Related papers: The statistical geometry of scale-free random tree…

We investigate scaling limits of trees built by uniform attachment with freezing, which is a variant of the classical model of random recursive trees introduced in a companion paper. Here vertices are allowed to freeze, and arriving…

The magnitude of Pearson correlation between two scalar random variables can be visually judged from the two-dimensional scatter plot of an independent and identically distributed sample drawn from the joint distribution of the two…

Geographical phenomena fall into two categories: scaleful phenomena and scale-free phenomena. The former bears characteristic scales, and the latter has no characteristic scale. The conventional quantitative and mathematical methods can…

Scale transformations have played an extremely successful role in studies of cosmological large-scale structure by relating the non-linear spectrum of cosmological density fluctuations to the linear primordial power at longer wavelengths.…

We present a novel type of weighted scale-free network model, in which the weight grows independently of the attachment of new nodes. The evolution of this network is thus determined not only by the preferential attachment of new nodes to…

We study random unrooted plane trees with $n$ vertices sampled according to the weights corresponding to the vertex-degrees. Our main result shows that if the generating series of the weights has positive radius of convergence, then this…

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 characterize the existence of certain geometric configurations in the fractal percolation limit set $A$ in terms of the almost sure dimension of $A$. Some examples of the configurations we study are: homothetic copies of finite sets,…

We investigate the properties of the spanning trees of various real-world and model networks. The spanning tree representing the communication kernel of the original network is determined by maximizing total weight of edges, whose weights…

We study a random tree, which was introduced by Ajazi et al. as part of a model of a neuronal network. Realising a scaling relation for the law of the tree, we can use elementary techniques to derive asymptotic results on the geometry as…

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…

We study nongeneric planar trees and prove the existence of a Gibbs measure on infinite trees obtained as a weak limit of the finite volume measures. It is shown that in the infinite volume limit there arises exactly one vertex of infinite…

We study the connections between volume growth, spectral properties and stochastic completeness of locally finite graphs. For a class of graphs with a very weak spherical symmetry we give a condition which implies both stochastic…

A scale-free network is grown in the Euclidean space with a global directional bias. On a vertical plane, nodes are introduced at unit rate at randomly selected points and a node is allowed to be connected only to the subset of nodes which…

In this paper we model the tomography of scale free networks by studying the structure of layers around an arbitrary network node. We find, both analytically and empirically, that the distance distribution of all nodes from a specific…

We study the large-scale topological and dynamical properties of real Internet maps at the autonomous system level, collected in a three years time interval. We find that the connectivity structure of the Internet presents average…

We study a percolation model on the square lattice, where clusters "freeze" (stop growing) as soon as their volume (i.e. the number of sites they contain) gets larger than N, the parameter of the model. A model where clusters freeze when…

We propose a new, direct, correlation-free approach based on central moments of profiles to the asymptotics of width (size of the most abundant level) in some random trees of logarithmic height. The approach is simple but gives precise…

The general relationship between an arbitrary frequency distribution and the expectation value of the frequency distributions of its samples is esablished. A set of combinations of expectation values whose value does not in general depend…

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…