Let Gn,p1 be a superposition of the random graph Gn,p and a one-dimensional lattice: the n
Nearby in the stack
vertices are set to be on a ring with fixed edges between the consecutive vertices, and with random independent edges given with probability
p
between any pair of vertices. Bootstrap percolation on a random graph is a process of spread of "activation" on a given realisation 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 study the size of the final active set in the limit when
n→∞
. The parameters of the model are
n
, the size
A0=A0(n)
of the initially active set and the probability
p=p(n)
of the edges in the graph. Bootstrap percolation process on
Gn,p
was studied earlier. Here we show that the addition of
n
local connections to the graph
Gn,p
leads to a more narrow critical window for the phase transition, preserving however, the critical scaling of parameters known for the model on
Gn,p
. We discover a range of parameters which yields percolation on