Symmetric Structure in Spatial and Social Processes

This paper presents an exploration into the effect of symmetric structure on the equilibrium properties of a widely applied class of ergodic Markov processes. An initial discussion of the notions concerning structure and process in this context sets the scene for the statement and proof of a simple theorem relating structures with a symmetry property to their associated steady-state relationships. The theorem has implications for models which arise in many disciplines. Symmetric structures and their related interpretation in terms of reversible processes occur in a variety of fields, and thus a series of examples pertaining to the spatial and social realms are used to illustrate these implications. The examples involve: access in spatial systems, diffusion based on random walks in spatial structures, economic exchange and equilibrium, social power and conflict resolution, design method in architecture and planning, and collective action in social systems. The themes running through all of these examples relate to balanced or equal movement, diffusion, exchange, or communication, which all imply reversible processes. The logic of the theorem is reinforced by each of these examples in intuitive terms, and by way of conclusion the many implications emerging from this analysis suggest directions for future research.