Game Theoretic Centrality Research Articles

Game theoretic centrality of a directed graph vertices

The paper considers a game theory approach to calculating the centrality value of the vertices in a directed graph, based on the number of vertex occurrences in fixed length paths. It is proposed to define vertex centrality as a solution of a cooperative game, where the characteristic function is given as the number of simple paths of fixed length in subgraphs corresponding to coalitions. The concept of integral centrality is introduced as the value of a definite integral of the payoff function. It is shown that this centrality measure satisfies the Boldi-Vigna axioms.

Game-Theoretic Centrality of Directed Graph Vertices

The paper considers a game theory approach to calculating the centrality value of the vertices in a directed graph, based on the number of vertex occurrences in fixed length paths. It is proposed to define vertex centrality as a solution of a cooperative game, where the characteristic function is given as the number of simple paths of fixed length in subgraphs corresponding to coalitions. The concept of integral centrality is introduced as the value of a definite integral of the payoff function. It is shown that this centrality measure satisfies the Boldi-Vigna axioms.

Game-Theoretic Centrality of Directed Graph Vertices

Game-Theoretic Centrality of Directed Graph Vertices

Game theoretic centrality: a novel approach to prioritize disease candidate genes by combining biological networks with the Shapley value

BackgroundComplex human health conditions with etiological heterogeneity like Autism Spectrum Disorder (ASD) often pose a challenge for traditional genome-wide association study approaches in defining a clear genotype to phenotype model. Coalitional game theory (CGT) is an exciting method that can consider the combinatorial effect of groups of variants working in concert to produce a phenotype. CGT has been applied to associate likely-gene-disrupting variants encoded from whole genome sequence data to ASD; however, this previous approach cannot take into account for prior biological knowledge. Here we extend CGT to incorporate a priori knowledge from biological networks through a game theoretic centrality measure based on Shapley value to rank genes by their relevance–the individual gene’s synergistic influence in a gene-to-gene interaction network. Game theoretic centrality extends the notion of Shapley value to the evaluation of a gene’s contribution to the overall connectivity of its corresponding node in a biological network.ResultsWe implemented and applied game theoretic centrality to rank genes on whole genomes from 756 multiplex autism families. Top ranking genes with the highest game theoretic centrality in both the weighted and unweighted approaches were enriched for pathways previously associated with autism, including pathways of the immune system. Four of the selected genes HLA-A, HLA-B, HLA-G, and HLA-DRB1–have also been implicated in ASD and further support the link between ASD and the human leukocyte antigen complex.ConclusionsGame theoretic centrality can prioritize influential, disease-associated genes within biological networks, and assist in the decoding of polygenic associations to complex disorders like autism.

On network analysis using non-additive integrals: extending the game-theoretic network centrality

There are large amounts of information that can be represented in terms of graphs. This includes social networks and internet. We can represent people and their interactions by means of graphs. Similarly, we can represent web pages (and sites) as well as links between pages by means of graphs. In order to study the properties of graphs, several indices have been defined. They include degree centrality, betweenness, and closeness. In this paper, we propose the use of Choquet and Sugeno integrals with respect to non-additive measures for network analysis. This is a natural extension of the use of game theory for network analysis. Recall that monotonic games in game theory are non-additive measures. We discuss the expected force, a centrality measure, in the light of non-additive integral network analysis. We also show that some results by Godo et al. can be used to compute network indices when the information associated with a graph is qualitative.

Efficient Computation of Semivalues for Game-Theoretic Network Centrality


 
 
 Some game-theoretic solution concepts such as the Shapley value and the Banzhaf index have recently gained popularity as measures of node centrality in networks. While this direction of research is promising, the computational problems that surround it are challenging and have largely been left open. To date there are only a few positive results in the literature, which show that some game-theoretic extensions of degree-, closeness- and betweenness-centrality measures are computable in polynomial time, i.e., without the need to enumerate the exponential number of all possible coalitions. In this article, we show that these results can be extended to a much larger class of centrality measures that are based on a family of solution concepts known as semivalues. The family of semivalues includes, among others, the Shapley value and the Banzhaf index. To this end, we present a generic framework for defining game-theoretic network centralities and prove that all centrality measures that can be expressed in this framework are computable in polynomial time. Using our framework, we present a number of new and polynomial-time computable game-theoretic centrality measures.
 
