We derive asymptotic properties for a stochastic dynamic network model in a stochastic dynamic population. In the model, nodes give birth to new nodes until they die, each node being equipped with a social index given at birth. During the life of a n
ode it creates edges to other nodes, nodes with high social index at higher rate, and edges disappear randomly in time. For this model we derive criterion for when a giant connected component exists after the process has evolved for a long period of time, assuming the node population grows to infinity. We also obtain an explicit expression for the degree correlation $rho$ (of neighbouring nodes) which shows that $rho$ is always positive irrespective of parameter values in one of the two treated submodels, and may be either positive or negative in the other model, depending on the parameters.
We study the component structure in random intersection graphs with tunable clustering, and show that the average degree works as a threshold for a phase transition for the size of the largest component. That is, if the expected degree is less than o
ne, the size of the largest component is a.a.s. of logarithmic order, but if the average degree is greater than one, a.a.s. a single large component of linear order emerges, and the size of the second largest component is at most of logarithmic order.
