相关论文: Depth properties of scaled attachment random recur…

The tree-depth is a parameter introduced under several names as a measure of sparsity of a graph. We compute asymptotic values of the tree-depth of random graphs. For dense graphs, p>> 1/n, the tree-depth of a random graph G is a.a.s.…

In this paper, we study the online nearest neighbor random tree in dimension $d\in \mathbb N$ (called $d$-NN tree for short) defined as follows. We fix the torus $\mathbb T^d_n$ of dimension $d$ and area $n$ and equip it with the metric…

Consider~\(n\) nodes distributed independently across~\(N\) cities contained with the unit square~\(S\) according to a distribution~\(f.\) Each city is modelled as an~\(r_n \times r_n\) square contained within~\(S\) and~\(MSTC_n\) denotes…

The depth-weighted tree DWT($f$) with weight function $f:\{0,1,2,\ldots\}\to (0,\infty)$ is a dynamic random tree grown from a root $r$ where vertices arrive consecutively and every new vertex attaches to a parent $u$ with probability…

We investigate the distribution of the depth of a node containing a specific key or, equivalently, the number of steps needed to retrieve an item stored in a randomly grown binary search tree. Using a representation in terms of mixed and…

We present a simple yet rigorous approach to the determination of the spectral dimension of random trees, based on the study of the massless limit of the Gaussian model on such trees. As a byproduct, we obtain evidence in favor of a new…

Consider a family of random ordered graph trees $(T_n)_{n\geq 1}$, where $T_n$ has $n$ vertices. It has previously been established that if the associated search-depth processes converge to the normalised Brownian excursion when rescaled…

We investigate extremal statistical properties such as the maximal and the minimal heights of randomly generated binary trees. By analyzing the master evolution equations we show that the cumulative distribution of extremal heights…

Let ${\mathbf T}_n$ be a uniformly random tree with vertex set $[n]=\{1,\ldots,n\}$, let $\Delta_{{\mathbf T}_n}$ be the largest vertex degree in ${\mathbf T}_n$, and let $\lambda_1({\mathbf T}_n),\ldots,\lambda_n({\mathbf T}_n)$ be the…

Local convergence of bounded degree graphs was introduced by Benjamini and Schramm. This result was extended further by Lyons to bounded average degree graphs. In this paper we study the convergence of random tree sequences with given…

The global structure of the minimal spanning tree (MST) is expected to be universal for a large class of underlying random discrete structures. However, very little is known about the intrinsic geometry of MSTs of most standard models, and…

Let (X,d_X) be an n-point metric space. We show that there exists a distribution D over non-contractive embeddings into trees f:X-->T such that for every x in X, the expectation with respect to D of the maximum over y in X of the ratio…

Bollob\'as-Riordan random pairing model of a preferential attachment graph $G_m^n$ is studied. Let $\{W_j\}_{j\le mn+1}$ be the process of sums of independent exponentials with mean $1$. We prove that the degrees of the first…

We study structural properties of trees grown by preferential attachment. In this mechanism, nodes are added sequentially and attached to existing nodes at a rate that is strictly proportional to the degree. We classify nodes by their depth…

Sparse residual tree (SRT) is an adaptive exploration method for multivariate scattered data approximation. It leads to sparse and stable approximations in areas where the data is sufficient or redundant, and points out the possible local…

We study a natural analogue of Ulam's problem for random rooted trees distributed according to a Plancherel-type measure. This probability measure is closely related to the classical Plancherel measure on integer partitions. For a…

We study a random recursive tree model featuring complete redirection called the random friend tree and introduced by Saram\"aki and Kaski. Vertices are attached in a sequential manner one by one by selecting an existing target vertex and…

We consider growing random recursive trees in random environment, in which at each step a new vertex is attached (by an edge of a random length) to an existing tree vertex according to a probability distribution that assigns the tree…

We study the gradient-based training of large-depth residual networks (ResNets) from standard random initializations. We show that infinite-depth ResNets behave as if they were infinitely wide, regardless of their actual width. More…

We study a new type of random minimum spanning trees. It is built on the complete graph where each vertex is given a weight, which is a positive real number. Then, each edge is given a capacity which is a random variable that only depends…