ﻻ يوجد ملخص باللغة العربية
We study the spectrum of a random multigraph with a degree sequence ${bf D}_n=(D_i)_{i=1}^n$ and average degree $1 ll omega_n ll n$, generated by the configuration model, and also the spectrum of the analogous random simple graph. We show that, when the empirical spectral distribution (ESD) of $omega_n^{-1} {bf D}_n $ converges weakly to a limit $ u$, under mild moment assumptions (e.g., $D_i/omega_n$ are i.i.d. with a finite second moment), the ESD of the normalized adjacency matrix converges in probability to $ uboxtimes sigma_{rm sc}$, the free multiplicative convolution of $ u$ with the semicircle law. Relating this limit with a variant of the Marchenko--Pastur law yields the continuity of its density (away from zero), and an effective procedure for determining its support. Our proof of convergence is based on a coupling between the random simple graph and multigraph with the same degrees, which might be of independent interest. We further construct and rely on a coupling of the multigraph to an inhomogeneous ErdH{o}s-Renyi graph with the target ESD, using three intermediate random graphs, with a negligible fraction of edges modified in each step.
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 $mathrm
Recent work has introduced sparse exchangeable graphs and the associated graphex framework, as a generalization of dense exchangeable graphs and the associated graphon framework. The development of this subject involves the interplay between the stat
In this paper we study the impact of random exponential edge weights on the distances in a random graph and, in particular, on its diameter. Our main result consists of a precise asymptotic expression for the maximal weight of the shortest weight pat
We prove a non-asymptotic concentration inequality for the spectral norm of sparse inhomogeneous random tensors with Bernoulli entries. For an order-$k$ inhomogeneous random tensor $T$ with sparsity $p_{max}geq frac{clog n}{n }$, we show that $|T-mat
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