A modified bootstrap percolation on a random graph coupled with a lattice.
Discrete Applied Mathematics(2019)
摘要
In this paper a random graph model GZN2,pd is introduced, which is a combination of fixed torus grid edges in (Z∕NZ)2
and some additional random ones. The random edges are called long, and the probability of having a long edge between vertices u,v∈(Z∕NZ)2 with graph distance d on the torus grid is pd=c∕Nd, where c is some constant. We show that, whp, the diameter D(GZN2,pd)=Θ(logN). Moreover, we consider a modified non-monotonous bootstrap percolation on GZN2,pd. We prove the presence of phase transitions in mean-field approximation and provide fairly sharp bounds on the error of the critical parameters.
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关键词
Graph diameter,Degree distribution,Bootstrap percolation,Phase transitions
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