Tomasz P. Michalak

Efficient computation of the shapley value for game-theoretic network centrality

The Shapley value--probably the most important normative payoff division scheme in coalitional games--has recently been advocated as a useful measure of centrality in networks. However, although this approach has a variety of real-world applications (including social and organisational networks, biological networks and communication networks), its computational properties have not been widely studied. To date, the only practicable approach to compute Shapley value-based centrality has been via Monte Carlo simulations which are computationally expensive and not guaranteed to give an exact answer. Against this background, this paper presents the first study of the computational aspects of the Shapley value for network centralities. Specifically, we develop exact analytical formulae for Shapley value-based centrality in both weighted and unweighted networks and develop efficient (polynomial time) and exact algorithms based on them. We empirically evaluate these algorithms on two real-life examples (an infrastructure network representing the topology of the Western States Power Grid and a collaboration network from the field of astrophysics) and demonstrate that they deliver significant speedups over the Monte Carlo approach. For instance, in the case of unweighted networks our algorithms are able to return the exact solution about 1600 times faster than the Monte Carlo approximation, even if we allow for a generous 10% error margin for the latter method.

Coalition structure generation

The coalition structure generation problem is a natural abstraction of one of the most important challenges in multi-agent systems: How can a number of agents divide themselves into groups in order to improve their performance? More precisely, the coalition structure generation problem focuses on partitioning the set of agents into mutually disjoint coalitions so that the total reward from the resulting coalitions is maximized. This problem is computationally challenging, even under quite restrictive assumptions. This has prompted researchers to develop a range of algorithms and heuristic approaches for solving the problem efficiently. This article presents a survey of these approaches. In particular, it surveys the main dynamic-programming approaches and anytime algorithms developed for coalition structure generation, and considers techniques specifically developed for a range of compact representation schemes for coalitional games. It also considers settings where there are constraints on the coalitions that are allowed to form, as well as settings where the formation of one coalition could influence the performance of other co-existing coalitions.

A shapley value-based approach to determine gatekeepers in social networks with applications

Inspired by emerging applications of social networks, we introduce in this paper a new centrality measure termed gatekeeper centrality. The new centrality is based on the well-known game-theoretic concept of Shapley value and, as we demonstrate, possesses unique qualities compared to the existing metrics. Furthermore, we present a dedicated approximate algorithm, based on the Monte Carlo sampling method, to compute the gatekeeper centrality. We also consider two well known applications in social network analysis, namely community detection and limiting the spread of mis-information; and show the merit of using the proposed framework to solve these two problems in comparison with the respective benchmark algorithms.

Defeating Terrorist Networks with Game Theory

This column discusses the problem of identifying key members of a terrorist network. Game-theoretic centrality measures offer solutions but also raise computational challenges. The authors present a survey of this work and show how some of the computational challenges can be overcome.

A centrality measure for networks with community structure based on a generalization of the owen value

There is currently much interest in the problem of measuring the centrality of nodes in networks/graphs; such measures have a range of applications, from social network analysis, to chemistry and biology. In this paper we propose the first measure of node centrality that takes into account the community structure of the underlying network. Our measure builds upon the recent literature on game-theoretic centralities, where solution concepts from cooperative game theory are used to reason about importance of nodes in the network. To allow for flexible modelling of community structures, we propose a generalization of the Owen value—a well-known solution concept from cooperative game theory to study games with a priori-given unions of players. As a result we obtain the first measure of centrality that accounts for both the value of an individual nodes relationships within the network and the quality of the community this node belongs to.

Implementation and Computation of a Value for Generalized Characteristic Function Games

Generalized characteristic function games are a variation of characteristic function games, in which the value of a coalition depends not only on the identities of its members, but also on the order in which the coalition is formed. This class of games is a useful abstraction for a number of realistic settings and economic situations, such as modeling relationships in social networks. To date, two main extensions of the Shapley value have been proposed for generalized characteristic function games: the Nowak-Radzik [1994] value and the Sánchez-Bergantiños [1997] value. In this context, the present article studies generalized characteristic function games from the point of view of implementation and computation. Specifically, the article makes two key contributions. First, building upon the mechanism by Dasgupta and Chiu [1998], we present a non-cooperative mechanism that implements both the Nowak-Radzik value and the Sánchez-Bergantiños value in Subgame-Perfect Nash Equilibria in expectations. Second, in order to facilitate an efficient computation supporting the implementation mechanism, we propose the Generalized Marginal-Contribution Nets representation for this type of game. This representation extends the results of Ieong and Shoham [2005] and Elkind et al. [2009] for characteristic function games and retains their attractive computational properties.

