Algorithms for Graph-Constrained Coalition Formation in the Real World

Coalition formation typically involves the coming together of multiple, heterogeneous, agents to achieve both their individual and collective goals. In this paper, we focus on a special case of coalition formation known as Graph-Constrained Coalition Formation (GCCF) whereby a network connecting the agents constrains the formation of coalitions. We focus on this type of problem given that in many real-world applications, agents may be connected by a communication network or only trust certain peers in their social network. We propose a novel representation of this problem based on the concept of edge contraction, which allows us to model the search space induced by the GCCF problem as a rooted tree. Then, we propose an anytime solution algorithm (CFSS), which is particularly efficient when applied to a general class of characteristic functions called $m+a$ functions. Moreover, we show how CFSS can be efficiently parallelised to solve GCCF using a non-redundant partition of the search space. We benchmark CFSS on both synthetic and realistic scenarios, using a real-world dataset consisting of the energy consumption of a large number of households in the UK. Our results show that, in the best case, the serial version of CFSS is 4 orders of magnitude faster than the state of the art, while the parallel version is 9.44 times faster than the serial version on a 12-core machine. Moreover, CFSS is the first approach to provide anytime approximate solutions with quality guarantees for very large systems of agents (i.e., with more than 2700 agents).Comment: Accepted for publication, cite as "in press

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. Fo

Distributed Cooperative Sensing in Cognitive Radio Networks: An Overlapping Coalition Formation Approach

Cooperative spectrum sensing has been shown to yield a significant performance improvement in cognitive radio networks. In this paper, we consider distributed cooperative sensing (DCS) in which secondary users (SUs) exchange data with one another instead of reporting to a common fusion center. In most existing DCS algorithms, the SUs are grouped into disjoint cooperative groups or coalitions, and within each coalition the local sensing data is exchanged. However, these schemes do not account for the possibility that an SU can be involved in multiple cooperative coalitions thus forming overlapping coalitions. Here, we address this problem using novel techniques from a class of cooperative games, known as overlapping coalition formation games, and based on the game model, we propose a distributed DCS algorithm in which the SUs self-organize into a desirable network structure with overlapping coalitions. Simulation results show that the proposed overlapping algorithm yields significant performance improvements, decreasing the total error probability up to 25% in the Q_m+Q_f criterion, the missed detection probability up to 20% in the Q_m/Q_f criterion, the overhead up to 80%, and the total report number up to 10%, compared with the state-of-the-art non-overlapping algorithm

Inequality and Network Structure

This paper explores the manner in which the structure of a social network constrains the level of inequality that can be sustained among its members. We assume that any distribution of value across the network must be stable with respect to coalitional deviations, and that players can form a deviating coalition only if they constitute a clique in the network. We show that if the network is bipartite, there is a unique stable payoff distribution that is maximally unequal in that it does not Lorenz dominate any other stable distribution. We obtain a complete ordering of the class of bipartite networks and show that those with larger maximum independent sets can sustain greater levels of inequality. The intuition behind this result is that networks with larger maximum independent sets are more sparse and hence offer fewer opportunities for coalitional deviations. We also demonstrate that standard centrality measures do not consistently predict inequality. We extend our framework by allowing a group of players to deviate if they are all within distance k of each other, and show that the ranking of networks by the extent of extremal inequality is not invariant in k.inequality;networks;coalitional deviations;power;centrality

Equilibrium in Labor Markets with Few Firms

We study competition between firms in labor markets, following a combinatorial model suggested by Kelso and Crawford [1982]. In this model, each firm is trying to recruit workers by offering a higher salary than its competitors, and its production function defines the utility generated from any actual set of recruited workers. We define two natural classes of production functions for firms, where the first one is based on additive capacities (weights), and the second on the influence of workers in a social network. We then analyze the existence of pure subgame perfect equilibrium (PSPE) in the labor market and its properties. While neither class holds the gross substitutes condition, we show that in both classes the existence of PSPE is guaranteed under certain restrictions, and in particular when there are only two competing firms. As a corollary, there exists a Walrasian equilibrium in a corresponding combinatorial auction, where bidders' valuation functions belong to these classes. While a PSPE may not exist when there are more than two firms, we perform an empirical study of equilibrium outcomes for the case of weight-based games with three firms, which extend our analytical results. We then show that stability can in some cases be extended to coalitional stability, and study the distribution of profit between firms and their workers in weight-based games

