Studies on generalized Yule models

We present a generalization of the Yule model for macroevolution in which, for the appearance of genera, we consider point processes with the order statistics property, while for the growth of species we use nonlinear time-fractional pure birth processes or a critical birth-death process. Further, in specific cases we derive the explicit form of the distribution of the number of species of a genus chosen uniformly at random for each time. Besides, we introduce a time-changed mixed Poisson process with the same marginal distribution as that of the time-fractional Poisson process.Comment: Published at https://doi.org/10.15559/18-VMSTA125 in the Modern Stochastics: Theory and Applications (https://vmsta.org/) by VTeX (http://www.vtex.lt/

Further properties of a random graph with duplications and deletions

We deal with a random graph model where at each step, a vertex is chosen uniformly at random, and it is either duplicated or its edges are deleted. Duplication has a given probability. We analyze the limit distribution of the degree of a fixed vertex and derive a.s. asymptotic bounds for the maximal degree. The model shows a phase transition phenomenon with respect to the probabilities of duplication and deletion. © 2016 Taylor & Franci

Further properties of a random graph with duplications and deletions

We deal with a random graph model where at each step, a vertex is chosen uniformly at random, and it is either duplicated or its edges are deleted. Duplication has a given probability. We analyze the limit distribution of the degree of a fixed vertex and derive a.s. asymptotic bounds for the maximal degree. The model shows a phase transition phenomenon with respect to the probabilities of duplication and deletion. © 2016 Taylor & Franci

Topological properties of P.A. random graphs with edge-step functions

In this work we investigate a preferential attachment model whose parameter is a function $f:\mathbb{N}\to[0,1]$ that drives the asymptotic proportion between the numbers of vertices and edges of the graph. We investigate topological features of the graphs, proving general bounds for the diameter and the clique number. Our results regarding the diameter are sharp when $f$ is a regularly varying function at infinity with strictly negative index of regular variation $-\gamma$. For this particular class, we prove a characterization for the diameter that depends only on $-\gamma$. More specifically, we prove that the diameter of such graphs is of order $1/\gamma$ with high probability, although its vertex set order goes to infinity polynomially. Sharp results for the diameter for a wide class of slowly varying functions are also obtained. The almost sure convergence for the properly normalized logarithm of the clique number of the graphs generated by slowly varying functions is also proved

Robustness of scale-free spatial networks

A growing family of random graphs is called robust if it retains a giant component after percolation with arbitrary positive retention probability. We study robustness for graphs, in which new vertices are given a spatial position on the $d$-dimensional torus and are connected to existing vertices with a probability favouring short spatial distances and high degrees. In this model of a scale-free network with clustering we can independently tune the power law exponent $\tau$ of the degree distribution and the rate $\delta d$ at which the connection probability decreases with the distance of two vertices. We show that the network is robust if $\tau<2+1/\delta$, but fails to be robust if $\tau>3$. In the case of one-dimensional space we also show that the network is not robust if $\tau<2+1/(\delta-1)$. This implies that robustness of a scale-free network depends not only on its power-law exponent but also on its clustering features. Other than the classical models of scale-free networks our model is not locally tree-like, and hence we need to develop novel methods for its study, including, for example, a surprising application of the BK-inequality.Comment: 34 pages, 4 figure

The dynamics of power laws: Fitness and aging in preferential attachment trees

Continuous-time branching processes describe the evolution of a population whose individuals generate a random number of children according to a birth process. Such branching processes can be used to understand preferential attachment models in which the birth rates are linear functions. We are motivated by citation networks, where power-law citation counts are observed as well as aging in the citation patterns. To model this, we introduce fitness and age-dependence in these birth processes. The multiplicative fitness moderates the rate at which children are born, while the aging is integrable, so that individuals receives a finite number of children in their lifetime. We show the existence of a limiting degree distribution for such processes. In the preferential attachment case, where fitness and aging are absent, this limiting degree distribution is known to have power-law tails. We show that the limiting degree distribution has exponential tails for bounded fitnesses in the presence of integrable aging, while the power-law tail is restored when integrable aging is combined with fitness with unbounded support with at most exponential tails. In the absence of integrable aging, such processes are explosive.Comment: 41 pages, 10 figure