We exploit a result by Nerman which shows that conditional limit theorems hold when a certain monotonicity condition is satisfied. Our main result is an application to vertex degrees in random graphs, where we obtain asymptotic normality for the number of vertices with a given degree in the random graph ${G(n,m)}$ with a fixed number of edges from the corresponding result for the random graph ${G(n,p)}$ with independent edges. We also give some simple applications to random allocations and to spacings. Finally, inspired by these results, but logically independent of them, we investigate whether a one-sided version of the Cramér--Wold theorem holds. We show that such a version holds under a weak supplementary condition, but not without it.
We consider random graphs with a given degree sequence and show, under weak technical conditions, asymptotic normality of the number of components isomorphic to a given tree, first for the random multigraph given by the configuration model and then, by a conditioning argument, for the simple uniform random graph with the given degree sequence. Such conditioning is standard for convergence in probability, but much less straightforward for convergence in distribution as here. The proof uses the method of moments, and is based on a new estimate of mixed cumulants in a case of weakly dependent var…
Random graphs with a given degree sequence are often constructed using the configuration model, which yields a random multigraph. We may adjust this multigraph by a sequence of switchings, eventually yielding a simple graph. We show that, assuming essentially a bounded second moment of the degree distribution, this construction with the simplest types of switchings yields a simple random graph with an almost uniform distribution, in the sense that the total variation distance is $o(1)$. This construction can be used to transfer results on distributional convergence from the configuration model…
Estimating Graph Dynamics from Population Observations
Mathematical Institute, Leiden University（莱顿大学数学研究所）；Korteweg-de Vries Institute for Mathematics, University of Amsterdam（阿姆斯特丹大学科特韦格-德弗里斯数学研究所）；Eurandom, Eindhoven University of Technology（埃因霍温理工大学欧拉姆德研究所）；Amsterdam Business School, Faculty of Economics and Business, University of Amsterdam（阿姆斯特丹大学经济与商业学院阿姆斯特丹商学院）
A weighted recursive tree is an evolving tree in which vertices are assigned random vertex-weights and new vertices connect to a predecessor with a probability proportional to its weight. Here, we study the maximum degree and near-maximum degrees in weighted recursive trees when the vertex-weights are almost surely bounded and their distribution function satisfies a mild regularity condition near zero. We are able to specify higher-order corrections to the first order growth of the maximum degree established in prior work. The accuracy of the results depends on the behaviour of the weight dist…
In this work we prove sufficient conditions for the Glauber dynamics corresponding to a sequence of (non-product) measures on finite product spaces to be rapidly mixing, i.e. that the mixing time with respect to the total variation distance satisfies $t_{mix} = O(N \log N)$, where $N$ is the system size. The proofs do not rely on coupling arguments, but instead use functional inequalities. As a byproduct, we obtain exponential decay of the relative entropy along the Glauber semigroup. These conditions can be checked in various examples, which include the exponential random graph models with su…
Let $T_n$ be a random recursive tree with $n$ nodes. List vertices of $T_n$ in decreasing order of degree as $v^1,\ldots,v^n$, and write $d^i$ and $h^i$ for the degree of $v^i$ and the distance of $v^i$ from the root, respectively. We prove that, as $n \to \infty$ along suitable subsequences, \[ \bigg(d^i - \lfloor \log_2 n \rfloor, \frac{h^i - μ\ln n}{\sqrt{σ^2\ln n}}\bigg) \to ((P_i,i \ge 1),(N_i,i \ge 1))\, , \] where $μ=1-(\log_2 e)/2$, $σ^2=1-(\log_2 e)/4$, $(P_i,i \ge 1)$ is a Poisson point process on $\mathbb{Z}$ and $(N_i,i \ge 1)$ is a vector of independent standard Gaussians. We addi…
