Giant strongly connected component of directed networks - art. no. 025101

We describe how to calculate the sizes of all giant connected components of a directed graph, including the strongly connected one. In particular, the World Wide Web is a directed network. The results are obtained for graphs with statistically uncorrelated vertices and an arbitrary joint in and out- degree distribution P(k(i),k(o)). We show that if P(k(i),k(o)) does not fac torize, the relative size of the giant strongly connected component deviate s from the product of the relative sizes of the giant in- and out-component s. The calculations of the relative sizes of all the giant components are d emonstrated using the simplest examples. We explain that the giant strongly connected component may be less resilient to random damage than the giant weakly connected one.