Strict Majority Bootstrap Percolation on Augmented Tori and Random Regular Graphs: Experimental Results

We study the strict majority bootstrap percolation process on graphs. Vertices may be active or passive. Initially, active vertices are chosen independently with probability p. Each passive vertex v becomes active if at least d deg(v)+1 2 e of its neighbors are active (and thereafter never changes its state). If at the end of the process all vertices become active then we say that the initial set of active vertices percolates on the graph. We address the problem of finding graphs for which percolation is likely to occur for small values of p. For that purpose we study percolation on two topologies. The first is an n× n toroidal grid augmented with a universal vertex. Also, each vertex v in the torus is connected to all nodes whose distance to v is less than or equal to a parameter r. The second family contains all random regular graphs of even degree, also augmented with a universal node. We compare our computational results to those obtained in previous publications for r-rings and random regular graphs.