Temporal connectivity in finite networks with non-uniform measures.

Soft random geometric graphs (SRGGs) have been widely applied to various models including those of wireless sensors, communication, and social and neural networks. SRGGs are constructed by randomly placing nodes in some space and making pairwise links probabilistically using a connection function that is system specific and usually decays with distance. In this paper we focus on the application of SRGGs to wireless communication networks where information is relayed in a multihop fashion, although the analysis is more general and can be applied elsewhere by using different distributions of nodes and/or connection functions. We adopt a general nonuniform density which can model the stationary distribution of different mobility models, with the interesting case being when the density goes to zero along the boundaries. The global connectivity properties of these nonuniform networks are likely to be determined by highly isolated nodes, where isolation can be caused by the spatial distribution or the local geometry (boundaries). We extend the analysis to temporal-spatial networks where we fix the underlying nonuniform distribution of points and the dynamics are caused by the temporal variations in the link set, and we explore the probability that a node near the corner is isolated at time T. This work allows for insight into how nonuniformity (caused by mobility) and boundaries impact the connectivity features of temporal-spatial networks. We provide a simple method for approximating these probabilities for a range of different connection functions and verify them against simulations. Boundary nodes are numerically shown to dominate the connectivity properties of these finite networks with nonuniform measure.