arXiv subjects

Rafael A. Rosales

Publications and source records attributed to Rafael A. Rosales.

11 recordsLinked to original sources

arXiv2026

Interacting vertex reinforced random walks on complete sub-graphs

This article introduces a model for interacting vertex-reinforced random walks, each taking values on a complete subgraph of a locally finite undirected graph. The transition probability for a walk to a given vertex depends on the cumulative proportion of visits by all walks that have access to that vertex. Proportions are modified by multiplication by a real valued interaction parameter and the addition of a parameter representing the intrinsic preference of the walk for the vertex. This model covers a wide range of interactions, including the cooperation (attraction) or competition (repulsion) of several walks at single vertices. We are principally concerned with strong laws for the proportion of visits to each vertex by all walks. We prove that this measure converges almost surely towards the set of fixed points of the transition probabilities. Almost sure convergence to a single fixed point is in fact the generic behaviour as we show this to hold for almost all parameter values of our model. Beyond almost sure convergence, our model provides a general framework that yields a detailed description of the limiting behaviour for any choice of interaction parameters and subgraph geometry. We illustrate this by analyzing interacting walks on complete graphs, stars, and cycles, chosen to highlight the model's broad applicability. The central contribution lies in offering a powerful tool to analyse diverse types of interactions mediated by the intersections of subgraphs. Importantly, our results provide not only convergence criteria, but also conditions under which the empirical proportions of the walks' visits fail to converge to certain boundary points of the space where their trajectory evolves.

math.PR↗

arXiv2026

The Clark-Kushner condition for interacting reinforced random walks on finite graphs

We establish the Clark-Kushner condition for a large class of interacting vertex-reinforced random walks on finite graphs, where the transition matrix $Q^i(x)$ of each walk depends on the joint vector $x$ of vertex occupation proportions and may have distinct rows. This allows one to study the dynamics of the vertex occupation measure by using the tools of stochastic approximation theory. However, the standard approach fails because the noise inputs are in our case not a martingale difference: they retain memory of the previous state. Using the solution of the Poisson equation for Markov chains, we decompose the noise into a martingale difference minus the increment of a bounded process -- a structure originating in Gordin's work on limit theorems for stationary processes. The key technical ingredient of our approach is a uniform geometric ergodicity bound derived from the Dobrushin contraction coefficient, which also controls the Lipschitz continuity of the solution of the Poisson equation. Our hypotheses require only that each $Q^i(x)$ be irreducible, aperiodic, and Lipschitz continuous in $x$; in particular, strictly positive entries are not assumed. Our results generalize and simplify previous arguments considered for single self-reinforced vertex-reinforced random walks.

math.PR↗

arXiv2025

Friedman vs Pólya

Suppose an urn contains initially any number of balls of two colours. One ball is drawn randomly and then put back with $α$ balls of the same colour and $β$ balls of the opposite colour. Both cases, $β=0$ and $β>0$ are well known and correspond respectively to Pólya's and Friedman's replacement schemes. We consider a mixture of both of these: with probability $p\in(0,1]$ balls are replaced according to Friedman's recipe and with probability $1-p$ according to the one by Pólya. Independently of the initial urn composition and independently of $α$, $β$, and the value of $p>0$, we show that the proportion of balls of one colour converges almost surely to $\frac12$. The latter is the limit behaviour obtained by using Friedman's scheme alone, i.e. when $p=1$. Our result follows by adapting an argument due to D. S. Ornstein.

math.PR↗

arXiv2024

Multiple colour interacting urns on complete graphs

We present a multiple colour generalisation of the model of graph interacting urns studied by Benaim et. al., Random Struct. Alg., 46: 614-634, 2015. We show that for complete graphs and for a broad class of reinforcement functions governing the addition of balls in the urns, the process of colour proportions at each urn converges almost surely to the fixed points of the reinforcement function.

math.PR↗

arXiv2021

A mixture model for determining SARS-Cov-2 variant composition in pooled samples

Despite of the fast development of highly effective vaccines to control the current COVID$-$19 pandemic, the unequal distribution and availability of these vaccines worldwide and the number of people infected in the world lead to the continuous emergence of SARS-CoV-2 (Severe Acute Respiratory Syndrome coronavirus 2) variants of concern. It is likely that real-time genomic surveillance will be continuously needed as an unceasing monitoring tool, necessary to follow the spillover of the disease spread and the evolution of the virus. In this context, new genomic variants of SARS-CoV-2 that may emerge as a response to selective pressure, including variants refractory to current vaccines, makes genomic surveillance programs tools of utmost importance. Here propose a statistical model for the estimation of the relative frequencies of SARS-CoV-2 variants in pooled samples. This model is built by considering a previously defined selection of genomic polymorphisms that characterize SARS-CoV-2 variants. The methods described here support both raw sequencing reads for polymorphisms-based markers calling and predefined markers in the VCF format. Results obtained by using simulated data show that our method is quite effective in recovering the correct variant proportions. Further, results obtained by considering longitudinal data from wastewater samples of two locations in Switzerland agree well with those describing the epidemiological evolution of COVID-19 variants in clinical samples of these locations. Our results show that the described method can be a valuable tool for tracking the proportions of SARS-CoV-2 variants.