 

Axiomatic Characterization of Game-Theoretic Centrality


 
 
 One of the fundamental research challenges in network science is centrality analysis, i.e., identifying the nodes that play the most important roles in the network. In this article, we focus on the game-theoretic approach to centrality analysis. While various centrality indices have been recently proposed based on this approach, it is still unknown how general is the game-theoretic approach to centrality and what distinguishes some game-theoretic centralities from others. In this article, we attempt to answer this question by providing the first axiomatic characterization of game-theoretic centralities. Specifically, we show that every possible centrality measure can be obtained following the game-theoretic approach. Furthermore, we study three natural classes of game-theoretic centrality, and prove that they can be characterized by certain intuitive properties pertaining to the well-known notion of Fairness due to Myerson.
 
 

DON and Shapley Value for Allocation among Cooperating Agents in a Network: Conditions for Equivalence

In a paper appearing in a recent issue of this journal ( Studies in Microeconomics), the authors explored a new method to allocate a divisible resource efficiently among cooperating agents located at the vertices of a connected undirected network. It was shown in that article that maximizing social welfare of the agents produces Pareto optimal allocations, referred to as dominance over neighbourhood (DON), capturing the notion of dominance over neighbourhood in terms of network degree. In this article, we show that the allocation suggested by the method competes well with current cooperative game-theoretic power centrality measures. We discuss the conditions under which DON turns exactly equivalent to a recent ‘fringe-based’ Shapley Value formulation for fixed networks, raising the possibility of such solutions being both Pareto optimal in a utilitarian social welfare maximization sense as well as fair in the Shapley value sense.

Parallelization of game theoretic centrality algorithms

Communication has become a lot easier with the advent of easy and cheap means of reaching people across the globe. This has allowed the development of large networked communities and, with the technology available to track them, has opened up the study of social networks at unprecedented scales. This has necessitated the scaling up of various network analysis algorithms that have been proposed earlier in the literature. While some algorithms can be readily adapted to large networks, in many cases the adaptation is not trivial. In this work, we explore the scaling up of a class of node centrality algorithms based on cooperative game theory. These were proposed earlier as an efficient alternatives to traditional measure of information diffusion centrality. We present here distributed versions of these algorithms in a Map-Reduce framework, currently the most popular distributed computing paradigm. We empirically demonstrate the scaling behavior of our algorithm on very large synthetic networks thereby establishing the utility of these methods in settings such as online social networks.

Ranking terrorists in networks: A sensitivity analysis of Al Qaeda's 9/11 attack

All over the world, intelligence services are collecting data concerning possible terrorist threats. This information is usually transformed into network structures in which the nodes represent the individuals in the data set and the links possible connections between these individuals. Unfortunately, it is nearly impossible to keep track of all individuals in the resulting complex network. Therefore, Lindelauf et al. (2013) introduced a methodology that ranks terrorists in a network. The rankings that result from this methodology can be used as a decision support system to efficiently allocate the scarce surveillance means of intelligence agencies. Moreover, usage of these rankings can improve the quality of surveillance which can in turn lead to prevention of attacks or destabilization of the networks under surveillance.The methodology introduced by Lindelauf et al. (2013) is based on a game theoretic centrality measure, which is innovative in the sense that it takes into account not only the structure of the network but also individual and coalitional characteristics of the members of the network. In this paper we elaborate on this methodology by introducing a new game theoretic centrality measure that better takes into account the operational strength of connected subnetworks.Moreover, we perform a sensitivity analysis on the rankings derived from this new centrality measure for the case of Al Qaeda's 9/11 attack. In this sensitivity analysis we consider firstly the possible additional information available about members of the network, secondly, variations in relational strength and, finally, the absence or presence of a small percentage of links in the network. We also introduce a case specific method to compare the different rankings that result from the sensitivity analysis and show that the new centrality measure is robust to small changes in the data.

Efficient Computation of Semivalues for Game-Theoretic Network Centrality

Solution concepts from cooperative game theory, such as the Shapley value or the Banzhaf index, have recently been advocated as interesting extensions of standard measures of node centrality in networks. While this direction of research is promising, the computation of game-theoretic centrality can be challenging. In an attempt to address the computational issues of game-theoretic network centrality, we present a generic framework for constructing game-theoretic network centralities. We prove that all extensions that can be expressed in this framework are computable in polynomial time. Using our framework, we present the first game-theoretic extensions of weighted and normalized degree centralities, impact factor centrality,distance-scaled and normalized betweenness centrality,and closeness and normalized closeness centralities.

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.

