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One-way flow networks: the role of heterogeneity.

Article Excerpt
Abstract I study a one-way flow connections model in which players are heterogeneous with respect to values and the costs of establishing a link. I show that values and costs heterogeneity are equally important in determining the level of connectedness and the architecture of equilibrium networks. I also show that when asymmetries are independent of the potential partner there are distributions of costs and values for which centrality is a distincitive feature of equilibrium networks. This sharply contrasts with the homogeneous case.

Keywords Network formation * Heterogeneity

JEL Classification Numbers D85 * C72

1 Introduction

The role of social and economic networks in shaping individual behavior and aggregate phenomena has been widely documented in recent years. (1) This has lead scholars in different disciplines to investigate the structural properties that networks exhibit in reality. The most stable empirical finding is that networks have very asymmetric architectures. Specifically, they exhibit high level of centrality: there are few nodes having many links, while the majority of nodes maintain few links. The connections model is the primary model used to explain the strategic formation of networks. (2) Variants of this model have been proposed in order to analyze different social and economic situations. Nevertheless, much of the work has explored the formation of undirected networks: a link induces benefits to both parties, i.e., two-way flow networks. In reality many networks are directed: the flow of benefits is directed only towards the investor of the link, i.e., one-way flow networks. For example, the World Wide Web is a directed network: nodes are agents maintaining a web page and links are hyperlinks that point from one web page to the other. (3)

Bala and Goyal (2000) analyze a non-cooperative model of network formation where players are homogeneous and there is one-way information flow. They show that if a player's payoffs are increasing in the number of other players accessed and decreasing in the number of links formed, a strict Nash network is either a wheel (a connected network in which each player creates and receives one link) or the empty network (with no links). The intuition for this result is as follows. Consider a minimally connected network where player 1 initiates a link with players 2 and 3, and each of these players has a link with player 1. Under the assumption of homogeneous costs of linking and values this network is not a strict equilibrium: player 2 is indifferent between maintaining the link with 1 and switching to player 3, instead. A generalization of this argument implies that a connected strict equilibrium is symmetric and has a wheel architecture. It is worth emphasizing two aspects of this result. The first aspect is that while centrality appears to be a crucial property of directed networks, equilibrium networks are symmetric when players are homogeneous. (4) Secondly, the findings of Bala and Goyal (2000) depend on the assumption of homogenous players. To observe this, assume that player 1 is just slightly cheaper to be linked with than players 2 and 3, ceteris paribus. This small introduction of heterogeneity implies that the network described in the example above becomes a strict equilibrium.

In the current paper, I study the role played by heterogeneous players in shaping the equilibrium architecture of directed networks. Players are heterogeneous in terms of the costs of linking and the values of accessing other players. Ex-ante asymmetries across players arise quite naturally in reality. For instance, in the context of information networks it is often the case that some individuals are more interested in particular issues and therefore better informed, which makes them more valuable contacts. Similarly, individuals differ in communication and social skills and it seems natural that forming links is cheaper for some individuals as compared to others. (5)

[FIGURE 1 OMITTED]

I start with a setting where values and costs of linking are heterogeneous across players but the heterogeneity is not partner specific: the cost for player i to invest in a social tie is [c.sub.i], and the benefit to player i to access another player is [V.sub.i]. In addition, I assume that the length of the path connecting player i to j does not matter in defining the benefits. Here, I provide a complete equilibrium characterization: a connected equilibrium is a wheel network and an unconnected equilibrium network is a center-sponsored star, a wheel with local center-sponsored stars, a wheel with singletons or empty (Proposition 3.1). Figure 1 illustrates all strict equilibria in a society with four players. This result provides three main insights. The first is that the wheel architecture is robust to asymmetries which are independent of the potential partner.

Secondly, players' heterogeneity alters the level of connectedness of equilibrium networks. Nonempty unconnected equilibria have well defined properties: there is a set of players sharing a maximum amount of information while the remaining players are socially isolated (they do not access any information). In sharp contrast with the homogeneous setting these equilibria are asymmetric and central players may emerge: (1) the players maximally informed are connected in a wheel component; and the players socially isolated are either (2)a. singletons or b. spokes of center-sponsored stars. Third, I show that the property of centrality uniquely and only emerges when the distribution of costs of linking and values of accessing other players satisfies two conditions. On the one hand, there are players, the centers, who have a very low costs of linking as compared to the values of accessing other players. On the other hand, there are players, isolate players, who have a very high costs of forming links as compared to the values of accessing other agents.

I then turn to settings where heterogeneity also depends on the potential partner. I show that as far as the costs of linking are not partner-specific even if values vary freely, at equilibrium, every nonsingleton component has still a wheel architecture (Proposition 3.2). Differently, when the costs of linking are allowed to depend on the potential partner almost any minimal network is a strict equilibrium for some costs and values (Proposition 3.3). Thus, when information flow without frictions, it is the costs of forming links heterogeneity which shapes the architecture of equilibrium networks. Is this result an artifact of the frictionless information flow assumption? The answer to this question is positive. Indeed, I show that as far as a small amount of decay in the information flow is introduced, values' asymmetries are as important as costs' asymmetries in determining the architecture of equilibrium networks (Proposition 3.4).

This paper is a contribution to the theory of network formation. This is a very active area of research currently (see references in footnote 2). Most of the existing literature focuses on the two-way flow connections model and it assumes homogeneous players. I elaborate on the respective roles of values and costs of forming links heterogeneity in shaping equilibrium architectures in a one-way flow connections model. My findings indicate that values and costs heterogeneity are equally important in determining the level of connectedness and the architecture of equilibrium networks. Furthermore, under players' heterogeneity unconnected equilibrium networks are asymmetric and central players arise under well-defined conditions. The emergence of equilibrium networks with central players sharply contrasts with the homogeneous model.

The works that come closer to mine are Galeotti et al. (2005) and Kim and Wong (2003). Galeotti et al. (2005) study the role of heterogeneity in a two-way flow model, while I focus on the one-way flow network technology. It is worth noting that the two assumptions on the information flow provide individuals with very different incentives and this makes the two models belonging to two distinct class of games. Galeotti et al. (2005) confirm that equilibrium networks exhibit short distances and high centrality even in settings with substantial heterogeneity. My findings show that in directed networks, players' heterogeneity allows for the emergence of central players, a property which cannot be obtained in a homogeneous model.

Kim and Wong (2003) study a one-sided connections model with heterogeneous players where agents form...

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