We consider diffusion in random graphs with given vertex degrees. Our diffusion model can be viewed as a variant of a cellular automaton growth process: assume that each node can be in one of the two possible states, inactive or active. The parameters of the model are two given functions θ: N → N and α:N → [0,1]. At the beginning of the process, each node v of degree dv becomes active with probability α(dv) independently of the other vertices. Presence of the active vertices triggers a percolation process: if a node v is active, it remains active forever. And if it is inactive, it will become active when at least θ(dv) of its neighbors are active. In the case where α(d) = α and θ(d) = θ, for each d ∈ N, our diffusion model is equivalent to what is called bootstrap percolation. The main result of this paper is a theorem which enables us to find the final proportion of the active vertices in the asymptotic case, i.e., when n → ∞. This is done via analysis of the process on the multigraph counterpart of the graph model.
ASJC Scopus subject areas
- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics