has degree , then the giant component exists if and only if which is also written in the form of is the mean degree of the network. Similar expressions are also valid for directed graphs, in which case the degree distribution is two-dimensional. There are three types of connected components in directed graphs. For a randomly chosen vertex: out-component is a set of vertices that can be reached by recursively following all out-edges forward; in-component is a set of vertices that can be reached by recursively following all in-edges backward; weak component is a set of vertices that can be reached by recursively following all edges regardless of their direction. Criteria for giant component existence in directed and undirected configuration graphs Let a randomly chosen vertex has in-edges and out edges. By definition, the average number of in- and out-edges coincides so that . If is the generating function of the degree distribution for an undirected network, then can be defined as . For directed networks, generating function assigned to the joint probability distribution can be written with two valuables and as: , then one can define and . The criteria for giant component existence in directed and undirected random graphs are given in the table below: See also Erdős–Rényi model Fractals Graph theory Interdependent networks Percolation theory Percolation Complex Networks Network Science Scale free networks References Graph connectivity Random graphs