A Dynamical Framework for Social Network Construction

In its barest form, a network exists as a set of vertices and a set of edges connecting them. Mapping real-world objects to vertices and particular relations between them to edges, the structure of a real-world system can be investigated as a network. Real-world systems studied as networks include the Internet, the World Wide Web, organizational and business networks, transportation systems, distribution networks, food webs, metabolic networks, social networks of acquaintanceship or other relations, networks of coauthorship and citations in academic papers, and many others. A good review of the field and general techniques used in its study can be found in [1]. For systems whose low-level structure can be thoroughly described in terms of simple dyadic relationships, a network model provides a full structural description. The Internet, a system of routers that transfer information between one another over physically extant data channels, is a good example. Other systems can be expressed and fruitfully analyzed as networks, but with some degree of loss in conversion. As a model of a social structure, a friendship network amongst a group of people suffers from both the narrowness and breadth of its scope. A social structure is composed of much more than just friendships between its members, and friendship itself is a decidedly subjective relation, spanning a broad range of significance for even a single observer. A friendship network might give insight into the functioning of a society, but does not define or constrain that society in the same way as a network of data lines constrains the internet. When looking at the evolution and growth of a realworld network, then, it would seem crucial to consider what level of systematic detail the network represents. Systems like the internet whose core structure is easily represented as a network should admit network-based models of their evolution. Such models might reasonably evolve a representative network recursively, solely in response to the network itself. In contrast, a system where a network representation constitutes a narrow view of a much richer structure should not look to the network alone in describing the system’s – and therein the network’s – evolutionary trajectory. This is a tall order, one that has not been met in the literature. Generally, models of network growth include the continual addition of edges or vertices to some initial network substrate under some preferential attachment scheme. The particular preferential attachment rule is designed either to generate a graph with certain statistical characteristics, or to capture premises of the modeled system by dictating how an individual edge might form in response to its network neighborhood. Some of these models are well-conceived (see [2]), and capture many of the empirical network qualities of the systems they represent. In restricting their growth mechanisms to network-endemic qualities, though, such models deny the contribution of the rest of their system’s rich structure. To explore the sorts of networks that might arise from a system defined only secondarily by way of its network structure, we propose a dynamical model of a social system. In our model, agents’ network relationships define their dynamical coupling in a social space, which leads to a system affected – but no defined – by network structure. Observing the evolution of networks in this model, we evaluate its usefulness as a representation of real-world social systems and as a unique dynamical system.