I survey the recent literature on the formation of networks. I provide definitions of network games, a number of examples of models from the literature, and discuss some of what is known about the (in)compatibility of overall societal welfare with individual incentives to form and sever links.

This paper introduces a new model of exchange: networks, rather than markets, of buyers and sellers. It begins with the empirically motivated premise that a buyer and seller must have a relationship, a “link, ” to exchange goods. Networks- buyers, sellers, and the pattern of links connecting them- are common exchange environments. This paper develops a methodology to study network structures and explains why agents may form networks. In a model that captures characteristics of a variety of industries, the paper shows that buyers and sellers, acting strategically in their own self-interests, can form the network structures that maximize overall

The science of social networks is a central field of sociological study, a major application of random graph theory, and an emerging area of study by economists, statistical physicists and computer scientists. While these literatures are (slowly) becoming aware of each other, and on occasion drawing from one another, they are still largely distinct in their methods, interests, and goals. Here, my aim is to provide some perspective on the research from these literatures, with a focus on the formal modeling of social networks and the two major types of models: those based on random graphs and those based on game theoretic reasoning. I highlight some of the strengths, weaknesses, and potential synergies between these two network modeling approaches.

We study learning in a setting where agents receive independent noisy signals about the true value of a variable and then communicate in a network. They naïvely update beliefs by repeatedly taking weighted averages of neighbors’ opinions. We show that all opinions in a large society converge to the truth if and only if the influence of the most influential agent vanishes as the society grows. We also identify obstructions to this, including prominent groups, and provide structural conditions on the network ensuring efficient learning. Whether agents converge to the truth is unrelated to how quickly consensus is approached. (JEL D83, D85, Z13)

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We develop a model in which delinquents compete with each other in criminal activities but may benefit from being friends with other criminals by learning and acquiring proper know-how on the crime business. By taking the social network connecting agents as given, we study the subgame perfect Nash equilibrium of this game in which individuals decide first to work or to become a criminal and then the crime effort provided if criminals. We show that this game always has a pure strategy subgame perfect Nash equilibrium that we characterize. Ex ante identical individuals connected through a network can end up with very different equilibrium outcomes: either employed, or isolated criminal or criminals in networks. We also show that multiple equilibria with different number of active criminals and levels of involvement in crime activities may coexist and are only driven by the geometry of the pattern of links connecting criminals. Using the equilibrium concept of pairwise-stable networks, we then show that the multiplicity of equilibrium outcomes

We study the (perfect Bayesian) equilibrium of a model of learning over a general social network. Each individual receives a signal about the underlying state of the world, observes the past actions of a stochastically-generated neighborhood of individuals, and chooses one of two possible actions. The stochastic process generating the neighborhoods defines the network topology (social network). We characterize pure-strategy equilibria for arbitrary stochastic and deterministic social networks and characterize the conditions under which there will be asymptotic learning—convergence (in probability) to the right action as the social network becomes large. We show that when private beliefs are unbounded (meaning that the implied likelihood ratios are unbounded), there will be asymptotic learning as long as there is some minimal amount of “expansion in observations”. We also characterize conditions under which there will be asymptotic learning when private beliefs are bounded.

This paper studies a model of dynamic network formation when individuals are farsighted: players evaluate the desirability of a “current ” move in terms of its consequences on the entire discounted stream of payoffs. We define a concept of equilibrium which takes into account far-sighted behavior of agents and allows for limited cooperation amongst agents. We show that an equilibrium process of network formation exists. We also show that there are network structures in which no equilibrium strategy profile can sustain efficient networks. We then provide sufficient conditions under which the equilibrium process will yield efficient outcomes.

We study learning and influence in a setting where agents communicate according to an arbitrary social network and naïvely update their beliefs by repeatedly taking weighted averages of their neighbors ’ opinions. A focus is on conditions under which beliefs of all agents in large societies converge to the truth, despite their naïve updating. We show that this happens if and only if the influence of the most influential agent in the society is vanishing as the society grows. Using simple examples, we identify two main obstructions which can prevent this. By ruling out these obstructions, we provide general structural conditions on the social network that are sufficient for convergence to truth. In addition, we show how social influence changes when some agents redistribute their trust, and we provide a complete characterization of the social networks for which there is a convergence of beliefs. Finally, we survey some recent structural results on the speed of convergence and relate these to issues of segregation, polarization and propaganda.

... obtained by analysing isolated networks, many real-world networks do in fact interact with and depend on other networks. The set of extensive results for the limiting case of non-interacting networks holds only to the extent that ignoring the presence of other networks can be justified. Recently, an analytical framework for studying the percolation properties of interacting networks has been developed. Here we review this framework and the results obtained so far for connectivity properties of ‘networks of networks’ formed by interdependent random networks.