Spatial networks: growth models and dynamical processes

Abstract

Many inherently spatial systems have been represented using networks. This thesis contributes to the understanding of such networks by investigating the effect of imposing spatial constraints upon both the process network formation and dynamics that occur upon the network. Degree heterogeneity is a feature of several real world networks. However, edge length is typically constrained in a spatial network, preventing the formation of the high degree nodes that are characteristic of degree heterogeneity. We instead constrain the network to be planar, producing networks that have a scale-free degree distribution. This model turns out to be a variant of random Apollonian growth and a one parameter family of models which incorporates the planar model alongside existing Apollonian models is proposed. We identify the REDS model as a spatial model that does constrain edge length and exhibits a form of degree heterogeneity, albeit a weaker form than the scale-free distribution. REDS seeks to model social network formation by conceiving its nodes as agents who disburse a personal budget in order to maintain social bonds. We strengthen the model’s plausibility by introducing uncertainty into the agents’ budget expenditure decisions. The degree heterogeneity that was readily observed in the original model is now recovered only where decisions are subject to high levels of uncertainty. An evolutionary game is a process that lends itself to simulation upon a spatial network. This is due to the fact that a spatially constrained population is more likely to exhibit network reciprocity known to result in increased levels of co operation. We find those experiments within existing literature to be unsatisfactory in that network connectivity is assumed a priori. We address this issue by further extending the REDS model such that its nodes play prisoner’s dilemma with their network neighbours. The budget with which agents form connections is now earned by accumulating payoffs from the dilemma game. This allows for a network topology that is now endogenous to the model. This model is further distinguished from prior coevolutinary models by its agents’ ignorance of the details of their individual strategic interactions.

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