Emergence of the giant weak component in directed random graphs with arbitrary degree distributions
Abstract
The weak component generalizes the idea of connected components to directed graphs. In this paper, an exact criterion for the 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 predefined distribution. The analytic expression obtained for the evolving degree distribution is then combined with the weakcomponent criterion to obtain the exact time of the phase transition. The phasetransition time is obtained as a function of the distribution that bounds the degrees. Remarkably, when viewed from the steppolymerization formalism, the new results yield FloryStockmayer gelation theory and generalize it to a broader scope.
 Publication:

Physical Review E
 Pub Date:
 July 2016
 DOI:
 10.1103/PhysRevE.94.012315
 arXiv:
 arXiv:1607.03793
 Bibcode:
 2016PhRvE..94a2315K
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Probability;
 05C80;
 82B30
 EPrint:
 12 pages, 5 figures