1. A phase transition in the evolution of bootstrap percolation processes on preferential attachment graphs
- Author
-
Nikolaos Fountoulakis and Mohammed Amin Abdullah
- Subjects
Phase transition ,Bootstrap percolation ,General Mathematics ,Critical phenomena ,0102 computer and information sciences ,Preferential attachment ,Quantitative Biology::Other ,01 natural sciences ,Combinatorics ,010104 statistics & probability ,FOS: Mathematics ,Quantitative Biology::Populations and Evolution ,Mathematics - Combinatorics ,0101 mathematics ,Mathematics ,Random graph ,Primary 60K30, 60K35, 05C90 Secondary 05C80, 60C05 ,Applied Mathematics ,Probability (math.PR) ,Computer Graphics and Computer-Aided Design ,Graph ,Vertex (geometry) ,010201 computation theory & mathematics ,Bounded function ,Combinatorics (math.CO) ,Mathematics - Probability ,Software - Abstract
The theme of this paper is the analysis of bootstrap percolation processes on random graphs generated by preferential attachment. This is a class of infection processes where vertices have two states: they are either infected or susceptible. At each round every susceptible vertex which has at least $r\geq 2$ infected neighbours becomes infected and remains so forever. Assume that initially $a(t)$ vertices are randomly infected, where $t$ is the total number of vertices of the graph. Suppose also that $r < m$, where $2m$ is the average degree. We determine a critical function $a_c(t)$ such that when $a(t) \gg a_c(t)$, complete infection occurs with high probability as $t \rightarrow \infty$, but when $a(t) \ll a_c (t)$, then with high probability the process evolves only for a bounded number of rounds and the final set of infected vertices is asymptotically equal to $a(t)$., This paper is significantly different to the previous version
- Published
- 2017