Related papers: Geometry of weighted recursive and affine preferen…

We study a model of random weighted uniform spanning trees on the complete graph with $n$ vertices, where each edge is assigned a weight of $n^{1+\gamma}$ with probability $1/n$ and $1$ otherwise. Whenever $\gamma$ is large enough, we prove…

The growing probabilities of additional offspring with the beneficial reversal allele for various population sizes, $N$, sequence lengths, $L$, selective advantages, $s$, fitness parameters, $k$, and measuring parameters, $C$, were…

Preferential attachment networks are a type of random network where new nodes are connected to existing ones at random, and are more likely to connect to those that already have many connections. We investigate further a family of models…

We prove almost sure convergence of the maximum degree in an evolving graph model combining a growing number of local choices with sublinear preferential attachment. At each step in the growth of the graph, a new vertex is introduced. Then…

We consider an evolving preferential attachment random graph model where at discrete times a new node is attached to an old node, selected with probability proportional to a superlinear function of its degree. For such schemes, it is known…

Motivated by online recommendation systems, we study a family of random forests. The vertices of the forest are labeled by integers. Each non-positive integer $i\le 0$ is the root of a tree. Vertices labeled by positive integers $n \ge 1$…

Neural Networks and Decision Trees: two popular techniques for supervised learning that are seemingly disconnected in their formulation and optimization method, have recently been combined in a single construct. The connection pivots on…

We study rooted planar random trees with a probability distribution which is proportional to a product of weight factors $w_n$ associated to the vertices of the tree and depending only on their individual degrees $n$. We focus on the case…

Leaves, i.e., vertices of degree one, can play a significant role in graph structure, especially in sparsely connected settings in which leaves often constitute the largest fraction of vertices. We consider a leaf-based counterpart of the…

We study the basic preferential attachment process, which generates a sequence of random trees, each obtained from the previous one by introducing a new vertex and joining it to one existing vertex, chosen with probability proportional to…

The preferential attachment network with fitness is a dynamic random graph model. New vertices are introduced consecutively and a new vertex is attached to an old vertex with probability proportional to the degree of the old one multiplied…

Many networks exhibit scale free behavior where their degree distribution obeys a power law for large vertex degrees. Models constructed to explain this phenomena have relied on preferential attachment where the networks grow by the…

We introduce a model of evolving preferential attachment trees where vertices are assigned weights, and the evolution of a vertex depends not only on its own weight, but also on the weights of its neighbours. We study the distribution of…

We analyze the growth models for complex networks including preferential attachment (A.-L. Barabasi and R. Albert, Science 286, 509 (1999)) and fitness model (Caldarelli et al., Phys. Rev. Lett. 89, 258702 (2002)) and demonstrate that,…

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…

Complex networks in different areas exhibit degree distributions with heavy upper tail. A preferential attachment mechanism in a growth process produces a graph with this feature. We herein investigate a variant of the simple preferential…

Preferential attachment is often suggested to be the underlying mechanism of the growth of a network, largely due to that many real networks are, to a certain extent, scale-free. However, such attribution is usually made under debatable…

We investigate a self-interacting random walk, whose dynamically evolving environment is a random tree built by the walker itself, as it walks around. At time $n=1,2,\dots$, right before stepping, the walker adds a random number (possibly…

We study three preferential attachment models where the parameters are such that the asymptotic degree distribution has infinite variance. Every edge is equipped with a non-negative i.i.d. weight. We study the weighted distance between two…

Tree-child networks are a recently-described class of directed acyclic graphs that have risen to prominence in phylogenetics (the study of evolutionary trees and networks). Although these networks have a number of attractive mathematical…