Mirta Gordon

• Directrice de Recherches
• Professor Emeritus at CNRS and University of Grenoble

As a large-scale instance of dramatic collective behavior, the 2005 French riots started in a poor suburb of Paris, then spread in all of France, lasting about three weeks. Remarkably, although there were no displacements of rioters, the riot activity did traveled. Daily national police data to which we had access have allowed us to take advantage...

Mathematical Model Reveals How French Riots Spread via a Giant Wave of Contagious Violence. Violence can spread like a disease, say epidemiologists who have modeled the spread of riots through France in 2005. L’article montre que la propagation des émeutes a pris la forme d’une vague (une forme qui diffère des émeutes de Londres en 2011). En effet...

The dynamics of adoption of innovations is an important subject in many fields and areas, like technological development, industrial processes, social behavior, fashion or marketing. The number of adopters of a new technology generally increases following a kind of logistic function. However, empirical data provide evidences that this behavior may...

The dynamics of adoption of innovations is an important subject in many fields and areas, like technological development, industrial processes, social behavior, fashion or marketing. The number of adopters of a new technology generally increases following a kind of logistic function. However, empirical data provide evidences that this behavior may...

We address the question of whether stigmatized minorities are over-represented in delinquency in France, based on an exhaustive database of all the juveniles below 19 years old convicted for serious crimes (liable to imprisonment according to the French Law) in the period 1985-2005 in Is\`ere (about 1 million inhabitants), one of the French Departm...

In this article, we briefly review the models of social interactions concerning the pricing of goods with Bandwagon properties.

Whenever customers' choices (e.g. to buy or not a given good) depend on others choices (cases coined 'positive externalities' or 'bandwagon effect' in the economic literature), the demand may be multiply valued: for a same posted price, there is either a small number of buyers, or a large one -- in which case one says that the customers coordinate....

We introduce a simple agent-based model which allows us to analyze three stylized facts: a fat-tailed size distribution of companies, a `tent-shaped' growth rate distribution, the scaling relation of the growth rate variance with firm size, and the causality between them. This is achieved under the simple hypothesis that firms compete for a scarce...

We present a macroeconomic agent-based model that combines several mechanisms operating at the same timescale, while remaining mathematically tractable. It comprises enterprises and workers who compete in a job market and a commodity goods market. The model is stock-flow consistent; a bank lends money charging interest rates, and keeps track of equ...

Crime is the result of a rational distinctive balance between the benefits and costs of an illegal act. This idea was proposed by Becker more than forty years ago (Becker (1968) [1]). In this paper, we simulate a simple artificial society, in which agents earn fixed wages and can augment (or lose) wealth as a result of a successful (or not) act of...

Nous revenons sur le problème de l'application des Machines à Vecteurs Support (SVM) à la classification de graphes. Plusieurs mesures de similarité, appelées généralement noyaux, ont été proposées récemment. Cependant, certaines, comme le noyau d'affectation optimale (15), ne sont pas semidéfinies positives, ce qui limite leur applicabilité avec l...

This paper presents a computational model of the way humans inductively identify and aggregate concepts from the low-level stimuli they are exposed to. Based on the idea that humans tend to select the simplest structures, it implements a dynamic hierarchical chunking mechanism in which the decision whether to create a new chunk is based on an infor...

A network of automata is made up of distinct units (or automata) i ; i = 1,…,N. One assumes that an automaton may be in one of a (generally finite) number of internal states σ i . At (discrete) time v the overall state I(v) of the network is defined as the set of states that the units take at this very time : I(v) = {σ i 7,(v)}. The time evolution...

We introduce a general framework for modelling the dynamics of the propensity to offend in a population of (possibly interacting) agents. We consider that each agent has an ‘honesty index’ which parameterizes his probability of abiding by the law. This probability also depends on a composite parameter associated to the attractiveness of the crime o...

We are interested in the possible contributions of mathematical modelling of crime. We refer to numerous and quite recent papers that analyse and discuss empirical data in an attempt to discover stylised trends worthy of being understood through simple models. We summarise part of this literature and try to understand the reasons of important discr...

We study experimentally a coordination game with N heterogeneous individuals under different information treatments. We explore the effects of information on the emergence of Pareto-efficient outcomes, by means of a gradual decrease of the information content provided to the players in successive experiments. We observe that successful coordination...

