Dynamic random networks in dynamic populations

We consider a random network evolving in continuous time in which new nodes are born and old may die, and where undirected edges between nodes are created randomly and may also disappear. The node population is Markovian and so is the creation and deletion of edges, given the node population. Each node is equipped with a random social index and the intensity at which a node creates new edges is proportional to the social index, and the neighbour is either chosen uniformly or proportional to the its social index in a modification of the model. We derive properties of the network as time and the node population tends to infinity. The degree distribution is shown to have a mixed Poisson distribution which may exhibit a heavy tail if the social index distribution has a heavy tail.