Giant strongly connected component of directed networks

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 factorize, the relative size of the giant strongly connected component deviates from the product of the relative sizes of the giant in- and out-components. The calculations of the relative sizes of all the giant components are demonstrated 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.