nodes with the largest marginal influence on the target equilibrium.
• A randomized (1‑1/e)‑approximation algorithm when the objective function is submodular, providing provable performance guarantees.

The paper connects influence games to potential games (showing the existence of a potential function when all influence factors are non‑negative) and to polymatrix games (capturing the pairwise interaction structure). These connections allow the use of existing equilibrium‑computation techniques and provide theoretical insight into convergence properties.

Experimental evaluation comprises two parts. First, synthetic networks (Erdős‑Rényi, Barabási‑Albert, and random threshold/weight assignments) are used to benchmark runtime, approximation quality, and the number of PSNE found. Results confirm that exact algorithms are efficient on the special graph classes, while heuristics scale well on larger, dense graphs. Second, real‑world case studies involve the U.S. Senate and the U.S. Supreme Court. Using publicly available voting and opinion data, the authors construct influence graphs and define policy‑oriented goals (e.g., passing a debt‑ceiling bill). They demonstrate that very small MIN sets (often just two or three legislators or justices) can uniquely enforce the desired equilibrium, highlighting the practical relevance of the approach compared to traditional diffusion‑based influence maximization, which would require many more seed nodes.

In the related‑work discussion, the authors contrast their equilibrium‑centric model with diffusion models based on forward recursion or threshold cascades, emphasizing that PSNE abstracts away dynamics and permits negative, asymmetric influences. They also note that while influence maximization focuses on spread, MIN focuses on strategic control of outcomes.

The conclusion outlines future directions: learning influence weights from data, extending the model to multi‑action settings, and incorporating uncertainty or robustness into the MIN formulation. Overall, the paper provides a rigorous, game‑theoretic foundation for influence analysis, offers algorithmic tools for identifying key actors, and validates the methodology on both synthetic and high‑stakes political networks.