Nicolas Fraiman

Connectivity of inhomogeneous random graphs

We find conditions for the connectivity of inhomogeneous random graphs with intermediate density. Our results generalize the classical result for Gn, p, when p=clogn/n. We draw n independent points Xi from a general distribution on a separable metric space, and let their indices form the vertex set of a graph. An edge i, j is added with probability min1,i¾?Xi,Xjlogn/n, where i¾?i¾?0 is a fixed kernel. We show that, under reasonably weak assumptions, the connectivity threshold of the model can be determined.

Depth Properties of scaled attachment random recursive trees

We study depth properties of a general class of random recursive trees where each node i attaches to the random node iX_i and X_0, ..., X_n is a sequence of i.i.d. random variables taking values in [0,1). We call such trees scaled attachment random recursive trees (SARRT). We prove that the typical depth D_n, the maximum depth (or height) H_n and the minimum depth M_n of a SARRT are asymptotically given by D_n \sim \mu^{-1} \log n, H_n \sim \alpha_{\max} \log n and M_n \sim \alpha_{\min} \log n where \mu, \alpha_{\max} and \alpha_{\min} are constants depending only on the distribution of X_0 whenever X_0 has a density. In particular, this gives a new elementary proof for the height of uniform random recursive trees H_n \sim e \log n that does not use branching random walks.

Lines in hypergraphs

One of the De Bruijn-Erdős theorems deals with finite hypergraphs where every two vertices belong to precisely one hyperedge. It asserts that, except in the perverse case where a single hyperedge equals the whole vertex set, the number of hyperedges is at least the number of vertices and the two numbers are equal if and only if the hypergraph belongs to one of simply described families, near-pencils and finite projective planes. Chen and Chvátal proposed to define the line uv in a 3-uniform hypergraph as the set of vertices that consists of u, v, and all w such that {u;v;w} is a hyperedge. With this definition, the De Bruijn-Erdős theorem is easily seen to be equivalent to the following statement: If no four vertices in a 3-uniform hypergraph carry two or three hyperedges, then, except in the perverse case where one of the lines equals the whole vertex set, the number of lines is at least the number of vertices and the two numbers are equal if and only if the hypergraph belongs to one of two simply described families. Our main result generalizes this statement by allowing any four vertices to carry three hyperedges (but keeping two forbidden): the conclusion remains the same except that a third simply described family, complements of Steiner triple systems, appears in the extremal case.

Public goods games in populations with fluctuating size

Many mathematical frameworks of evolutionary game dynamics assume that the total population size is constant and that selection affects only the relative frequency of strategies. Here, we consider evolutionary game dynamics in an extended Wright-Fisher process with variable population size. In such a scenario, it is possible that the entire population becomes extinct. Survival of the population may depend on which strategy prevails in the game dynamics. Studying cooperative dilemmas, it is a natural feature of such a model that cooperators enable survival, while defectors drive extinction. Although defectors are favored for any mixed population, random drift could lead to their elimination and the resulting pure-cooperator population could survive. On the other hand, if the defectors remain, then the population will quickly go extinct because the frequency of cooperators steadily declines and defectors alone cannot survive. In a mutation-selection model, we find that (i) a steady supply of cooperators can enable long-term population survival, provided selection is sufficiently strong, and (ii) selection can increase the abundance of cooperators but reduce their relative frequency. Thus, evolutionary game dynamics in populations with variable size generate a multifaceted notion of what constitutes a traits long-term success.

Nonparametric statistics of dynamic networks with distinguishable nodes

The study of random graphs and networks had an explosive development in the last couple of decades. Meanwhile, statistical analysis of graph sequences is less developed. In this paper we focus on graphs with a fixed number of labeled nodes and study some statistical problems in a nonparametric framework. We introduce natural notions of center and a depth function for graphs that evolve in time. This allows us to develop several statistical techniques including testing, supervised and unsupervised classification, and a notion of principal component sets in the space of graphs. Some examples and asymptotic results are given, as well as a real data example. The literature of random graphs and networks has grown exponentially during the last fifteen years. A huge number of different research lines have been developed in order to study the behavior of several stochastic models and real data networks. Some important results among those lines include the existence of stationary measures in dynamic models (or static but growing in size), characterizations of thresholds for giant components and connectivity, analysis of the spread of epidemics over fixed networks, and the development of new topological measures to characterize network structure (modules, motifs, etc.). In contrast, the study of the statistical properties of such models is not yet well developed. In particular, research has focused on techniques for community detection (27, 40, 42, 37, 36, 39, 28, 10, 9, 23, 8, 7, 6, 5, 4, 3, 2, 1), estimation in the stochastic block model and spectral clustering (41, 31, 32, 34), among other problems. A general approach to fit probability models based on the method of moments was introduced by (29). Furthermore, (26) propose a semi parametric two sample test for dot product graphs. The bibliography mostly concentrates on problems where there is a unique static graph. On the other hand, the case of a sequence of graphs has received little attention. A relevant reference in this direction is (43) where the authors estimate parametric time varying networks. Our contribution here is among this line, one of the main interesting aspects of our proposal is that it is non-parametric. We discuss how some statistical methods can be adapted to analyze a random sample of networks or the stochastic dynamics of a unique network. The theory of random graphs is dominated by models where the label of each node is not relevant for the kind of properties that have been studied. Nevertheless, in the majority of real networks such like those modeling brain connections, financial markets, the internet, or protein interactions, the label of each node appears naturally and it is relevant. That is the reason why we consider important to develop some statistical methods in the space of

The Random Connection Model on the Torus

We study the diameter of a family of random graphs on the torus that can be used to model wireless networks. In the random connection model two points x and y are connected with probability g ( y−x ), where g is a given function. We prove that the diameter of the graph is bounded by a constant, which depends only on ‖ g ‖ 1 , with high probability as the number of vertices in the graph tends to infinity.