Information capacity of a perceptron

Brunel N., Nadal J.-P., and Toulouse G.

We study the information storage capacity of a simple perceptron in the error regime. For random unbiased patterns the geometrical analysis gives a logarithmic dependence of the information content in the asymptotic limit.
In that regime the statistical physics approach, when used at the simplest level of replica theory, does not give satisfactory results. However for perceptrons with finite stability, the information content can be simply calculated with statistical physics methods in a region above the critical storage level, for biased as well as for unbiased patterns.
(Copyright © Institute of Physics and IOP Publishing Limited)

A network model of the coupling of ion channels with secondary messenger in cell signaling

Plouraboué F., Atlan H., Weisbuch G. and Nadal J.-P.

We demonstrate that formal neural networks techniques allow to build the simplest models compatible with a limited but systematic set of experimental data. The experimental system under study is the growth of mouse macrophage like cell lines under the combined influence of two ion channels, the growth factor receptor and adenylate cyclase. We conclude that 3 components out of 4 can be described by linear multithreshold automata. The remaining component behavior being non-monotonous necessitate the introduction of a fifth hidden variable, or of non-linear interactions.

Duality Between Learning Machines:
A Bridge Between Supervised and Unsupervised Learning

Relevant parameters for a class of sequence-retrieving neural networks

Lefèvre O. and Nadal J.-P.

Reconsidering a recently introduced model of sequence-retrieving neural network, we introduce appropriate analogues of the well-known stabilities and show how these, together with two coupling parameters $\lambda$ and $\vartheta$, entirely control the dynamics in the case of strong dilution. The model is exactly solved and phase diagrams are drawn for two different choices of the synaptic matrices; they reveal a rich structure. We then briefly speculate as to the role of these parameters within a more general framework.
(Copyright © Les Editions de Physique 1993)

Information processing by a perceptron in an unsupervised learning task

Jean-Pierre Nadal and Nestor Parga

We study the ability of a simple neural network (a perceptron architecture, no hidden units, binary outputs) to process information in the context of an unsupervised learning task. The network is asked to provide the best possible neural representation of a given input distribution, according to some criterion taken from Information Theory. We compare various optimization criteria that have been proposed : maximum information transmission, minimum redundancy and closeness to factorial code.
We show that for the perceptron one can compute the maximal information that the code (the output neural representation) can convey about the input. We show that one can use Statistical Mechanics techniques, such as the replica techniques, to compute the typical mutual information between input and output distributions. More precisely, for a Gaussian input source with a given correlation matrix, we compute the typical mutual information when the couplings are chosen randomly. We determine the correlations between the synaptic couplings which maximize the gain of information. We analyse the results in the case of a one dimensional receptive field.

Optimal unsupervised learning

Tim L. H. Watkin and Jean-Pierre Nadal

We introduce an inferential approach to unsupervised learning which allows us to define an optimal learning strategy. Applying these ideas to a simple, previously studied model, we show that it is impossible to detect structure in data until a critical number of examples have been presented-an effect which will be observed in all problems with certain underlying symmetries. Thereafter, the advantage of optimal learning over previously studied learning algorithms depends critically upon the distribution of patterns; optimal learning may be exponentially faster. Models with more subtle correlations are harder to analyse, but in a simple limit of one such problem we calculate exactly the efficacy of an algorithm similar to some used in practice, and compare it to that of the optimal prescription.
(Copyright © Institute of Physics and IOP Publishing Limited)

Nonlinear neurons in the low-noise limit:
a factorial code maximizes information transfer

Jean-Pierre Nadal and Nestor Parga

We investigate the consequences of maximizing information transfer in a simple neural network (one input layer, one output layer), focussing on the case of non linear transfer functions. We assume that both receptive fields (synaptic efficacies) and transfer functions can be adapted to the environment.

The main result is that, for bounded and invertible transfer functions, in the case of a vanishing additive output noise, and no input noise, maximization of information (Linsker's infomax principle) leads to a factorial code - hence to the same solution as required by the redundancy reduction principle of Barlow, or, in the signal processing language, to Independent Component Analysis (ICA).

We show also that this result is valid for linear, more generally unbounded, transfer functions, provided optimization is performed under an additive constraint, that is which can be written as a sum of terms, each one being specific to one output neuron. Finally we study the effect of a non zero input noise. We find that, at first order in the input noise, assumed to be small as compared to the - small - output noise, the above results are still valid, provided the output noise is uncorrelated from one neuron to the other.

Trio learning: a new strategy for building hybrid trees

d'Alché-Buc F., Zwierski D. and Nadal J.-P.