We are interested in the possible contributions of mathematical modeling of crime. We refer to numerous and quite recent papers that analyze and discuss empirical data in an attempt to discover stylized trends worthy of being understood through simple models. We summarize part of this literature and try to understand the reasons of important discre...

Crime is an economically important activity, sometimes called the industry of crime. It may represent a mechanism of wealth distribution but also a social and economic charge because of the cost of the law enforcement system. Sometimes it may be less costly for the society to allow for some level of criminality. A drawback of such policy may lead t...

The problem of classifying sonar signals from rocks and mines first studied by Gorman and Sejnowski has become a benchmark against which many learning algorithms have been tested. We show that both the training set and the test set of this benchmark are linearly separable, although with different hyperplanes. Moreover, the complete set of learning...

Nous présentons de manière pédagogique les modèles de mémoire associative de Little et Hopfield. Nous donnons des exemples de l'intérêt des modèles mathématiques, transposables à d'autres disciplines. Finalement nous présentons un modèle d'interactions sociales et déduisons des conséquences inédites en sciences sociales, grâce à l'équivalence de ce...

A efficient incremental learning algorithm for classification tasks, called NetLines, well adapted for both binary and real-valued input patterns is presented. It generates small compact feedforward neural networks with one hidden layer of binary units and binary output units. A convergence theorem ensures that solutions with a finite number of hid...

We consider a model of socially interacting individuals that make a binary choice in a context of positive additive endogenous externalities. It encompasses as particular cases several models from the sociology and economics literature. We extend previous results to the case of a general distribution of idiosyncratic preferences, called here Idiosy...

This paper applies machine learning techniques to student modeling. It presents a method for discovering high-level student behaviors from a very large set of low-level traces corresponding to problem-solving actions in a learning environment. Basic actions are encoded into sets of domain-dependent attribute-value patterns called cases. Then a doma...

Basic evidences on non-profit making and other forms of benevolent-based organizations reveal a rough partition of members between some {\em pure consumers} of the public good (free-riders) and {\em benevolent} individuals (cooperators). We study the relationship between the community size and the level of cooperation in a simple model where the ut...

This paper summarizes the effects of social influences in a monopoly market with heterogeneous agents. The market equilibria are presented in the limiting case of global influence. Considering static profit maximization there may exist two different regimes: to sell either to a large fraction of customers at a low price, or to a small fraction of t...

We study the implications of social interactions and individual learning features on consumer demand in a simple market model. We consider a social system of interacting heterogeneous agents with learning abilities. Given a fixed price, agents repeatedly decide whether or not to buy a unit of a good, so as to maximize their expected utilities. This...

Notre approche consiste à concevoir un modèle cognitif de l'apprentissage inductif de concepts basé uniquement sur le principe de simplicité. Nous utilisons une formulation mathématique de ce principe appelée Minimum Description Length, dans le but de créer un modèle informatique capable de générer automatiquement des hypothèses à partir d'un ensem...

Many researchers consider interactive learning environments to be interesting solutions for overcoming the limits of classical one-to-many teaching methods. However, these environments should incorporate accurate representations of student knowledge to provide relevant guidance. In a problem-solving environment, one way to build and update this stu...

We consider a model of socially interacting individuals that make a binary choice in a context of positive additive endogenous externalities. It encompasses as particular cases several models from the sociology and economics literature. We extend previous results to the case of a general distribution of idiosyncratic preferences, called here Idiosy...

We consider a social system of interacting heterogeneous agents with learning abilities, a model close to Random Field Ising Models, where the random field corresponds to the idiosyncratic willingness to pay. Given a fixed price, agents decide repeatedly whether to buy or not a unit of a good, so as to maximize their expected utilities. We show tha...

This paper presents a general method for identifying student intermediate mental steps from sequences of actions stored by problem solving-based learning environments, in order to provide feedback to teachers on knowledge that statistically seems to be used by a particular student. When many intermediate mental steps are possible, ambiguity is remo...

We introduce a test, named π-subsumption, which computes partial subsumptions between a hypothesis h and an example e, as well as a measure, the subsumption index, which quantifies the covering degree between h and e. The behavior of this measure is studied on the phase transition problem.

