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A limit theorem for a sequence of diffusion processes on graphs is proved in a case when vary both parameters of the processes (the drift and diffusion coefficients on every edge and the asymmetry coefficients in every vertex), and configuration of graphs, where the processes are set on. The explicit formulae for the parameters of asymmetry for the vertices of the limiting graph are given in the case, when, in the pre-limiting graphs, some groups of vertices form knots contracting into a points.
We study a stochastic compartmental susceptible-infected (SI) epidemic process on a configuration model random graph with a given degree distribution over a finite time interval $[0,T],$ for some $ T>0$. In this setting, we split the population of gr
For a difference approximations of multidimensional diffusion, the truncated local limit theorem is proved. Under very mild conditions on the distribution of the difference terms, this theorem provides that the transition probabilities of these appro
We consider bootstrap percolation and diffusion in sparse random graphs with fixed degrees, constructed by configuration model. Every node has two states: it is either active or inactive. We assume that to each node is assigned a nonnegative (integer
Any (measurable) function $K$ from $mathbb{R}^n$ to $mathbb{R}$ defines an operator $mathbf{K}$ acting on random variables $X$ by $mathbf{K}(X)=K(X_1, ldots, X_n)$, where the $X_j$ are independent copies of $X$. The main result of this paper concerns
Let $mathcal{G} = {G_1 = (V, E_1), dots, G_m = (V, E_m)}$ be a collection of $m$ graphs defined on a common set of vertices $V$ but with different edge sets $E_1, dots, E_m$. Informally, a function $f :V rightarrow mathbb{R}$ is smooth with respect t