Spreading Processes and Large Components in Ordered, Directed Random Graphs

TL;DR

This paper analyzes a directed random graph model with ordered vertices, establishing a sharp threshold for the emergence of a large reachable component related to spreading processes on networks.

Contribution

It proves the existence of a sharp threshold at p* = log(n)/n for the transition from small to large reachable components in ordered directed random graphs.

Findings

Threshold at p* = log(n)/n for large component emergence

Transition from o(n) to Ω(n) size of reachable component

Applicable to spreading processes in directed networks

Abstract

Order the vertices of a directed random graph \math{v_1,...,v_n}; edge \math{(v_i,v_j)} for \math{i<j} exists independently with probability \math{p}. This random graph model is related to certain spreading processes on networks. We consider the component reachable from \math{v_1} and prove existence of a sharp threshold \math{p^*=\log n/n} at which this reachable component transitions from \math{o(n)} to \math{\Omega(n)}.