Growing a Network on a Given Substrate

Conventional studies of network growth models mainly look at the steady state degree distribution of the graph. Often long time behavior is considered, hence the initial condition is ignored. In this contribution, the time evolution of the degree distribution is the center of attention. We consider two specific growth models; incoming nodes with uniform and preferential attachment, and the degree distribution of the graph for arbitrary initial condition is obtained as a function of time. This allows us to characterize the transient behavior of the degree distribution, as well as to quantify the rate of convergence to the steady-state limit.