Coalition structure generation over graphs

We give the analysis of the computational complexity of coalition structure generation over graphs. Given an undirected graph G = (N,E) and a valuation function v : P(N) → R over the subsets of nodes, the problem is to find a partition of N into connected subsets, that maximises the sum of the components values. This problem is generally NP-complete; in particular, it is hard for a defined class of valuation functions which are independent of disconnected members — that is, two nodes have no effect on each others marginal contribution to their vertex separator. Nonetheless, for all such functions we provide bounds on the complexity of coalition structure generation over general and minor free graphs. Our proof is constructive and yields algorithms for solving corresponding instances of the problem. Furthermore, we derive linear time bounds for graphs of bounded treewidth. However, as we show, the problem remains NP-complete for planar graphs, and hence, for any Kk minor free graphs where k ≥ 5. Moreover, a 3-SAT problem with m clauses can be represented by a coalition structure generation problem over a planar graph with O(m2) nodes. Importantly, our hardness result holds for a particular subclass of valuation functions, termed edge sum, where the value of each subset of nodes is simply determined by the sum of given weights of the edges in the induced subgraph

Social Data Offloading in D2D-Enhanced Cellular Networks by Network Formation Games

Recently, cellular networks are severely overloaded by social-based services, such as YouTube, Facebook and Twitter, in which thousands of clients subscribe a common content provider (e.g., a popular singer) and download his/her content updates all the time. Offloading such traffic through complementary networks, such as a delay tolerant network formed by device-to-device (D2D) communications between mobile subscribers, is a promising solution to reduce the cellular burdens. In the existing solutions, mobile users are assumed to be volunteers who selfishlessly deliver the content to every other user in proximity while moving. However, practical users are selfish and they will evaluate their individual payoffs in the D2D sharing process, which may highly influence the network performance compared to the case of selfishless users. In this paper, we take user selfishness into consideration and propose a network formation game to capture the dynamic characteristics of selfish behaviors. In the proposed game, we provide the utility function of each user and specify the conditions under which the subscribers are guaranteed to converge to a stable network. Then, we propose a practical network formation algorithm in which the users can decide their D2D sharing strategies based on their historical records. Simulation results show that user selfishness can highly degrade the efficiency of data offloading, compared with ideal volunteer users. Also, the decrease caused by user selfishness can be highly affected by the cost ratio between the cellular transmission and D2D transmission, the access delays, and mobility patterns

Hedonic Coalition Formation for Distributed Task Allocation among Wireless Agents

Autonomous wireless agents such as unmanned aerial vehicles or mobile base stations present a great potential for deployment in next-generation wireless networks. While current literature has been mainly focused on the use of agents within robotics or software applications, we propose a novel usage model for self-organizing agents suited to wireless networks. In the proposed model, a number of agents are required to collect data from several arbitrarily located tasks. Each task represents a queue of packets that require collection and subsequent wireless transmission by the agents to a central receiver. The problem is modeled as a hedonic coalition formation game between the agents and the tasks that interact in order to form disjoint coalitions. Each formed coalition is modeled as a polling system consisting of a number of agents which move between the different tasks present in the coalition, collect and transmit the packets. Within each coalition, some agents can also take the role of a relay for improving the packet success rate of the transmission. The proposed algorithm allows the tasks and the agents to take distributed decisions to join or leave a coalition, based on the achieved benefit in terms of effective throughput, and the cost in terms of delay. As a result of these decisions, the agents and tasks structure themselves into independent disjoint coalitions which constitute a Nash-stable network partition. Moreover, the proposed algorithm allows the agents and tasks to adapt the topology to environmental changes such as the arrival/removal of tasks or the mobility of the tasks. Simulation results show how the proposed algorithm improves the performance, in terms of average player (agent or task) payoff, of at least 30.26% (for a network of 5 agents with up to 25 tasks) relatively to a scheme that allocates nearby tasks equally among agents.Comment: to appear, IEEE Transactions on Mobile Computin

Coalitional Game Theory for Communication Networks: A Tutorial

Game theoretical techniques have recently become prevalent in many engineering applications, notably in communications. With the emergence of cooperation as a new communication paradigm, and the need for self-organizing, decentralized, and autonomic networks, it has become imperative to seek suitable game theoretical tools that allow to analyze and study the behavior and interactions of the nodes in future communication networks. In this context, this tutorial introduces the concepts of cooperative game theory, namely coalitional games, and their potential applications in communication and wireless networks. For this purpose, we classify coalitional games into three categories: Canonical coalitional games, coalition formation games, and coalitional graph games. This new classification represents an application-oriented approach for understanding and analyzing coalitional games. For each class of coalitional games, we present the fundamental components, introduce the key properties, mathematical techniques, and solution concepts, and describe the methodologies for applying these games in several applications drawn from the state-of-the-art research in communications. In a nutshell, this article constitutes a unified treatment of coalitional game theory tailored to the demands of communications and network engineers.Comment: IEEE Signal Processing Magazine, Special Issue on Game Theory, to appear, 2009. IEEE Signal Processing Magazine, Special Issue on Game Theory, to appear, 200