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A general random graph evolution mechanism is defined. The evolution is a combination of the preferential attachment model and the interaction of N vertices (N>=3). A vertex in the graph is characterized by its degree and its weight. The…

The spread of infectious disease in a human community or the proliferation of fake news on social media can be modeled as a randomly growing tree-shaped graph. The history of the random growth process is often unobserved but contains…

A random network is grown by introducing at unit rate randomly selected nodes on the Euclidean space. A node is randomly connected to its $i$-th predecessor of degree $k_i$ with a directed link of length $\ell$ using a probability…

We consider Gibbs distributions on finite random plane trees with bounded branching. We show that as the order of the tree grows to infinity, the distribution of any finite neighborhood of the root of the tree converges to a limit. We…

We survey recent developments about random real trees, whose prototype is the Continuum Random Tree (CRT) introduced by Aldous in 1991. We briefly explain the formalism of real trees, which yields a neat presentation of the theory and in…

We consider a continuous-time random walk on a regular tree of finite depth and study its favorite points among the leaf vertices. For the walk started from a leaf vertex and stopped upon hitting the root we prove that, in the limit as as…

Consider the d-dimensional lattice Z^d where each vertex is ``open'' or ``closed'' with probability p or 1-p, respectively. An open vertex v is connected by an edge to the closest open vertex w such that the dth co-ordinates of v and w…

We are interested in the asymptotics of random trees built by linear preferential attachment, also known in the literature as Barab\'asi-Albert trees or plane-oriented recursive trees. We first prove a conjecture of Bubeck, Mossel \& R\'acz…

G. Edelman, O. Sporns, and G. Tononi have introduced the neural complexity of a family of random variables, defining it as a specific average of mutual information over subfamilies. We show that their choice of weights satisfies two natural…

The statistical property of a growing scale-free network is studied based on an earlier model proposed by Krapivsky, Rodgers, and Redner [Phys. Rev. Lett. 86, 5401 (2001)], with the additional constraints of forbidden of self-connection and…

We show that the structure of a growing tree preserves an information on the shape of an initial graph. For the exponential trees, evidence of this kind of memory is provided by means of the iterative equations, derived for the moments of…

A degree sequence is a sequence ${\bf s}=(N_i,i\geq 0)$ of non-negative integers satisfying $1+\sum_i iN_i=\sum_i N_i<\infty$. We are interested in the uniform distribution $\mathbb{P}_{{\bf s}}$ on rooted plane trees whose degree sequence…

Understanding the patterns and processes of diversification of life in the planet is a key challenge of science. The Tree of Life represents such diversification processes through the evolutionary relationships among the different taxa, and…

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…

We prove that the uniform unlabelled unrooted tree with n vertices and vertex degrees in a fixed set converges in the Gromov-Hausdorff sense after a suitable rescaling to the Brownian continuum random tree. This proves a conjecture by…

We consider random binary trees that appear as the output of certain standard algorithms for sorting and searching if the input is random. We introduce the subtree size metric on search trees and show that the resulting metric spaces…

Diversification is nested, and early models suggested this could lead to a great deal of evolutionary redundancy in the Tree of Life. This result is based on a particular set of branch lengths produced by the common coalescent, where…

We introduce a random graph model based on k-trees, which can be generated by applying a probabilistic preferential attachment rule, but which also has a simple combinatorial description. We carry out a precise distributional analysis of…

Dealing with datasets of very high dimension is a major challenge in machine learning. In this paper, we consider the problem of feature selection in applications where the memory is not large enough to contain all features. In this…

When data is plentiful, the loss achieved by well-trained neural networks scales as a power-law $L \propto N^{-\alpha}$ in the number of network parameters $N$. This empirical scaling law holds for a wide variety of data modalities, and may…