Cooperative Game Theory Approaches for Network Partitioning (1707.03587v2)

Abstract: The paper is devoted to game-theoretic methods for community detection in networks. The traditional methods for detecting community structure are based on selecting denser subgraphs inside the network. Here we propose to use the methods of cooperative game theory that highlight not only the link density but also the mechanisms of cluster formation. Specifically, we suggest two approaches from cooperative game theory: the first approach is based on the Myerson value, whereas the second approach is based on hedonic games. Both approaches allow to detect clusters with various resolution. However, the tuning of the resolution parameter in the hedonic games approach is particularly intuitive. Furthermore, the modularity based approach and its generalizations can be viewed as particular cases of the hedonic games.

• The paper introduces cooperative game theory methods for network partitioning using the Myerson value and hedonic games frameworks.
• The Myerson value approach allocates link contributions through direct and indirect connections with a tunable parameter to overcome resolution limits.
• The hedonic games method employs additively separable preference functions to form stable network coalitions, validated by experiments like Zachary's Karate Club.

Cooperative Game Theory Approaches for Network Partitioning

The study presents an exploration of network partitioning through the lens of cooperative game theory, specifically employing the Myerson value and hedonic games frameworks. The authors, Avrachenkov et al., propose methodologies that not only consider the density of links within the network but also unravel the underlying mechanisms facilitating cluster formation.

Introduction

The relevance of community detection in networks cannot be overstated. Traditional methods have broadly relied on spectral clustering, random walks, statistical physics, and modularity. However, these approaches may overlook the dynamic forces behind network cluster formation. The paper posits that cooperative game theory is particularly well-suited to fill this gap, providing a robust theoretical foundation to explain the emergence of network clusters.

Myerson Value Approach

The first proposed approach leverages the Myerson value, a concept in cooperative game theory that allocates value among players based on network constraints. The Myerson value effectively incorporates the contributions of all coalitions, ensuring a fair distribution in networked games.

Methodology

The authors define the characteristic function using discounted paths and demonstrate how to compute the Myerson value efficiently via allocation rules:

• Direct Connections: Two directly connected players receive half of the link's value.
• Indirect Connections: For longer paths, the value is shared among the path participants, proportionate to the path length.

Using illustrative examples, the paper elucidates the application of the Myerson value for network partitioning. For instance, in a six-node graph, the partitioning results hinge on a tunable parameter $r$. By adjusting $r$, one can modulate the resolution of detected clusters, thus addressing problems like the resolution limit.

Hedonic Games Approach

The second approach is predicated on hedonic games, where players form coalitions based on individual preferences. By introducing a parametric value function, the authors facilitate the detection of clusters with varying resolution levels.

Methodology

A crucial aspect of hedonic games is the additively separable preference function, defined as:

$$v_{ij} = \begin{cases} 1 - \alpha & \text{if } (i, j) \in E, \ -\alpha & \text{if } (i, j) \notin E, \end{cases}$$

where $\alpha \in [0, 1]$. The potential function then aggregates the preference values within coalitions to determine stable partitions.

The paper proves that as $\alpha$ approaches zero, the grand coalition maximizes the potential. Conversely, as $\alpha$ nears one, the network decomposes into maximal cliques. This dynamic tuning of $\alpha$ allows for flexible granularity in clustering.

Evaluation and Applications

Several example networks, including Zachary's Karate Club, demonstrate the effectiveness of the proposed methodologies. The Myerson value and hedonic games approaches yield partitions that align closely with observational or expected outcomes, indicating the robustness of these techniques in practical scenarios.

The paper also offers a novel game-theoretic interpretation of modularity. By computing the potential using a value function adjusted for the configuration model, the authors align their hedonic game framework with modularity optimization. Thus, maximizing modularity is equivalent to achieving a Nash-stable partition in the hedonic games context.

Conclusion and Future Directions

The proposed methods provide significant insights into the forces driving network partitioning while offering tunable resolution to better detect community structures. Future research avenues include applying these approaches to larger social networks and enhancing computational methods for practical implementation. The integration of Monte Carlo techniques could further refine these methods, making them more scalable and efficient for real-world applications.

Overall, this work bridges cooperative game theory and network science, presenting a sophisticated toolkit for researchers and practitioners aiming to unravel the complexities of networked systems.