Epidemic processes on digraphs of random mappings

Abstract

A random mapping (T n ;q) of a finite set V, V = {1,2,…,n}, into itself assigns independently to each i ∊ V its unique image j ∊ V with probability q if i = j and with probability P = (1-q)/(n−1) if i ≠ j. Three versions of epidemic processes on a random digraph G T representing (T n ;q) are studied. The exact probability distributions of the total number of infected elements as well as the threshold functions for these epidemic processes are determined.