Sparsely coded neural networks

Claude Meunier and Jean-Pierre Nadal

Review paper on sparse coding in (auto)associative memories - in particular in Willshaw et al (1969) and Hopfield (1982) type models.

Maximization of mutual information in a linear noisy network: a detailed study

Campa A., Del Giudice P., Parga N. and Nadal J.-P.

We consider a linear, one-layer feedforward neural network performing a coding task. The goal of the network is to provide a statistical neural representation that convey as much information as possible on the input stimuli in noisy conditions. We determine the family of synaptic couplings that maximizes the mutual information between input and output distribution. Optimization is performed under different constraints on the synaptic efficacies. We analyze the dependence of the solutions on input and output noises. This work goes beyond previous studies of the same problem in that:
(i) we perform a detailed stability analysis in order to find the global maxima of the mutual information;
(ii) we examine the properties of the optimal synaptic configurations under different constraints;
(iii) we do not assume translational invariance of the input data, as it is usually done when input are assumed to be visual stimuli.

Asymptotic performance of a constructive algorithm

Florence d'Alché-Buc and Jean-Pierre Nadal

A formal approach to market organization: choice functions,
mean field approximation and maximum entropy principle

Nadal J.-P., Weisbuch G., Chenevez O. and Kirman A.

Redundancy Reduction and Independent Component Analysis:
Conditions on Cumulants and Adaptive Approaches

Jean-Pierre Nadal and Nestor Parga

In the context of both sensory coding and signal processing, building factorized codes has been shown to be an efficient strategy. In a wide variety of situations, the signal to be processed is a linear mixture of statistically independent sources. Building a factorized code is then equivalent to performing blind source separation. Thanks to the linear structure of the data, this can be done, in the language of signal processing, by finding an appropriate linear filter, or equivalently, in the language of neural modeling, by using a simple feedforward neural network.
In this article, we discuss several aspects of the source separation problem. We give simple conditions on the network output that, if satisfied, guarantee that source separation has been obtained. Then we study adaptive approaches, in particular those based on redundancy reduction and maximization of mutual information. We show how the resulting updating rules are related to the BCM theory of synaptic plasticity. Eventually we briefly discuss extensions to the case of nonlinear mixtures. Throughout this article, we take care to put into perspective our work with other studies on source separation and redundancy reduction. In particular we review algebraic solutions, pointing out their simplicity but also their drawbacks.
(Copyright © The MIT Press)

Information processing by a noisy binary channel

Korutcheva E., Parga N. and Nadal J.-P.

We study the information processing properties of a binary channel receiving data from a gaussian source. A systematic comparison with linear processing is done. A remarkable property of the binary sytem is that, as the ratio $\alpha$ between the number of output and input units increases, binary processing becomes equivalent to linear processing with a quantization output noise that depends on $\alpha$. In this regime , that holds up to $O( \alpha^{-4})$ , information processing occurs as if populations of $\alpha$ binary units cooperate to represent one $\alpha$-bit output unit. Unsupervised learning of a noisy environment by optimization of the parameters of the binary channel is also considered.

Self-Similarity Properties of Natural Images Resemble Those of Turbulent Flows

Antonio Turiel, Germ´n Mato, Néstor Parga and Jean-Pierre Nadal

We show that the statistics of an edge type variable in natural images exhibits self-similarity properties which resemble those of local energy dissipation in turbulent flows. Our results show that self-similarity and extended self-similarity hold remarkably for the statistics of the local edge variance, and that the very same models can be used to predict all of the associated exponents. These results suggest using natural images as a laboratory for testing more elaborate scaling models of interest for the statistical description of turbulent flows. The properties we have exhibited are relevant for the modeling of the early visual system: They should be included in models designed for the prediction of receptive fields.
[© 1998 by The American Physical Society]

Nonlinear feedforward networks with stochastic outputs:
infomax implies redundancy reduction

Jean-Pierre Nadal, Nicolas Brunel and Nestor Parga

Mutual information, Fisher information and population coding

Nicolas Brunel and Jean-Pierre Nadal

In the context of parameter estimation and model selection, it is only quite recently that a direct link between the Fisher information and information theoretic quantities has been exhibited. We give an interpretation of this link within the standard framework of information theory. We show that in the context of population coding, the mutual information between the activity of a large array of neurons and a stimulus to which the neurons are tuned is naturally related to the Fisher information.
In the light of this result we consider the optimization of the tuning curves parameters in the case of neurons responding to a stimulus represented by an angular variable.
(Copyright © The MIT Press)

Unsupervised and supervised learning:
the mutual information between parameters and observations

Didier Herschkowitz and Jean-Pierre Nadal

Recent works in parameter estimation and neural coding have demonstrated that optimal performance are related to the mutual information between parameters and data. In this paper:

Blind Source Separation with Time Dependent Mixtures

Nestor Parga and Jean-Pierre Nadal

We address the problem of blind source separation in the case of a time dependent mixture matrix. For a slowly and smoothly varying mixture matrix, we propose a systematic expansion which leads to a practical algebraic solution when stationary and ergodic properties hold for the sources.
[© 2000 Elsevier Science B. V.]

