arXiv subjects

Irene Crimaldi

Publications and source records attributed to Irene Crimaldi.

At least 37 records · Page 2Linked to original sources

arXiv2020

The Rescaled Polya Urn: local reinforcement and chi-squared goodness of fit test

Motivated by recent studies of big samples, this work aims at constructing a parametric model which is characterized by the following features: (i) a "local" reinforcement, i.e. a reinforcement mechanism mainly based on the last observations, (ii) a random persistent fluctuation of the predictive mean, and (iii) a long-term convergence of the empirical mean to a deterministic limit, together with a chi-squared goodness of fit result. This triple purpose has been achieved by the introduction of a new variant of the Eggenberger-Polya urn, that we call the "Rescaled" Polya urn. We provide a complete asymptotic characterization of this model, pointing out that, for a certain choice of the parameters, it has properties different from the ones typically exhibited from the other urn models in the literature. Therefore, beyond the possible statistical application, this work could be interesting for those who are concerned with stochastic processes with reinforcement.

math.ST↗

arXiv2019

Collaboration and followership: a stochastic model for activities in social networks

In this work we investigate how future actions are influenced by the previous ones, in the specific contexts of scientific collaborations and friendships on social networks. We are not interested in modeling the process of link formation between the agents themselves, we instead describe the activity of the agents, providing a model for the formation of the bipartite network of actions and their features. Therefore we only require to know the chronological order in which the actions are performed, and not the order in which the agents are observed. Moreover, the total number of possible features is not specified a priori but is allowed to increase along time, and new actions can independently show some new entry features or exhibit some of the old ones. The choice of the old features is driven by a degree-fitness method. With this term we mean that the probability that a new action shows one of the old features does not solely depend on the "popularity" of that feature (i.e. the number of previous actions showing it), but is also affected by some individual traits of the agents or the features themselves, synthesized in certain quantities, called "fitnesses" or "weights", that can have different forms and different meaning according to the specific setting considered. We show some theoretical properties of the model and provide statistical tools for the parameters' estimation. The model has been tested on three different datasets and the numerical results are provided and discussed.

physics.soc-ph↗

arXiv2018

Interacting reinforced stochastic processes: statistical inference based on the weighted empirical means

This work deals with a system of interacting reinforced stochastic processes, where each process $X^j=(X_{n,j})_n$ is located at a vertex $j$ of a finite weighted direct graph, and it can be interpreted as the sequence of "actions" adopted by an agent $j$ of the network. The interaction among the dynamics of these processes depends on the weighted adjacency matrix $W$ associated to the underlying graph: indeed, the probability that an agent $j$ chooses a certain action depends on its personal "inclination" $Z_{n,j}$ and on the inclinations $Z_{n,h}$, with $h\neq j$, of the other agents according to the entries of $W$. The best known example of reinforced stochastic process is the Polya urn. The present paper characterizes the asymptotic behavior of the weighted empirical means $N_{n,j}=\sum_{k=1}^n q_{n,k} X_{k,j}$, proving their almost sure synchronization and some central limit theorems in the sense of stable convergence. By means of a more sophisticated decomposition of the considered processes adopted here, these findings complete and improve some asymptotic results for the personal inclinations $Z^j=(Z_{n,j})_n$ and for the empirical means $\overline{X}^j=(\sum_{k=1}^n X_{k,j}/n)_n$ given in recent papers (e.g. [arXiv:1705.02126, Bernoulli, Forth.]; [arXiv:1607.08514, Ann. Appl. Probab., 27(6):3787-3844, 2017]; [arXiv:1602.06217, Stochastic Process. Appl., 129(1):70-101, 2019]). Our work is motivated by the aim to understand how the different rates of convergence of the involved stochastic processes combine and, from an applicative point of view, by the construction of confidence intervals for the common limit inclination of the agents and of a test statistics to make inference on the matrix $W$, based on the weighted empirical means. In particular, we answer a research question posed in [arXiv:1705.02126, Bernoulli, Forth.]