q-bio.GN↗

arXiv2021

Two repelling random walks on $\mathbb Z$

We consider two interacting random walks on $\mathbb{Z}$ such that the transition probability of one walk in one direction decreases exponentially with the number of transitions of the other walk in that direction. The joint process may thus be seen as two random walks reinforced to repel each other. The strength of the repulsion is further modulated in our model by a parameter $β\geq 0$. When $β= 0$ both processes are independent symmetric random walks on $\mathbb{Z}$, and hence recurrent. We show that both random walks are further recurrent if $β\in (0,1]$. We also show that these processes are transient and diverge in opposite directions if $β> 2$. The case $β\in (1,2]$ remains widely open. Our results are obtained by considering the dynamical system approach to stochastic approximations.

math.PR↗

arXiv2020

Vertex reinforced random walks with exponential interaction on complete graphs

We describe a model for $m$ vertex reinforced interacting random walks on complete graphs with $d\geq 2$ vertices. The transition probability of a random walk to a given vertex depends exponentially on the proportion of visits made by all walks to that vertex. The individual proportion of visits is modulated by a strength parameter that can be set equal to any real number. This model covers a large variety of interactions including different vertex repulsion and attraction strengths between any two random walks as well as self-reinforced interactions. We show that the process of empirical vertex occupation measures defined by the interacting random walks converges (a.s.) to the limit set of the flow induced by a smooth vector field. Further, if the set of equilibria of the field is formed by isolated points, then the vertex occupation measures converge (a.s.) to an equilibrium of the field. These facts are shown by means of the construction of a strict Lyapunov function. We show that if the absolute value of the interaction strength parameters are smaller than a certain upper bound, then, for any number of random walks ($m\geq 2$) on any graph ($d \geq 2$), the vertex occupation measure converges toward a unique equilibrium. We provide two additional examples of repelling random walks for the cases $m=d=2$ and $m=3$, $d=2$. The latter is used to study some properties of three exponentially repelling random walks on $\mathbb{Z}$.

math.PR↗

arXiv2016

Topological dynamics of piecewise λ-affine maps

Let $-1<λ<1$ and $f:[0,1)\to\mathbb{R}$ be a piecewise $λ$-affine map, that is, there exist points $0=c_0<c_1<\cdots <c_{n-1}<c_n=1$ and real numbers $b_1,\ldots,b_n$ such that $f(x)=λx+b_i$ for every $x\in [c_{i-1},c_i)$. We prove that, for Lebesgue almost every $δ\in\mathbb{R}$, the map $f_δ=f+δ\,({\rm mod}\,1)$ is asymptotically periodic. More precisely, $f_δ$ has at most $2n$ periodic orbits and the $ω$-limit set of every $x\in [0,1)$ is a periodic orbit.

math.DS↗

arXiv2014

Piecewise contractions defined by iterated function systems

Let $ϕ_1,\ldots,ϕ_n:[0,1]\to (0,1)$ be Lipschitz contractions. Let $I=[0,1)$, $x_0=0$ and $x_n=1$. We prove that for Lebesgue almost every $(x_1,...,x_{n-1})$ satisfying $0<x_1<\cdots <x_{n-1}<1$, the piecewise contraction $f:I\to I$ defined by $x\in [x_{i-1},x_i)\mapsto ϕ_i(x)$ is asymptotically periodic. More precisely, $f$ has at least one and at most $n$ periodic orbits and the $ω$-limit set $ω_f(x)$ is a periodic orbit of $f$ for every $x\in I$.

math.DS↗

arXiv2013

Asymptotically periodic piecewise contractions of the interval

We consider the iterates of a generic injective piecewise contraction of the interval defined by a finite family of contractions. Let $ϕ_i:[0,1]\to (0,1)$, $1\le i\le n$, be $C^2$-diffeomorphisms with $\sup_{x\in (0,1)} \vert Dϕ_i(x)\vert<1$ whose images $ϕ_1([0,1]), \ldots, ϕ_n([0,1])$ are pairwise disjoint. Let $0<x_1<\cdots<x_{n-1}<1$ and let $I_1,\ldots, I_n$ be a partition of the interval $[0,1)$ into subintervals $I_i$ having interior $(x_{i-1},x_i)$, where $x_0=0$ and $x_n=1$. Let $f_{x_1,\ldots,x_{n-1}}$ be the map given by $x\mapsto ϕ_i(x)$ if $x\in I_i$, for $1\le i\le n$. Among other results we prove that for Lebesgue almost every $(x_1,\ldots,x_{n-1})$, the piecewise contraction $f_{x_1,\ldots,x_{n-1}}$ is asymptotically periodic.

math.DS↗

arXiv2013

Identification of cromosomal translocation hotspots via scan statistics

The detection of genomic regions unusually rich in a given pattern is an important undertaking in the analysis of next generation sequencing data. Recent studies of chromosomal translocations in activated B lymphocytes have identified regions that are frequently translocated to c-myc oncogene. A quantitative method for the identification of translocation hotspots was crucial to this study. Here we improve this analysis by using a simple probabilistic model and the framework provided by scan statistics to define the number and location of translocation breakpoint hotspots. A key feature of our method is that it provides a global chromosome-wide significance level to clustering, as opposed to previous methods based on local criteria. Whilst being motivated by a specific application, the detection of unusual clusters is a widespread problem in bioinformatics. We expect our method to be useful in the analysis of data from other experimental approaches such as of ChIP-seq and 4C-seq.

q-bio.GN↗