  This paper provides time-dependent expressions for the expected degree
distribution of a given network that is subject to growth, as a function of
time. We consider both uniform attachment, where incoming nodes form links to
existing nodes selected uniformly at random, and preferential attachment, when
probabilities are assigned proportional to the degrees of the existing nodes.
We consider the cases of single and multiple links being formed by each
newly-introduced node. The initial conditions are arbitrary, that is, the
solution depends on the degree distribution of the initial graph which is the
substrate of the growth. Previous work in the literature focuses on the
asymptotic state, that is, when the number of nodes added to the initial graph
tends to infinity, rendering the effect of the initial graph negligible. Our
contribution provides a solution for the expected degree distribution as a
function of time, for arbitrary initial condition. Previous results match our
results in the asymptotic limit.