math.PR↗

arXiv2018

Networks of reinforced stochastic processes: asymptotics for the empirical means

This work deals with systems of interacting reinforced stochastic processes, where each process $X^j=(X_{n,j})_n$ is located at a vertex $j$ of a finite weighted direct graph, and it can be interpreted as the sequence of "actions" adopted by an agent $j$ of the network. The interaction among the evolving dynamics of these processes depends on the weighted adjacency matrix $W$ associated to the underlying graph: indeed, the probability that an agent $j$ chooses a certain action depends on its personal "inclination" $Z_{n,j}$ and on the inclinations $Z_{n,h}$, with $h\neq j$, of the other agents according to the elements of $W$. Asymptotic results for the stochastic processes of the personal inclinations $Z^j=(Z_{n,j})_n$ have been subject of studies in recent papers (e.g. Aletti, Crimaldi, and Ghiglietti [arXiv:1607.08514, Ann. Appl. Probab., 27(6):3787-3844, 2017]; Crimaldi, Dai Pra, Louis, and Minelli [arXiv:1602.06217, Forthcoming in Stochastic Process. Appl.]); while the asymptotic behavior of quantities based on the stochastic processes $X^j$ of the actions has never been studied yet. In this paper, we fill this gap by characterizing the asymptotic behavior of the empirical means $N_{n,j}=\sum_{k=1}^n X_{k,j}/n$, proving their almost sure synchronization and some central limit theorems in the sense of stable convergence. Moreover, we discuss some statistical applications of these convergence results concerning confidence intervals for the random limit toward which all the processes of the system converge and tools to make inference on the matrix $W$.

math.PR↗

arXiv2016

Synchronization of Reinforced Stochastic Processes with a Network-based Interaction

Randomly evolving systems composed by elements which interact among each other have always been of great interest in several scientific fields. This work deals with the synchronization phenomenon, that could be roughly defined as the tendency of different components to adopt a common behavior. We continue the study of a model of interacting stochastic processes with reinforcement, that recently has been introduced in Crimaldi et al. (2016, arXiv:1602.06217). Generally speaking, by reinforcement we mean any mechanism for which the probability that a given event occurs has an increasing dependence on the number of times that events of the same type occurred in the past. The particularity of systems of such stochastic processes is that synchronization is induced along time by the reinforcement mechanism itself and does not require a large-scale limit. We focus on the relationship between the topology of the network of the interactions and the long-time synchronization phenomenon. After proving the almost sure synchronization, we provide some CLTs in the sense of stable convergence that establish the convergence rates and the asymptotic distributions for both convergence to the common limit and synchronization. The obtained results lead to the construction of asymptotic confidence intervals for the limit random variable and of statistical tests to make inference on the topology of the network given the observation of the reinforced stochastic processes positioned at the vertices.

math.PR↗

arXiv2016

Synchronization and functional central limit theorems for interacting reinforced random walks

We obtain Central Limit Theorems in Functional form for a class of time-inhomogeneous interacting random walks on the simplex of probability measures over a finite set. Due to a reinforcement mechanism, the increments of the walks are correlated, forcing their convergence to the same, possibly random, limit. Random walks of this form have been introduced in the context of urn models and in stochastic approximation. We also propose an application to opinion dynamics in a random network evolving via preferential attachment. We study, in particular, random walks interacting through a mean-field rule and compare the rate they converge to their limit with the rate of synchronization, i.e. the rate at which their mutual distances converge to zero. Under certain conditions, synchronization is faster than convergence.

math.PR↗

arXiv2016

Homophily and Triadic Closure in Evolving Social Networks

We present a new network model accounting for multidimensional assortativity. Each node is characterized by a number of features and the probability of a link between two nodes depends on common features. We do not fix a priori the total number of possible features. The bipartite network of the nodes and the features evolves according to a stochastic dynamics that depends on three parameters that respectively regulate the preferential attachment in the transmission of the features to the nodes, the number of new features per node, and the power-law behavior of the total number of observed features. Our model also takes into account a mechanism of triadic closure. We provide theoretical results and statistical estimators for the parameters of the model. We validate our approach by means of simulations and an empirical analysis of a network of scientific collaborations.

