Using Self-Consistent Integral Equations to Determine Percolation in Nonreciprocal Random Networks

Nonreciprocal random (NR) networks, for which the probability of node i connecting to node j differs from the probability of node j connecting to node i, are an important generalisation of directed Erdős–Rényi networks. They can be fruitfully used to study the effects of network structure such as node distinguishability on network properties and are suitable for modelling trade or competitive networks. In order to study their percolation properties, we have developed self-consistent integral equations which can be used to analytically determine both the critical threshold for percolation and the size of the giant strongly connected component (GSCC) in directed random networks. These equations are solved exactly for both unstructured intransitive NR networks and structured transitive NR networks thereby demonstrating that transitive structure substantially limits the circumstances under which percolation can occur in NR networks. Simulations confirm the validity of these solutions.