The Emergence of Complex Exchange Networks

The celebrated ”small-world” and ”scale-free” network structures have recently been prominently proposed as models of large networks in a variety of substantive realms. However, it remains unclear to what degree the mechanisms of ”preferential attachment” and ”random rewiring” that have been argued to underlie the emergence of these networks are sociologically plausible. First, these mechanisms require the sociologically implausible assumption of globally available knowledge about structural positions. Second, individual goals and cognitions that motivate actors’ decisions to make or break ties remain implicit. In this paper we present a computational model of the emergence of complex networks (such as small-world or power-law) that is grounded in social exchange theory and assumptions of bounded rationality. More specifically, we draw on previous work by Flache and Hegselmann and assume that agents seek to find and keep attractive exchange partners in a population where actors differ in attractiveness, for example in their expertise in knowledge exchange. Agents pursue their goal under imperfect, local information without initial knowledge about others or the global network structure. Moreover, agents in our model are strongly adaptive, i.e. knowledge is only acquired in the course of interaction and simple search heuristics are applied. Our results suggest that under a large range of conditions a segregated network arises in which exchange partners are highly similar in attractiveness. Moreover, we find that the topological structure of the emergent social network depends heavily upon the harshness of the exchange game (i.e. the potential damage agents can suffer from exploitation and the restrictiveness of access to potential exchange partners) and the learning capacities of the agents (memory size, exploration probability, knowledge exchange mechanism). In particular, we show that it depends on the combination of these conditions whether star-like, small-world or scale-free structures emerge. 1 Macrostructures and Microprocesses in Complex Network Emergence The celebrated ”small-world” Watts and Strogatz (1998) and ”scale-free” Barabási and Albert (1999) network structures have recently been prominently proposed as models of large networks in a variety of substantive realms, including networks of social relationships. It has been claimed that empirical examples of complex network structures can be found in the global patterns of website references in Polytechnic University of Catalonia, Barcelona, Spain. Email: {jmpujol,sanguesa,jdelgado}@lsi.upc. es University of Groningen, Groningen, The Netherlands. Email: a.flache@ppsw.rug. nl 1 Macrostructures and Microprocesses in Complex Network Emergence 2 the World Wide Web, acquaintance relationships between people, networks of sex contacts, networks of scientific collaborations, or contacts between terrorists. Interest in these so-called ”complex networks” has been particularly stimulated, because on the one hand theoretical research suggests that these structures are highly efficient in terms of certain global network features, and on the other hand these networks may be the emergent, yet unintended result of simple processes of network change that operate at the level of the individual nodes or ”agents” 1. In particular, complex networks have shown to be very efficient in terms of information propagation Walsh (1999); Delgado (2002), and their connectedness is very robust against randomly distributed hazards that may destroy single nodes in the network Albert and Barabási (2000). The two most prominent models of the individual level mechanisms that have been proposed to underlie the emergence of complex networks are the mechanisms of stochastic rewiring Watts and Strogatz (1998), described in the seminal work by Watts and Strogatz, and the mechanism of preferential attachment and uniform growth, proposed by Barabási and Albert (1999). This work has inspired a range of subsequent models, particularly of the emergence of power-law networks (for a recent overview cf. Albert and Barabási (2002)). The model of stochastic rewiring has been designed to show how relatively few random changes in a network can transform an initially fully locally clustered network with large (or even infinite, cf. Watts (1999)) average path-length, into a ”small-world” structure that is still locally clustered, yet characterized by short average distances between the nodes. Local clustering refers here to a large degree of overlap between the ego-networks of adjacent nodes, as in small villages where residents are mainly connected to other residents and have few external contacts. Watts and Strogatz (1998) showed that a small probability of ”rewiring” a randomly selected tie to some randomly selected new target in the network is sufficient to bring about a radical change in the global features of the network, from a pattern with low speed of information propagation to a structure that is highly efficient for information diffusion through network ties. The main reason is that a single rewired tie does not change fundamentally the local structures of ego-networks, but it is sufficient to bridge the distance between two otherwise distant, or even disconnected, local units. Later elaborations proposed social processes such as geographical migration within or between countries as examples of this mechanism Watts (1999). While the work by Watts and others was extremely important to demonstrate how efficient network structures may emerge from simple local processes, the authors have paid little attention to the question why and under what conditions the mechanism of stochastic random rewiring may be a plausible result of social behavior, i.e. of the individual motives and cognitions that lead social actors to make or break relationships. It is, for example, questionable whether geographical migration may produce stochastic random rewiring. Geographical migration is often characterized by locally correlated patterns of behavior, where the experience that one pioneer migrant makes at his new residence is communicated back to old acquaintances, potentially triggering a new wave of migration to the same location. By contrast, random migration would spread new ties across the entire population, presumably with very different consequences for the network structures that may evolve. Accordingly, we believe that the conditions under which small-world structures may plausibly arise in social networks can be better identified when network changes are explicitly derived from sociologically plausible assumptions about the individual goals and behavioral heuristics that underlie network changes at the microlevel. 1The modelling approach we apply in this paper is called ”agent based modelling” (cf. Macy and Willer (2002) for a recent overview of agent based modelling in the social sciences). To keep our terminology consistent with the conventions of agent-based-modelling, we use in this paper the term ”agent” to refer to the individual actors that constitute the nodes of networks 1 Macrostructures and Microprocesses in Complex Network Emergence 3 The model of preferential attachment also leaves the individual level decision process widely implicit. Barabási and Albert developed their model to explain how in a growing network, like the network of page references in the world wide web, a power-law degree distribution emerges, where, broadly, the number of nodes that have a particular degree decreases with the degree to the power of some constant. The mechanism of preferential attachment assumes that new actors who enter the network prefer to relate to those existing nodes that already have a high degree. A plausible reason for preferential attachment, which generalizes beyond the example of the internet, may be that actors strive for social status. new actors entering a group may expect that they maximize their own social status by relating to the most popular peers, generating preferential attachment. While the mechanism of preferential attachment may seem better sociologically motivated than random rewiring, it relies on two assumptions that seem implausible for a large range of social networks. First, the assumption that actors have public and full access to reliable knowledge about the structure of the whole network, in particular the indegree of any other member of the network. Second, the assumption that actors are cognitively capable to process this information accurately. These assumptions are required to obtain a parametric and analytical solution that can reproduce some empirically observed networks. However, we argue that due to the behavioral implausibility of the micromechanism, the model of preferential attachment reproduces rather than explains complex network structures. This criticism extends to recent more sophisticated models of preferential attachment, like the one proposed by Walsh (2001). In this model, a new node that is added to the network is attached to a node in the existent network following a probability distribution that depends on the degree of the nodes present in the network. While this model is very successful in generating graphs with the power-law property, it also assumes that agents know the complete distribution of degrees of the graph and process this information accurately when choosing to whom they want to relate. Several researchers have addressed the issue of behavioral models underlying complex network dynamics. Transitive linking Krapivsky and Redner (2001); Ebel et al. (2002) was in particular proposed as a mechanism where agents use solely information about their local ties. In a nutshell, transitive linking assumes that new ties are most likely be formed between actors who have a common acquaintance in the network. This work could show that preferential attachment behavior may arise from the transitive linking process, even when individual actors have no global