cs.SI↗

arXiv2015

Asymptotics for randomly reinforced urns with random barriers

An urn contains black and red balls. Let $Z_n$ be the proportion of black balls at time $n$ and $0\leq L L$, then $b_n$ is replaced together with a random number $R_n$ of red balls. Otherwise, no additional balls are added, and $b_n$ alone is replaced. In this paper, we assume $R_n=B_n$. Then, under mild conditions, it is shown that $Z_n\overset{a.s.}\longrightarrow Z$ for some random variable $Z$, and \begin{gather*} D_n:=\sqrt{n}\,(Z_n-Z)\longrightarrow\mathcal{N}(0,σ^2)\quad\text{conditionally a.s.} \end{gather*} where $σ^2$ is a certain random variance. Almost sure conditional convergence means that \begin{gather*} P\bigl(D_n\in\cdot\mid\mathcal{G}_n\bigr)\overset{weakly}\longrightarrow\mathcal{N}(0,\,σ^2)\quad\text{a.s.} \end{gather*} where $P\bigl(D_n\in\cdot\mid\mathcal{G}_n\bigr)$ is a regular version of the conditional distribution of $D_n$ given the past $\mathcal{G}_n$. Thus, in particular, one obtains $D_n\longrightarrow\mathcal{N}(0,σ^2)$ stably. It is also shown that $L<Z<U$ a.s. and $Z$ has non-atomic distribution.

math.PR↗

arXiv2015

Central limit theorems for a hypergeometric randomly reinforced urn

We consider a variant of the randomly reinforced urn where more balls can be simultaneously drawn out and balls of different colors can be simultaneously added. More precisely, at each time-step, the conditional distribution of the number of extracted balls of a certain color given the past is assumed to be hypergeometric. We prove some central limit theorems in the sense of stable convergence and of almost sure conditional convergence, which are stronger than convergence in distribution. The proven results provide asymptotic confidence intervals for the limit proportion, whose distribution is generally unknown. Moreover, we also consider the case of more urns subjected to some random common factors.

math.PR↗

arXiv2015

Fluctuation Theorems for Synchronization of Interacting Polya's urns

We consider a model of N two-colors urns in which the reinforcement of each urn depends also on the content of all the other urns. This interaction is of mean-field type and it is tuned by a parameter $α$ in [0,1]; in particular, for $α=0$ the N urns behave as N independent Polya's urns. As shown in [9], for $α>0$ urns synchronize, in the sense that the fraction of balls of a given color converges a.s. to the same (random) limit in all urns. In this paper we study fluctuations around this synchronized regime. The scaling of these fluctuations depends on the parameter $α$. In particular the standard scaling $t^{-1/2}$ appears only for $α>1/2$. For $α\geq 1/2$ we also determine the limit distribution of the rescaled fluctuations. We use the notion of stable convergence, which is stronger than convergence in distribution.

math.PR↗

arXiv2015

Central limit theorems for an Indian buffet model with random weights

The three-parameter Indian buffet process is generalized. The possibly different role played by customers is taken into account by suitable (random) weights. Various limit theorems are also proved for such generalized Indian buffet process. Let $L_n$ be the number of dishes experimented by the first $n$ customers, and let $\overline{K}_n=(1/n)\sum_{i=1}^nK_i$ where $K_i$ is the number of dishes tried by customer $i$. The asymptotic distributions of $L_n$ and $\overline{K}_n$, suitably centered and scaled, are obtained. The convergence turns out to be stable (and not only in distribution). As a particular case, the results apply to the standard (i.e., nongeneralized) Indian buffet process.

math.PR↗

arXiv2014

A Network Model characterized by a Latent Attribute Structure with Competition

The quest for a model that is able to explain, describe, analyze and simulate real-world complex networks is of uttermost practical as well as theoretical interest. In this paper we introduce and study a network model that is based on a latent attribute structure: each node is characterized by a number of features and the probability of the existence of an edge between two nodes depends on the features they share. Features are chosen according to a process of Indian-Buffet type but with an additional random "fitness" parameter attached to each node, that determines its ability to transmit its own features to other nodes. As a consequence, a node's connectivity does not depend on its age alone, so also "young" nodes are able to compete and succeed in acquiring links. One of the advantages of our model for the latent bipartite "node-attribute" network is that it depends on few parameters with a straightforward interpretation. We provide some theoretical, as well experimental, results regarding the power-law behaviour of the model and the estimation of the parameters. By experimental data, we also show how the proposed model for the attribute structure naturally captures most local and global properties (e.g., degree distributions, connectivity and distance distributions) real networks exhibit. keyword: Complex network, social network, attribute matrix, Indian Buffet process

