ﻻ يوجد ملخص باللغة العربية
Consider a random regular graph with degree $d$ and of size $n$. Assign to each edge an i.i.d. exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the shortest-weight paths between a fixed vertex and all the other vertices, as well as between any pair of vertices. Namely, for any fixed $d geq 3$, we show that the longest of these shortest-weight paths has about $hat{alpha}log n$ edges where $hat{alpha}$ is the unique solution of the equation $alpha log(frac{d-2}{d-1}alpha) - alpha = frac{d-3}{d-2}$, for $alpha > frac{d-1}{d-2}$.
We study some percolation problems on the complete graph over $mathbf N$. In particular, we give sharp sufficient conditions for the existence of (finite or infinite) cliques and paths in a random subgraph. No specific assumption on the probability,
A localized method to distribute paths on random graphs is devised, aimed at finding the shortest paths between given source/destination pairs while avoiding path overlaps at nodes. We propose a method based on message-passing techniques to process g
Consider random $d$-regular graphs, i.e., random graphs such that there are exactly $d$ edges from each vertex for some $dge 3$. We study both the configuration model version of this graph, which has occasional multi-edges and self-loops, as well as
A bootstrap percolation process on a graph G is an infection process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round every uninfected node which has at least r infected neighbours becomes infected
For each $n ge 1$, let $mathrm{d}^n=(d^{n}(i),1 le i le n)$ be a sequence of positive integers with even sum $sum_{i=1}^n d^n(i) ge 2n$. Let $(G_n,T_n,Gamma_n)$ be uniformly distributed over the set of simple graphs $G_n$ with degree sequence $mathrm