Cutoff for non-backtracking random walks on sparse random graphs

A finite ergodic Markov chain is said to exhibit cutoff if its distance to stationarity remains close to 1 over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Discovered in the context of card shuffling (Aldous-Diaconis, 1986), this phenomenon is now believed to be rather typical among fast mixing Markov chains. Yet, establishing it rigorously often requires a challengingly detailed understanding of the underlying chain. Here we consider non-backtracking random walks on random graphs with a given degree sequence. Under a general sparsity condition, we establish the cutoff phenomenon, determine its precise window, and prove that the (suitably rescaled) cutoff profile approaches a remarkably simple, universal shape

Comparing mixing times on sparse random graphs

It is natural to expect that nonbacktracking random walk will mix faster than simple random walks, but so far this has only been proved in regular graphs. To analyze typical irregular graphs, let $G$ be a random graph on $n$ vertices with minimum degree 3 and a degree distribution that has exponential tails. We determine the precise worst-case mixing time for simple random walk on $G$, and show that, with high probability, it exhibits cutoff at time $\mathbf{h}^{-1} \log n$, where $\mathbf{h}$ is the asymptotic entropy for simple random walk on a Galton--Watson tree that approximates $G$ locally. (Previously this was only known for typical starting points.) Furthermore, we show that this asymptotic mixing time is strictly larger than the mixing time of nonbacktracking walk, via a delicate comparison of entropies on the Galton-Watson tree

Estimating graph parameters with random walks

An algorithm observes the trajectories of random walks over an unknown graph $G$, starting from the same vertex $x$, as well as the degrees along the trajectories. For all finite connected graphs, one can estimate the number of edges $m$ up to a bounded factor in $O\left(t_{\mathrm{rel}}^{3/4}\sqrt{m/d}\right)$ steps, where $t_{\mathrm{rel}}$ is the relaxation time of the lazy random walk on $G$ and $d$ is the minimum degree in $G$. Alternatively, $m$ can be estimated in $O\left(t_{\mathrm{unif}} +t_{\mathrm{rel}}^{5/6}\sqrt{n}\right)$, where $n$ is the number of vertices and $t_{\mathrm{unif}}$ is the uniform mixing time on $G$. The number of vertices $n$ can then be estimated up to a bounded factor in an additional $O\left(t_{\mathrm{unif}}\frac{m}{n}\right)$ steps. Our algorithms are based on counting the number of intersections of random walk paths $X,Y$, i.e. the number of pairs $(t,s)$ such that $X_t=Y_s$. This improves on previous estimates which only consider collisions (i.e., times $t$ with $X_t=Y_t$). We also show that the complexity of our algorithms is optimal, even when restricting to graphs with a prescribed relaxation time. Finally, we show that, given either $m$ or the mixing time of $G$, we can compute the "other parameter" with a self-stopping algorithm

Cutoff for a Stratified Random Walk on the Hypercube

We consider the random walk on the hypercube which moves by picking an ordered pair (i,j) of distinct coordinates uniformly at random and adding the bit at location i to the bit at location j, modulo 2. We show that this Markov chain has cutoff at time (3/2)n*log(n) with window of size n, solving a question posed by Chung and Graham (1997)

Inference in balanced community modulated recursive trees

We introduce a random recursive tree model with two communities, called balanced community modulated random recursive tree, or BCMRT in short. In this setting, pairs of nodes of different type appear sequentially. Each one of them decides independently to attach to their own type with probability 1-q, or to the other type with probability q, and then chooses its parent uniformly within the set of existing nodes with the selected type. We find that the limiting degree distributions coincide for different q. Therefore, as far as inference is concerned, other statistics have to be studied. We first consider the setting where the time-labels of the nodes, i.e. their time of arrival, are observed but their type is not. In this setting, we design a consistent estimator for q and provide bounds for the feasibility of testing between two different values of q. Moreover, we show that if q is small enough, then it is possible to cluster in a way correlated with the true partition, even though the algorithm is exponential in time. In the unlabelled setting, i.e. when only the tree structure is observed, we show that it is possible to test between different values of q in a strictly better way than by random guessing. This follows from a delicate analysis of the sum-of-distances statistic

Application of Malliavin Calculus and Wiener chaos to option pricing theory.