cs.SI↗

arXiv2014

Cluster analysis of weighted bipartite networks: a new copula-based approach

In this work we are interested in identifying clusters of "positional equivalent" actors, i.e. actors who play a similar role in a system. In particular, we analyze weighted bipartite networks that describes the relationships between actors on one side and features or traits on the other, together with the intensity level to which actors show their features. The main contribution of our work is twofold. First, we develop a methodological approach that takes into account the underlying multivariate dependence among groups of actors. The idea is that positions in a network could be defined on the basis of the similar intensity levels that the actors exhibit in expressing some features, instead of just considering relationships that actors hold with each others. Second, we propose a new clustering procedure that exploits the potentiality of copula functions, a mathematical instrument for the modelization of the stochastic dependence structure. Our clustering algorithm can be applied both to binary and real-valued matrices. We validate it with simulations and applications to real-world data.

physics.data-an↗

arXiv2013

A generalized telegraph process with velocity driven by random trials

We consider a random trial-based telegraph process, which describes a motion on the real line with two constant velocities along opposite directions. At each epoch of the underlying counting process the new velocity is determined by the outcome of a random trial. Two schemes are taken into account: Bernoulli trials and classical Pólya urn trials. We investigate the probability law of the process and the mean of the velocity of the moving particle. We finally discuss two cases of interest: (i) the case of Bernoulli trials and intertimes having exponential distributions with linear rates (in which, interestingly, the process exhibits a logistic stationary density with non-zero mean), and (ii) the case of Pólya trials and intertimes having first Gamma and then exponential distributions with constant rates.

math.PR↗

arXiv2012

An Anscombe-type theorem

Let (X_n) be a sequence of random variables (with values in a separable metric space) and (N_n) a sequence of random indices. Conditions for X_{N_n} to converge stably (in particular, in distribution) are provided. Some examples, where such conditions work but those already existing fail, are given as well. Key words and phrases: Anscombe theorem, Exchangeability, Random indices, Random sums, Stable convergence

math.PR↗

arXiv2012

The Evolution of Complex Networks: A New Framework

We introduce a new framework for the analysis of the dynamics of networks, based on randomly reinforced urn (RRU) processes, in which the weight of the edges is determined by a reinforcement mechanism. We rigorously explain the empirical evidence that in many real networks there is a subset of "dominant edges" that control a major share of the total weight of the network. Furthermore, we introduce a new statistical procedure to study the evolution of networks over time, assessing if a given instance of the nework is taken at its steady state or not. Our results are quite general, since they are not based on a particular probability distribution or functional form of the weights. We test our model in the context of the International Trade Network, showing the existence of a core of dominant links and determining its size.

physics.soc-ph↗

arXiv2010

Rate of convergence of predictive distributions for dependent data

This paper deals with empirical processes of the type \[C_n(B)=\sqrt{n}\{μ_n(B)-P(X_{n+1}\in B\mid X_1,...,X_n)\},\] where $(X_n)$ is a sequence of random variables and $μ_n=(1/n)\sum_{i=1}^nδ_{X_i}$ the empirical measure. Conditions for $\sup_B|C_n(B)|$ to converge stably (in particular, in distribution) are given, where $B$ ranges over a suitable class of measurable sets. These conditions apply when $(X_n)$ is exchangeable or, more generally, conditionally identically distributed (in the sense of Berti et al. [Ann. Probab. 32 (2004) 2029--2052]). By such conditions, in some relevant situations, one obtains that $\sup_B|C_n(B)|\stackrel{P}{\to}0$ or even that $\sqrt{n}\sup_B|C_n(B)|$ converges a.s. Results of this type are useful in Bayesian statistics.

math.ST↗

arXiv2009

Central Limit Theorems for Multicolor Urns with Dominated Colors

An urn contains balls of d colors. At each time, a ball is drawn and then replaced together with a random number of balls of the same color. Assuming that some colors are dominated by others, we prove central limit theorems. Some statistical applications are discussed.

math.PR↗