La modélisation automatique des connaissances de l’apprenant est nécessaire dès lors que l’on cherche à accompagner un apprentissage dans le cadre des EIAH. Dans ce rapport, nous présentons un travail en cours dans lequel des techniques de classification automatique permettent d’engendrer des modèles d’élèves interprétables par un enseignant ou un...

We study the mean field dynamics of a model introduced by Phan et al [Wehia, 2005] of a polymorphic social community. The individuals may choose between three strategies: either not to join the community or, in the case of joining it, to cooperate or to behave as a free-rider. Individuals' preferences have an idiosyncratic component and a social co...

We explore the effects of social influence in a simple market model in which a large number of agents face a binary choice: to buy/not to buy a single unit of a product at a price posted by a single seller (monopoly market). We consider the case of positive externalities: an agent is more willing to buy if other agents make the same decision. We co...

In this paper, we consider a discrete choice model where heterogeneous agents are subject to mutual influences. We explore some consequences on the market's behaviour, in the simplest case of a uniform willingness to pay distribution. We exhibit a first-order phase transition in the profit optimization by the monopolist: if the social influence is...

Basic evidences on non prot making and other forms of benevolent-based organizations reveal a rough partition within these groups between some pure consumers of the public good (free-riders) and benevolent individuals (cooperators). This polymorphic conguration seems to be a stable form of organization in a variety of situations. We study the relat...

This article shows how statistical physics may contribute to the modelling of collective phenomena in economics and social science. The main topic here is the study of the global (aggregate) behavior of a large population, when the agents make choices under social influence. We present several examples, starting from pioneering works in economics a...

We study social organizations with possible coexistence at equilibrium of cooperating individuals and pure consumers (free-riders). We investigate this polymorphic equilibrium using a game-theoretic approach and a statistical physics analysis of a simple model. The agents face a binary decision problem: whether to contribute or not to the public go...

Jean-François Nicaud (responsable Du Projet)

This Chapter extends some economic models that take advantage of a formalism inspired from statistical mechanics to account for social inuence in individual decisions. Starting with a framework suggested by Durlauf, Blume and Brock, we introduce three classes of models shifting progressively from rational towards adaptive expectations. We discuss t...

In this lecture we introduce the concepts and tools of Statistical Mechanics using the Ising model of condensed matter physics, proposed about one century ago to explain magnetism, as a paradigm. The evocative power of this model brought the techniques of Statistical Mechanics to other fields, like brain models in Cognitive Science, Machine Learnin...

We use panel data from the National Longitudinal Survey of Youth (NLSY) to examine how body weight changes with age for a cohort moving through early adulthood, to investigate how the age-obesity gradient differs with socioeconomic status (SES) and to study channels for these SES disparities. Our results show first that weight increases with age an...

Nous présentons une analyse de comportements d'élèves qui ont résolu des exercices d'algèbre avec le logiciel Aplusix. Nous avons réalisé un logiciel produisant des statistiques, à partir des protocoles (fichiers enregistrant les interactions élève-logiciel), au niveau de la justesse des étapes de calcul et des résolutions. Nous avons particulièrem...

We investigate the interaction of learning and evolution in a changing environment. A stable learning capability is regarded as an emergent adaptive system evolved by natural selection of genetic variants. We consider the evolution of an asexual population. Each genotype can have 'fixed' and 'flexible' alleles. The former express themselves as syna...

In this paper, we explore the effects of localised externalities introduced through interaction structures upon the properties of the simplest market model: the discrete choice model with a single homogeneous product and a single seller (the monopoly case). The resulting market is viewed as a complex interactive system with a communication network....

The N-dimensional parity problem is frequently a difficult classification task for Neural Networks. We found an expression for the minimum number of errors ?f as function of N for this problem, performed by a perceptron. We verified this quantity experimentally for N=1,...,15 using an optimal train perceptron. With a constructive approach we solved...

We present a model of spiking neuron that emulates the output of the usual static neurons with sigmoidal activation functions. It allows for hardware implementations of standard feedforward networks, trained off-line with any classical learning algorithm (i.e. back-propagation and its variants). The model is validated on hand-written digits recogni...

Typical learning curves for Soft Margin Classifiers (SMCs) learning both realizable and unrealizable tasks are determined using the tools of Statistical Mechanics. We derive the analytical behaviour of the learning curves in the regimes of small and large training sets. The generalization errors present different decay laws towards the asymptotic v...

