4 research outputs found
Bootstrap percolation on a graph with random and local connections
Let be a superposition of the random graph and a
one-dimensional lattice: the vertices are set to be on a ring with fixed
edges between the consecutive vertices, and with random independent edges given
with probability 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 active
neighbours become active as well. We study the size of the final active set in
the limit when . The parameters of the model are , the
size of the initially active set and the probability of
the edges in the graph.
Bootstrap percolation process on was studied earlier. Here we show
that the addition of local connections to the graph leads to a
more narrow critical window for the phase transition, preserving however, the
critical scaling of parameters known for the model on . We discover a
range of parameters which yields percolation on but not on
.Comment: 38 pages, 2 figure
On varieties of graphs
In this paper, we introduce the notion of a variety of graphs closed under isomorphic images, subgraph identifications and induced subgraphs (induced connected subgraphs) firstly and next closed under isomorphic images, subgraph identifications, circuits and cliques. The structure of the corresponding lattices is investigated
On varieties of graphs
In this paper, we introduce the notion of a variety of graphs closed under isomorphic images, subgraph identifications and induced subgraphs (induced connected subgraphs) firstly and next closed under isomorphic images, subgraph identifications, circuits and cliques. The structure of the corresponding lattices is investigated