[Paper Review] Power indices of influence games and new centrality measures for social networks

Xavier Molinero, Fabián Riquelme+1 more|arXiv (Cornell University)|Jun 28, 2013

Complex Network Analysis Techniques44 references8 citations

TL;DR

This paper introduces four new centrality measures—Banzhaf, Shapley-Shubik, effort, and satisfaction—derived from influence games in social networks, where influence spreads based on node thresholds. It demonstrates that these measures offer distinct, often divergent, centrality rankings compared to traditional metrics, providing new insights into structural influence and decision-making roles in networks.

ABSTRACT

In social network analysis, there is a common perception that influence is relevant to determine the global behavior of the society and thus it can be used to enforce cooperation by targeting an adequate initial set of individuals or to analyze global choice processes. Here we propose centrality measures that can be used to analyze the relevance of the actors in process related to spread of influence. In [39] it was considered a multiagent system in which the agents are eager to perform a collective task depending on the perception of the willingness to perform the task of other individuals. The setting is modeled using a notion of simple games called influence games. Those games are defined on graphs were the nodes are labeled by their influence threshold and the spread of influence between its nodes is used to determine whether a coalition is winning or not. Influence games provide tools to measure the importance of the actors of a social network by means of classic power indices and provide a framework to consider new centrality criteria. In this paper we consider two of the most classical power indices, i.e., Banzhaf and Shapley-Shubik indices, as centrality measures for social networks in influence games. Although there is some work related to specific scenarios of game-theoretic networks, here we use such indices as centrality measures in any social network where the spread of influence phenomenon can be applied. Further, we define new centrality measures such as satisfaction and effort that, as far as we know, have not been considered so far. We also perform a comparison of the proposed measures with other three classic centrality measures, degree, closeness and betweenness, considering three social networks. We show that in some cases our measurements provide centrality hierarchies similar to those of other measures, while in other cases provide different hierarchies.

Motivation & Objective

• To bridge social network analysis with cooperative game theory by applying power indices to influence spread in networks.
• To define novel centrality measures—effort and satisfaction—based on influence game dynamics, which have not been previously considered.
• To evaluate the performance and distinctiveness of these new measures against classic centrality metrics in real-world social networks.
• To demonstrate that power indices and new measures provide different, and sometimes more insightful, centrality rankings than degree, closeness, and betweenness in specific network structures.

Proposed method

• Model social networks as influence games where nodes have influence thresholds and influence spreads along edges based on reachability.
• Apply classic power indices—Banzhaf and Shapley-Shubik—to measure node importance based on their pivotal role in winning coalitions.
• Define effort centrality as the minimal influence required to make a node’s activation decisive in reaching the quota.
• Define satisfaction centrality as the proportion of losing coalitions in which a node is pivotal for winning when added.
• Use real-world networks (Student Government, Monkeys’ interaction, and another) to compute and compare all centrality measures.
• Perform quantitative comparison across networks to assess consistency and divergence in centrality rankings.

Experimental results

Research questions

• RQ1 How do Banzhaf and Shapley-Shubik power indices compare to traditional centrality measures in identifying influential nodes in influence-based networks?
• RQ2 To what extent do the new measures—effort and satisfaction—capture structural roles not reflected by degree, closeness, or betweenness centrality?
• RQ3 In which network configurations do the new centrality measures produce significantly different rankings than classical measures?
• RQ4 Can effort and satisfaction centrality effectively identify key influencers in edge- and vertex-labeled directed graphs, where standard measures are limited?
• RQ5 How do the centrality rankings of influential nodes (e.g., prime minister, advisors) align with their actual roles in decision-making processes within the network?

Key findings

• In the Student Government network with quota q=6, the Banzhaf and closeness centrality measures produced completely different node rankings, with no overlap in the top-ranked nodes.
• Nodes 1, 3, and 10 had high closeness centrality due to high accessibility but low Banzhaf centrality, indicating they are not pivotal in winning coalitions.
• The prime minister (node 2) had low Banzhaf and Shapley-Shubik centrality due to being influenced by many others, despite being a central figure in the network structure.
• The satisfaction centrality measure identified the prime minister as highly influential because he belonged to many losing coalitions, indicating his pivotal role in coalition formation.
• Effort centrality highlighted nodes like minister 6 as less central due to high activation cost, even if they were structurally connected, showing sensitivity to influence thresholds.
• In the Monkeys’ interaction network, effort centrality produced a distinct ranking from traditional measures, indicating its ability to detect nodes with high influence cost despite moderate connectivity.