Rigorous Bounds to Retarded Learning Using an elegant approach, Herschkowitz and Opper [1] established rigorous bounds on the information inferable (learnable) from a set of data (m points [ N) when the latter are drawn from a distribution Px P 0 x 3 exp2V l, where P 0 x is a spherical normal law and exp2V l is a modulation along an unknown anisotr...

We show that the lower bound to the critical fraction of data needed to infer (learn) the orientation of the anisotropy axis of a probability distribution, determined by Herschkowitz and Opper [Phys.Rev.Lett. 86, 2174 (2001)], is not always valid. If there is some structure in the data along the anisotropy axis, their analysis is incorrect, and lea...

We study the typical learning properties of the recently introduced soft margin classifiers (SMCs), learning realizable and unrealizable tasks, with the tools of statistical mechanics. We derive analytically the behavior of the learning curves in the regime of very large training sets. We obtain exponential and power laws for the decay of the gener...

In this paper, we study the typical learning properties of the recently proposed support vector machines (SVMs). The generalization error on linearly separable tasks, the capacity, the typical number of support vectors, the margin and the robustness or noise tolerance of a class of SVMs are determined in the framework of statistical mechanics. The...

We present a simple model in order to discuss the interaction of the genetic and behavioral systems throughout evolution. This considers a set of adaptive perceptrons in which some of their synapses can be updated through a learning process. This framework provides an extension of the well-known Hinton and Nowlan model by blending together some lea...

The effect of substrate distortion upon commensurate-incommensurate (C-I) transitions in physisorbed layers is considered. For appropriate values of the elastic constants, the misfit parameter becomes proportional to the square root of the chemical potential near the transition. For other values of the elastic constants, a first-order transition re...

It is shown that elastic strains or other harmonic fields can mediate oscillating indirect interactions if direct interactions are not limited to nearest neighbours. In particular the mean-field approximation of the Ising model with competing interactions can be interpreted at low temperature in terms of walls (or 'solitons') which interact through...

The authors consider a physisorbed monolayer of rare gas atoms on graphite near the commensurate-incommensurate (C-I) phase transition. They calculate the energy associated with a small rotation of a wall, to second order in the rotation angle. The wall will be titled with respect to the symmetry axis of the substrate if the two-dimensional compres...

For pt.I see ibid., vol.18, p.3919 (1985). The authors calculate wall and wall crossing energies numerically as a function of VG, the modulation of the substrate potential, at T=0. The wall energy is negative if VG<or=11K. The wall profile does not agree with that predicted by the analytic theory, because of anharmonicity in the Kr-Kr interaction....

The role of anharmonicity on the commensurate (C) phase of Kr and Xe monolayers adsorbed on graphite is analysed. If the usual interatomic interaction parameters are accepted, Kr monolayers at low temperature should be in an incommensurate (I) phase. Thermal expansion stabilises the commensurate phase at temperatures higher than TIC, the temperatur...

We study the typical properties of polynomial support vector machines within a statistical mechanics approach that takes into account the number of high order features relative to the input space dimension. We analyze the effect of different features’ normalizations on the generalization error, for different kinds of learning tasks. If the normaliz...

The storage capacity of an incremental learning algorithm for the parity machine, the Tilinglike Learning Algorithm, is analytically determined in the limit of a large number of hidden perceptrons. Different learning rules for the simple perceptron are investigated. The usual Gardner-Derrida one leads to a storage capacity close to the upper bound,...

In this article we study the eects of introducing structure in the input distribution of the data to be learnt by a simple perceptron. We determine the learning curves within the framework of Statistical Mechanics. Stepwise generalization occurs as a function of the number of examples when the distribution of patterns is highly anisotropic. Althoug...

Upper and lower bounds for the typical storage capacity of a constructive algorithm, the tilinglike learning algorithm for the parity machine (Biehl M and Opper M 1991 Phys. Rev. A 44 6888), are determined in the asymptotic limit of large training set sizes. The properties of a perceptron with threshold, learning a training set of patterns having a...

The learning properties of finite size polynomial Support Vector Machines are analyzed in the case of realizable classification tasks. The normalization of the high order features acts as a squeezing factor, introducing a strong anisotropy in the patterns distribution in feature space. As a function of the training set size, the corresponding gener...