Hiding individuals and communities in a social network

The Internet and social media have fuelled enormous interest in social network analysis. New tools continue to be developed and used to analyse our personal connections, with particular emphasis on detecting communities or identifying key individuals in a social network. This raises privacy concerns that are likely to exacerbate in the future. With this in mind, we ask the question ‘Can individuals or groups actively manage their connections to evade social network analysis tools?’ By addressing this question, the general public may better protect their privacy, oppressed activist groups may better conceal their existence and security agencies may better understand how terrorists escape detection. We first study how an individual can evade ‘node centrality’ analysis while minimizing the negative impact that this may have on his or her influence. We prove that an optimal solution to this problem is difficult to compute. Despite this hardness, we demonstrate how even a simple heuristic, whereby attention is restricted to the individual’s immediate neighbourhood, can be surprisingly effective in practice; for example, it could easily disguise Mohamed Atta’s leading position within the World Trade Center terrorist network. We also study how a community can increase the likelihood of being overlooked by community-detection algorithms. We propose a measure of concealment—expressing how well a community is hidden—and use it to demonstrate the effectiveness of a simple heuristic, whereby members of the community either ‘unfriend’ certain other members or ‘befriend’ some non-members in a coordinated effort to camouflage their community.Waniek and colleagues show that individuals and communities can disguise themselves from detection online by standard social network analysis tools through simple changes to their social network connections.

Efficient algorithms for game-theoretic betweenness centrality

Betweenness centrality measures the ability of different nodes to control the flow of information in a network. In this article, we extend the standard definition of betweenness centrality using Semivalues-a family of solution concepts from cooperative game theory that includes, among others, the Shapley value and the Banzhaf power index. Any Semivalue-based betweenness centrality measure (such as, for example, the Shapley value-based betweenness centrality measure) has the advantage of evaluating the importance of individual nodes by considering the roles they each play in different groups of nodes. Our key result is the development of a general polynomial-time algorithm to compute the Semivalue-based betweenness centrality measure, and an even faster algorithm to compute the Shapley value-based betweenness centrality measure, both for weighted and unweighted networks. Interestingly, for the unweighted case, our algorithm for computing the Shapley value-based centrality has the same complexity as the best known algorithm for computing the standard betweenness centrality due to Brandes 15. We empirically evaluate our measures in a simulated scenario where nodes fail simultaneously. We show that, compared to the standard measure, the ranking obtained by our measures reflects more accurately the influence that different nodes have on the functionality of the network. We propose a betweenness centrality based on the Shapley value and Semivalue.We develop polynomial algorithms for our game-theoretic metrics.We evaluate our measures in scenarios where simultaneous node failures occur.Our measures obtain better results than standard betweenness centrality.We provide an empirical evaluation of algorithms on real-life and random graphs.

The Stochastic Shapley Value for coalitional games with externalities

A long debated but still open question in the game theory literature is that of how to extend the Shapley Value to coalitional games with externalities. While previous work predominantly focused on developing alternative axiomatizations, in this article we propose a novel approach which centers around the coalition formation process and the underlying probability distribution from which a suitable axiomatization naturally follows. Specifically, we view coalition formation in games with externalities as a discrete-time stochastic process. We focus, in particular, on the Chinese Restaurant Process – a well-known stochastic process from probability theory. We show that reformulating Shapleys coalition formation process as the Chinese Restaurant Process yields in games with externalities a unique value with various desirable properties. We then generalize this result by proving that all values that satisfy the direct translation of Shapleys axioms to games with externalities can be obtained using our approach based on stochastic processes.

Marginality Approach to Shapley Value in Games with Externalities

One of the long-debated issues in coalitional game theory is how to extend the Shapley value to games with externalities. In particular, when externalities occur, a direct translation of Shapleys axioms does not imply a unique value. In this paper we study the marginality approach to this problem, based on the idea of an alpha-parametrized definition of the marginal contribution, where alpha is a vector of weights associated with an agent joining/leaving a coalition. We prove that all values that satisfy the direct translation of Shapleys axioms can be obtained using the marginality approach. Moreover, we show that every such value can be uniquely derived using marginality approach by choosing appropriate weights alpha. Next, we analyze how properties of a value translate to the requirements on the definition of the marginal contribution (i.e. weights alpha). Building upon this analysis, we show that under certain conditions, two other axiomatizations of the Shapley value (i.e., Youngs marginality axiomatization and Myersons axiomatization based on the concept of balanced contributions), translated to games with externalities using the proper definition of the alpha-parametrized marginal contribution, are equivalent to Shapleys axiomatization.