Growing directed networks: organization and dynamics

We study the organization and dynamics of growing directed networks. These networks are built by adding nodes successively in such a way that each new node has K directed links to the existing ones. The organization of a growing directed network is analyzed in terms of the number of ‘descendants’ of each node in the network. We show that the distribution P(S) of the size, S, of the descendant cluster is described generically by a power-law, P(S)∼S−η, where the exponent η depends on the value of K as well as the strength of preferential attachment. We determine that, in the case of growing random directed networks without any preferential attachment, η is given by 1+1/K. We also show that the Boolean dynamics of these networks is stable for any value of K. However, with a small fraction of reversal in the direction of the links, the dynamics of growing directed networks appears to operate on ‘the edge of chaos’ with a power-law distribution of the cycle lengths. We suggest that the growing directed network may serve as another paradigm for the emergence of the scale-free features in network organization and dynamics.