Isolation probabilities in dynamic soft random geometric graphs

We consider soft random geometric graphs, constructed by distributing points (nodes) randomly according to a Poisson Point Process, and forming links between pairs of nodes with a probability that depends on their mutual distance, the "connection function." Each node has a probability of being isolated depending on the locations of the other nodes; we give analytic expressions for the distribution of isolation probabilities. Keeping the node locations fixed, the links break and reform over time, making a dynamic network; this is a good model of a wireless ad-hoc network with communication channels undergoing rapid fading. We use the above isolation probabilities to investigate the distribution of the time to transmit information to all the nodes, finding good agreement with numerics.