We examine how diffusion and learning processes are influenced by network properties, focusing on density and homophily – the tendency of agents to associate disproportionately with those sharing similar traits. Homophily does not affect the speed of diffusions that travel along shortest paths; their rate is determined only by the size of the society and the number of links per agent. In contrast, homophily substantially slows learning based on repeated averaging of neighbors ’ information. Our analysis shows that changing a network can have widely different effects on information flow depending on the details of the transmission process and we provide general tools for analyzing such changes.

We study the convergence times of dynamics in games involving graphical relationships of players. Our model of local interaction games generalizes a variety of recently studied games in game theory and distributed computing. In a local interaction game each agent is a node embedded in a graph and plays the same 2-player game with each neighbor. He can choose his strategy only once and must apply his choice in each game he is involved in. This represents a fundamental model of decision making with local interaction and distributed control. Furthermore, we introduce a generalization called 2-type interaction games, in which one 2-player game is played on edges and possibly another game is played on non-edges. For the popular case with symmetric 2×2 games, we show that several dynamics converge in polynomial time. This includes arbitrary sequential better response dynamics, as well as concurrent dynamics resulting from a distributed protocol that does not rely on global knowledge. We supplement these results with an experimental comparison of sequential and concurrent dynamics. 1

Price of anarchy and price of stability are the primary notions for measuring the efficiency (i.e. the social welfare) of the outcome of a game. Both of these notions focus on extreme cases: one is defined as the inefficiency ratio of the worst-case equilibrium and the other as the best one. Therefore, studying these notions often results in discovering equilibria that are not necessarily the most likely outcomes of the dynamics of selfish and non-coordinating agents. The current paper studies the inefficiency of the equilibria that are most stable in the presence of noise. In particular, we study two variations of non-cooperative games: atomic congestion games and selfish load balancing. The noisy best-response dynamics in these games keeps the joint action profile around a particular set of equilibria that minimize the potential function. The inefficiency ratio in the neighborhood of these “stable ” equilibria is much better than the price of anarchy. Furthermore, the dynamics reaches these equilibria in polynomial time. Our observations show that in the game environments where a small noise is present, the system as a whole works better than what a pessimist may predict. They also suggest that in congestion games, introducing a small noise in the payoff of the agents may improve the social welfare.

We examine how diffusion, best response, and learning processes are influenced by social network structure. We focus on the impact of two network properties: link density and homophily – the tendency of agents to associate disproportionately with those having similar traits. Homophily does not affect the speed of processes that travel along shortest paths; their rate is determined by the size of the society and the number of links per agent. In contrast, homophily substantially slows the convergence of processes based on weighted averaging of neighbors ’ behaviors or beliefs, including some best response dynamics and learning processes.

We study a simple game-theoretic model for the spread of an innovation in a network. The diffusion of the innovation is modeled as the dynamics of a coordination game in which the adoption of a common strategy between players has a higher payoff. Classical results in game theory provide a simple condition for the innovation to spread through the network. The present paper characterizes the rate of convergence as a function of graph structure. In particular, we derive a dichotomy between well-connected (e.g. random) graphs that show slow convergence and poorly connected, low dimensional graphs that show fast convergence.

We examine how diffusion, best response, and learning processes are influenced by social network structure. We focus on the impact of two network properties: link density and homophily – the tendency of agents to associate disproportionately with those having similar traits. Homophily does not affect the speed of processes that travel along shortest paths; their rate is determined by the size of the society and the number of links per agent. In contrast, homophily substantially slows the convergence of processes based on weighted averaging of neighbors ’ behaviors or beliefs, including some best response dynamics and learning processes.

Logit Dynamics [Blume, Games and Economic Behavior, 1993] is a randomized best response dynamics for strategic games: at every time step a player is selected uniformly at random and she chooses a new strategy according to a probability distribution biased toward strategies promising higher payoffs. This process defines an ergodic Markov chain, over the set of strategy profiles of the game, whose unique stationary distribution is the long-term equilibrium concept for the game. However, when the mixing time of the chain is large (e.g., exponential in the number of players), the stationary distribution loses its appeal as equilibrium concept, and the transient phase of the Markov chain becomes important. In several cases it happens that on a time-scale shorter than mixing time the chain is “quasistationary”, meaning that it stays close to some small set of the state space, while in a time-scale multiple of the mixing time it jumps from one quasi-stationaryconfiguration to another; this phenomenon is usually called “metastability”. In this paper we give a quantitative definition of “metastable probability distributions ” for a Markov chain and we study the metastability of the Logit dynamics for some classes of coordination games. In particular, we study no-risk-dominant coordination games on the clique (which is equivalent to the well-known Glauber dynamics for the Ising model) and coordination games on a ring (both the risk-dominant and norisk-dominant case). We also describe a simple “artificial” game that highlights the distinctive features of our metastability notion based on distributions. 1

We analyze the Schelling model of segregation in which a society of n individuals live in a ring. Each individual is one of two races and is only satisfied with his location so long as at least half his 2w nearest neighbors are of the same race as him. In the dynamics, randomly-chosen unhappy individuals successively swap locations. We consider the average size of monochromatic neighborhoods in the final stable state. Our analysis is the first rigorous analysis of the Schelling dynamics. We note that, in contrast to prior approximate analyses, the final state is nearly integrated: the average size of monochromatic neighborhoods is independent of n and polynomial in w. 1