Title: Temporal connectivity in finite networks with non-uniform measures,
Abstract: Soft Random Geometric Graphs (SRGGs) have been widely applied to various
models including those of wireless sensor, communication, 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 multi hop 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 non-uniform 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 non-uniform 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 non-uniform
distribution of points and the dynamics are caused by the temporal variations
in the link set, and explore the probability a node near the corner is isolated
at time $T$. This work allows for insight into how non-uniformity (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 non-uniform measure.