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Inspired by a concept in comparative genomics, we investigate properties of randomly chosen members of G_1(m,n,t), the set of bipartite graphs with $m$ left vertices, n right vertices, t edges, and each vertex of degree at least one. We give asymptotic results for the number of such graphs and the number of $(i,j)$ trees they contain. We compute the thresholds for the emergence of a giant component and for the graph to be connected.
We consider the random walk attachment graph introduced by Saram{a}ki and Kaski and proposed as a mechanism to explain how behaviour similar to preferential attachment may appear requiring only local knowledge. We show that if the length of the rando
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
Consider a collection of random variables attached to the vertices of a graph. The reconstruction problem requires to estimate one of them given `far away observations. Several theoretical results (and simple algorithms) are available when their join
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
In this paper we study the impact of random exponential edge weights on the distances in a random graph and, in particular, on its diameter. Our main result consists of a precise asymptotic expression for the maximal weight of the shortest weight pat