Asbtract:  We consider a random network in which nodes, as well as edges, are born and die. The node population is Markovian and so is the edge-network (given the node population). Each node is equipped with a random social index affecting the rate of creating neighbours. We derive the asymptotic population size, age-distribution and degree-distribution for the network, give some numerical illustrations, and discuss possible extensions of the model. (Joint work with Mathias Lindholm)