Interfaces and the edge percolation map of random directed networks

TL;DR

This paper reexamines directed network percolation focusing on edges, revealing five key giant components and interfaces that connect different network regions, with analytical solutions validated against simulations.

Contribution

It introduces a formal framework for understanding edge-based percolation in directed networks, including new equations for component sizes and analysis of correlations.

Findings

Analytical solutions match simulations for various network types

Identifies five distinct giant edge components and interfaces

Provides insights into network structure and functionality

Abstract

The traditional node percolation map of directed networks is reanalyzed in terms of edges. In the percolated phase, edges can mainly organize into five distinct giant connected components, interfaces bridging the communication of nodes in the strongly connected component and those in the in- and out-components. Formal equations for the relative sizes in number of edges of these giant structures are derived for arbitrary joint degree distributions in the presence of local and two-point correlations. The uncorrelated null model is fully solved analytically and compared against simulations, finding an excellent agreement between the theoretical predictions and the edge percolation map of synthetically generated networks with exponential or scale-free in-degree distribution and exponential out-degree distribution. Interfaces, and their internal organization giving place from "hairy ball" percolation landscapes to bottleneck straits, could bring new light to the discussion of how structure is interwoven with functionality, in particular in flow networks.