# Explosive Percolation on Directed Networks Due to Monotonic Flow of Activity

## TL;DR

This paper investigates explosive percolation in directed, rank-ordered networks, revealing a discontinuous emergence of large strongly connected components at high densities and the effects of biased link formation.

## Contribution

It introduces a simple monotonic percolation process on directed, rank-ordered networks and analyzes its unusual discontinuous transition behavior and component structure.

## Key findings

- Discontinuous emergence of a dominant strongly connected component.
- Large-scale connectivity occurs at very high edge density.
- Multiple giant strongly connected components can coexist before percolation.

## Abstract

An important class of real-world networks have directed edges, and in addition, some rank ordering on the nodes, for instance the "popularity" of users in online social networks. Yet, nearly all research related to explosive percolation has been restricted to undirected networks. Furthermore, information on such rank ordered networks typically flows from higher ranked to lower ranked individuals, such as follower relations, replies and retweets on Twitter. Here we introduce a simple percolation process on an ordered, directed network where edges are added monotonically with respect to the rank ordering. We show with a numerical approach that the emergence of a dominant strongly connected component appears to be discontinuous. Large scale connectivity occurs at very high density compared with most percolation processes, and this holds not just for the strongly connected component structure but for the weakly connected component structure as well. We present analysis with branching processes which explains this unusual behavior and gives basic intuition for the underlying mechanisms. We also show that before the emergence of a dominant strongly connected component, multiple giant strongly connected components may exist simultaneously. By adding a competitive percolation rule with a small bias to link uses of similar rank, we show this leads to formation of two distinct components, one of high ranked users, and one of low ranked users, with little flow between the two components.