Related papers: percolation on finite graphs
Consider a uniform expanders family G_n with a uniform bound on the degrees. It is shown that for any p and c>0, a random subgraph of G_n obtained by retaining each edge, randomly and independently, with probability p, will have at most one…
We consider the Bernoulli bond percolation process (with parameter $p$) on infinite graphs and we give a general criterion for bounded degree graphs to exhibit a non-trivial percolation threshold based either on a single isoperimetric…
We prove that critical percolation on any quasi-transitive graph of exponential volume growth does not have a unique infinite cluster. This allows us to deduce from earlier results that critical percolation on any graph in this class does…
We investigate fast diffusions on finite directed graphs. We prove results in a way dual to presented in Bobrowski, A. Ann. Henri Poincar\'e (2012) 13(6): 1501-1510 and Bobrowski, A., Morawska, K. DCDS-B (2012), 17(7): 2313-2327, and obtain…
Bounds on the minimum degree and on the number of vertices at- taining it have been much studied for finite edge-/vertex-minimally k- connected/k-edge-connected graphs. We give an overview of the results known for finite graphs, and show…
We study the limit theory of large threshold graphs and apply this to a variety of models for random threshold graphs. The results give a nice set of examples for the emerging theory of graph limits.
A certain class of directed metric graphs is considered. Asymptotics for a number of possible endpoints of a random walk at large times is found.
We study the random graph obtained by random deletion of vertices or edges from a random graph with given vertex degrees. A simple trick of exploding vertices instead of deleting them, enables us to derive results from known results for…
We study combinatorial parameters of a recently introduced bootstrap percolation problem in finite projective planes. We present sharp results on the size of the minimum percolating sets and the maximal non-percolating sets. Additional…
In this work we study the asymptotic traffic flow in Gromov's hyperbolic graphs. We prove that under certain mild hypotheses the traffic flow in a hyperbolic graph tends to pass through a finite set of highly congested nodes. These nodes…
In this paper, the leading term of the asymptotics of the number of possible final positions of a random walk on a directed Hamiltonian metric graph is found. Consideration of such dynamical systems could be motivated by problems of…
The aim of this paper is twofold. First, we study eigenvalues and eigenvectors of the adjacency matrix of a bond percolation graph when the base graph is finite and well approximated locally by an infinite regular graph. We relate…
We introduce a set of iterative equations that exactly solves the size distribution of components on small arbitrary graphs after the random removal of edges. We also demonstrate how these equations can be used to predict the distribution…
In this note, we study non-transitive graphs and prove a number of results when they satisfy a coarse version of transitivity. Also, for each finitely generated group $G$, we produce continuum many pairwise non-quasi-isometric regular…
We study the existence of periodic colorings and orientations in locally finite graphs. A coloring or orientation of a graph $G$ is periodic if the resulting colored or oriented graph is quasi-transitive, meaning that $V(G)$ has finitely…
In this work we establish that finite directed graphs give rise to semiflows on the power set of their nodes. We analyze the topological dynamics for semiflows on finite directed graphs by characterizing Morse decompositions, recurrence…
We study site and bond percolation on directed simple random graphs with a given degree distribution and derive the expressions for the critical value of the percolation probability above which the giant strongly connected component emerges…
In this note we study the phase transition for percolation on quasi-transitive graphs with quasi-transitively inhomogeneous edge-retention probabilities. A quasi-transitive graph is an infinite graph with finitely many different "types" of…
A broader definition of generalized truncations of graphs is introduced followed by an exploration of some standard concepts and parameters with regard to generalized truncations.
We study transitivity properties of graphs with more than one end. We completely classify the distance-transitive such graphs and, for all $k \geq 3$, the $k$-CS-transitive such graphs.