## Thomas Vallier (15/09/14)

Benedikt Magnússon, september 11, 2014

Málstofa í stærðfræði

### Fyrirlesari: Thomas Vallier, University of Iceland Titill: Bootstrap percolation on the random graph $$G_{n,p}$$

Staðsetning: V-157, VRII
Tími: Mánudagur 15. september, frá 15:00 til 16:00.

### Ágrip:

Bootstrap percolation on the random graph $$G_{n,p}$$ is a process of spread of “activation” on a given realization of the graph with a given number of initially active nodes. At each step those vertices which have not been active but have at least $$r ≥ 2$$ active neighbours become active as well.

We consider the n vertices with global connections inherited from the structure of the graph $$G_{n,p}$$, meaning that any two vertices share an edge with probability $$p$$ independently of the others.

The presentation is based on the article ”Bootstrap percolation on the random graph $$G_{n,p}$$” by Janson, Luczak, Turova and Vallier.

Among other results, they study the size $$A^*$$ of the final active set depending on the number of vertices active at the origin as a function of n (the number of vertices) and $$p = p(n)$$ (the probability of connections) which is written $$a_0(n, p) = a_0$$. The model exhibits a sharp phase transition: depending on the parameters of the model the final size of activation with a high probability is either $$n − o(n)$$ or it is $$o(n)$$.

I will give a pictorial introduction to the model and explain briefly the approach of the authors to derive the threshold for bootstrap percolation on $$G_{n,p}$$.