Emergence of the giant weak-component in directed random graphs with arbitrary degree distributions

The weak component generalizes the idea of connected components to directed graphs. In this paper, an exact criterion for existence of the giant weak component is derived for directed graphs with arbitrary bivariate degree distributions. In addition we consider a random process for evolving directed graphs with bounded degrees. The bounds are not the same for different vertices but satisfy a pre-defined distribution. The analytic expression obtained for the evolving degree distribution is then combined with the weak-component criterion to obtain the exact time of the phase transition. The phase-transition time is obtained as a function of the distribution that bounds the degrees. Remarkably, when viewed from the step polymerization formalism, the new results yield Flory-Stockmayer gelation theory and generalize it to a broader scope.