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

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