Nonreciprocal random networks and their percolation properties

Abstract: We study the effects of nonreciprocity and network structure on percolation. To this end, we investigate nonreciprocal random networks - directed networks for which the probability of a link occurring from node i to node j differs from the probability of the reverse link occurring from node j to node i. We analytically determine the degree and percolation properties of such networks with exactly two types of link probability, demonstrating that whether the networks are structured such that the nodes are not statistically indistinguishable has profound effects on these measures, both quantitively and in how such networks need to be approached. In particular, we develop a technique for solving the percolation problem which can be applied to both structured and unstructured networks. The method entails writing self-consistent integral and differential equations for the probability that each node will belong to the network's giant component. Exact solutions to these equations are obtained and simulations which confirm our analytic predictions are presented.