Ranking Terrorists in Networks: A Sensitivity Analysis of Al Qaeda's 9/11 Attack

All over the world intelligence services are collecting data concerning possible terrorist threats. This information is usually transformed into network structures in which the nodes represent the individuals in the data set and the links possible connections between these individuals. Unfortunately, it is nearly impossible to keep track of all individuals in the resulting complex network. Therefore, Lindelauf et al. (2013) introduced a methodology that ranks terrorists in a network. The rankings that result from this methodology can be used as a decision support system to efficiently allocate the scarce surveillance means of intelligence agencies. Moreover, usage of these rankings can improve the quality of surveillance which can in turn lead to prevention of attacks or destabilization of the networks under surveillance. The methodology introduced by Lindelauf et al. (2013) is based on a game theoretic centrality measure, which is innovative in the sense that it takes into account not only the structure of the network but also individual and coalitional characteristics of the members of the network. In this paper we elaborate on this methodology by introducing a new game theoretic centrality measure that better takes into account the operational strength of connected subnetworks. Moreover, we perform a sensitivity analysis on the rankings derived from this new centrality measure for the case of Al Qaeda’s 9/11 attack. In this sensitivity analysis we consider firstly the possible additional information available about members of the network, secondly, variations in relational strength and, finally, the absence or presence of a small percentage of links in the network. We also introduce a case specific method to compare the different rankings that result from the sensitivity analysis and show that the new centrality measure is robust to small changes in the data.

Cooperative game theoretic centrality analysis of terrorist networks: The cases of Jemaah Islamiyah and Al Qaeda

The identification of key players in a terrorist organization aids in preventing attacks, the efficient allocation of surveillance measures, and the destabilization of the corresponding network. In this paper, we introduce a game theoretic approach to identify key players in terrorist networks. In particular we use the Shapley value as a measure of importance in cooperative games that are specifically designed to reflect the context of the terrorist organization at hand. The advantage of this approach is that both the structure of the terrorist network, which usually reflects a communication and interaction structure, as well as non-network features, i.e., individual based parameters such as financial means or bomb building skills, can be taken into account. The application of our methodology to the analysis results in rankings of the terrorists in the network. We illustrate our methodology through two case studies: Jemaah Islamiyah’s Bali bombing and Al Qaedas 9/11 attack, which lead to new insights in the operational networks responsible for these attacks.

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 th...

Leaderless Covert Networks: A Quantitative Approach

Lindelauf et al. (2009a) introduced a quantitative approach to investigate optimal structures of covert networks. This approach used an objective function which is based on the secrecy versus information trade-off these organizations face. Sageman (2008) hypothesized that covert networks organize according to leaderless principles, i.e., clear leaders can not be identified. This flat organizational structure is quite robust to destabilization tactics which target the most important persons in a network. There exist several centrality measures to express the importance of persons in a network. The most recent one introduced in the field of covert networks is a game theoretical centrality measure which takes into account both the structure of the covert network, which usually reflects a communication structure, as well as non-network features, which represent individual parameters like financial means or bomb building skills, see Lindelauf et al. (2011). The question we try to answer in this chapter is whether there is a relationship between the quality of a covert network based on their optimality with respect to the trade-off between secrecy and information and the variance of the game theoretic centrality measures of the respective individuals in the network. The leaderless hypothesis seems to suggest that good covert networks do not have a high distinction between centrality of the individuals, i.e., they are leaderless. We investigate this by looking at homogeneous networks and heterogeneous networks in which the links between individuals are weighted. We find that (approximate) optimal networks have low variance in game-theoretic centrality, i.e., we find evidence that supports the leaderless hypothesis. However, if the networks are heterogeneous in the sense that, for instance, certain individuals communicate much more often than others, we find that the variance increases significantly. Finally, we look at the Jemaah Islamiyah 2002 Bali bombing. We find that the operational network used to conduct and to coordinate the bombing not only facilitated both secrecy and efficiency but also adhered to the leaderless principle.

Game Theoretic Centrality Analysis of Terrorist Networks: The Cases of Jemaah Islamiyah and Al Qaeda

The identification of key players in a terrorist network can lead to prevention of attacks, due to efficient allocation of surveillance means or isolation of key players in order to destabilize the network. In this paper we introduce a game theoretic approach to identify key players in terrorist networks. The advantage of this approach is that both the structure of the terrorist network, which usually reflects a communication structure, as well non-network features, which represent individual parameters like financial means or bomb building skills, can be taken into account. The application of our methodology results in rankings of the terrorists in the network. We illustrate our methodology by two case studies: Jemaah Islamiyah’s Bali bombing and Al Qaeda’s 9/11 attack, which has led to new insights in the operational networks responsible for these attacks.