Upper and lower bounds for the typical storage capacity of a constructive algorithm, the Tilinglike Learning Algorithm for the Parity Machine [M. Biehl and M. Opper, Phys. Rev. A {\bf 44} 6888 (1991)], are determined in the asymptotic limit of large training set sizes. The properties of a perceptron with threshold, learning a training set of patter...

The Microelectronics and Photonics Testbed (MPTB) is a scientific satellite carrying twenty-four experiments on-board in a high radiation orbit since November 1997. The first objective of this paper is to summarize one year flight results, telemetred from one of its experiments, a digital “neural board” programmed to perform texture analysis by mea...

PCNN (pulse coupled neural networks) and more generally spiking-neuron models seem to meet the real-time and robustness constraints necessary in on-board pattern recognition applications. However, efficient learning algorithms are still lacking for such networks. We consider a feedforward network of spiking neurons. The weights and biases are obtai...

We present a theoretical study of the properties of a class of Support Vector Machines within the framework of Statistical Mechanics. We determine their capacity, the margin, the number of support vec-tors and the distribution of distances of the patterns to the separating hyperplane in feature-space.

In this article we study the effects of introducing structure in the input distribution of the data to be learnt by a simple perceptron. We determine the learning curves within the framework of Statistical Mechanics. Stepwise generalization occurs as a function of the number of examples when the distribution of patterns is highly anisotropic. Altho...

We consider the optimal performance that may be reached in the problem of learning the symmetry-breaking direction of a cloud of P = N points in a N -dimensional space. The perfor-mance is measured through the overlap R opt between the true symmetry-breaking direction and the learnt one. Depending on the problem, the learning curves Ropt() may pres...

This article presents a new incremental learning algorithm for classification tasks, called Net Lines, which is well adapted for both binary and real-valued input patterns. It generates small, compact feedforward neural networks with one hidden layer of binary units and binary output units. A convergence theorem ensures that solutions with a finite...

. We study the classification of sonar targets first introduced by Gorman & Sejnowski (1988). We discovered that not only the training set and the test set of this benchmark are both linearly separable, although by different hyperplanes, but that the complete set of patterns, training and test patterns together, is also linearly separable. The dist...

We study the typical learning properties of the recently proposed Support Vectors Machines. The generalization error on linearly separable tasks, the capacity, the typical number of Support Vectors, the margin, and the robustness or noise tolerance of a class of Support Vector Machines are determined in the framework of Statistical Mechanics. The r...

We determine the optimal performance of learning the orientation of the symmetry axis of a set of P N points that are uniformly distributed in all the directions but one on the N-dimensional space. The components along the symmetry breaking direction, of unitary vector B, are sampled from a mixture of two Gaussians of variable separation and width....

We determine the optimal performance of learning the orientation of the symmetry axis of a set of P = alpha N points that are uniformly distributed in all the directions but one on the N-dimensional sphere. The components along the symmetry breaking direction, of unitary vector B, are sampled from a mixture of two gaussians of variable separation a...

We determine the cost function that minimizes the generalization error of a perceptron learning a realizable task. This cost endows the perceptron with the optimal (Bayesian) generalization performance. The distribution of distances of the training patterns of the optimal generalizer is determined.

We study numerically the properties of the Bayesian perceptron through a gradient descent on the optimal cost function. The theoretical distribution of stabilities is deduced. It predicts that the optimal generalizer lies close to the boundary of the space of error-free solutions. The numerical simulations are in good agreement with the theoretical...

We present a convergence theorem for incremental learning algorithms, valid for real-valued input patterns. The upper bound to the number of hidden units is equal to P-1, where P is the number of patterns in the training set

In this paper, we compare three neural learning models, on the sonar target classification problem of Gorman and Sejnowski, in two ways. First, we compare the performance on training and testing sets, in the usual sense. Second, we investigate in the database, in order to compare the examples which are hard to be learned and the patterns which are...

We study the numerical performances of Minimerror, a recently introduced learning algorithm for the perceptron that has analytically been shown to be optimal both on learning linearly and nonlinearly separable functions. We present its implementation on learning linearly separable boolean functions. Numerical results are in excellent agreement with...

We analyse the properties of a new learning algorithm for binary perceptrons based on the minimization of a temperature-dependent differentiable cost function. We show that learning at finite temperature increases the stabilities of learned patterns, endowing the perceptron with robustness, at the price of accepting a small fraction of errors in th...