This dissertation provides a contribution to the option pricing literature by means of some recent developments in probability theory, namely the Malliavin Calculus and the Wiener chaos theory. It concentrates on the issue of faster convergence of Monte Carlo and Quasi-Monte Carlo simulations for the Greeks, on the topic of the Asian option as well as on the approximation for convexity adjustment for fixed income derivatives. The first part presents a new method to speed up the convergence of Monte- Carlo and Quasi-Monte Carlo simulations of the Greeks by means of Malliavin weighted schemes. We extend the pioneering works of Fournie et al. (1999), (2000) by deriving necessary and sufficient conditions for a function to serve as a weight function and by providing the weight function with minimum variance. To do so, we introduce its generator defined as its Skorohod integrand. On a numerical example, we find evidence of spectacular efficiency of this method for corridor options, especially for the gamma calculation. The second part brings new insights on the Asian option. We first show how to price discrete Asian options consistent with different types of underlying densities, especially non-normal returns, by means of the Fast Fourier Transform algorithm. We then extends Malliavin weighted schemes to continuous time Asian options. In the last part, we first prove that the Black Scholes convexity adjustment (Brotherton-Ratcliffe and Iben (1993)) can be consistently derived in a martingale framework. As an application, we examine the convexity bias between CMS and forward swap rates. However, for more complicated term structures assumptions, this approach does not hold any more. We offer a solution to this, thanks to an approximation formula, in the case of multi-factor lognormal zero coupon models, using Wiener chaos theory

Extraction de nanofibrilles de cellulose à structure et propriétés contrôlées : caractérisation, propriétés rhéologiques et application nanocomposites

The cellulose nanofibrils (CNF), obtained by TEMPO oxidation of native cellulose microfibrils as colloidal aqueous suspensions, are biosourced nanoparticles having rheological and optical properties well adapted for the conception of new nanomaterials with high performance.The main purpose of this study was to control and optimize the conditions for preparing these NFCs extracted from date palm tree by examining the oxidation time and the number of passes through the homogenizer..The success of the reaction was demonstrated by FT-IR spectroscopy. The rate of the carboxylic groups has been calculated by conductometric titration and ranged between 221 and 772 mol / g of anhydroglucose. Morphological studies show that oxidized CNFs are very individualized by introducing negative charges on their surfaces that induce electrostatic repulsion forces between the fibrils. Particular attention has been given to the viscoelasticity of oxidized-TEMPO CNF suspensions whose monitoring was carried out by a rheometer ARES-G2TA. These nanocharges were incorporated in a thermoplastic (PVAc) and nanocomposite materials obtained were characterized by SEM, TGA, DSC, DMA and mechanical testing.Les nanofibrilles de cellulose (NFC), obtenus par oxydation TEMPO des microfibrilles de cellulose native sous forme de suspensions colloïdales aqueuses, sont des nanoparticules biosourcées ayant des propriétés rhéologiques et optiques particulièrement séduisantes pour la conception de nanomatériaux à haute performance. Le but principal de cette étude était de contrôler et optimiser les conditions de préparation de ces NFCs extraites du rachis de palmier dattier en examinant le temps d'oxydation et le nombre de passe à travers l'homogéinsateur.La réussite de la réaction a été démontrée par spectroscopies FT-IR. Le taux de groupements carboxyliques a été calculé par dosage conductimétrique et était compris entre 221 et 772 µmol/g d'anhydroglucose. Les études morphologiques montrent que NFCs oxydées sont assez bien individualisés grâce à l'introduction des charges négatives à leur surface qui induisent des forces de répulsion électrostatique entre les fibrilles. Une attention particulière a été accordée à la viscoélasticité des suspensions NFC oxydées TEMPO dont le suivi a été réalisé par un rhéomètre ARES-G2TA. Ces nanocharges ont ensuite été incorporées au sein d'un thermoplastique (PVAc), puis les matériaux nanocomposites obtenus ont été caractérisés par MEB, ATG, DSC, DMA et par des tests mécaniques