Analysis of geophysical time series and information theory: Independent Component Analysis

Filipe Aires, Alain Chédin and Jean-Pierre Nadal

Résumé :
Dans le but d'identifier les causes physiques de la variabilité d'un système dynamique, la communauté géophysique utilise de façon intensive les techniques statistiques d'extraction de composantes. Un algorithme récemment développé, fondé sur la théorie de l'information, est introduit dans ce travail : l'analyse en composantes indépendantes (ACI). Cette technique présente deux avantages majeurs sur les techniques classiques. Premièrement, elle a pour but d'extraire des composantes statistiquement indépendantes, là où les techniques classiques cherchent uniquement la décorrélation. Deuxièmement, l'hypothèse linéaire pour le mélange des composantes n'est pas requise. Cette nouvelle technique est présenté dans le contexte de l'analyse de séries temporelles géophysiques. L'algorithme ACI est appliqué à l'étude de la variabilité de la température de surface de l'océan (TSO) tropical, avec une attention particulière pour l'analyse des liens entre le phénomène El Niño/Southern Oscillation (Enso) et la variabilité de la TSO Atlantique.
(© 1999 - Académie des Sciences/ Éditions Scientifiques et Médicales Elsevier SAS)

Independent component analysis of multivariate time series. Application to the tropical SST variability

Filipe Aires, Alain Chédin and Jean-Pierre Nadal

With the aim of identifying the physical causes of variability of a given dynamical system, the geophysical community has made an extensive use of classical component extraction techniques such as principal component analysis (PCA) or rotational techniques (RT). We introduce a recently developed algorithm based on information theory: independent component analysis (ICA). This new technique presents two major advantages over classical methods. First, it aims at extracting statistically independent components where classical techniques search for decorrelated components (i.e., a weaker constraint). Second, the linear hypothesis for the mixture of components is not required. In this paper, after having briefly summarized the essentials of classical techniques, we present the new method in the context of geophysical time series analysis. We then illustrate the ICA algorithm by applying it to the study of the variability of the tropical sea surface temperature (SST), with a particular emphasis on the analysis of the links between El Niño Southern Oscillation (ENSO) and Atlantic SST variability. The new algorithm appears to be particularly efficient in describing the complexity of the phenomena and their various sources of variability in space and time.
(© 2000 by the American Geophysical Union)

Meet, Discuss and Segregate!

Gerard Weisbuch, Guillaume Deffuant, Frederic Amblard, Jean-Pierre Nadal

We present a model of opinion dynamics in which agents adjust continuous opinions as a result of random binary encounters whenever their difference in opinion is below a given threshold. High thresholds yield convergence of opinions toward an average opinion, whereas low thresholds result in several opinion clusters. The model is further generalized to network interactions, threshold heterogeneity, adaptive thresholds, and binary strings of opinions.
(Copyright © 2002 Wiley Periodicals, Inc., A Wiley Company)

Rigorous Bounds to Retarded Learning

Arnaud Buhot, Mirta B. Gordon, Jean-Pierre Nadal

Orientational minimal redundancy wavelets: from edge detection to perception

Antonio Turiel, Jean-Pierre Nadal and Nestor Parga

Natural images are complex but very structured objects and, in spite of its com- plexity, the sensory areas in the neocortex in mammals are able to devise learned strategies to encode them endciently. How is this goal achieved? In this paper, we will discuss the multiscaling approach, which has been recently used to derive a redundancy reducing wavelet basis. This kind of representation can be statistically learned from the data and is optimally adapted for image coding; besides, it presents some remarkable features found in the visual pathway. We will show that the introduction of oriented wavelets is necessary to provide a complete description, which stresses the role of the wavelets as edge detectors.
(Copyright © 2003 Elsevier Science Ltd.)

Is there something out there? Infering space from sensorimotor dependencies

David Philipona, Kevin O'Regan and Jean-Pierre Nadal

This letter suggests that in biological organisms, the perceived structure of reality, in particular the notions of body, environment, space, object, and attribute, could be a consequence of an effort on the part of brains to account for the dependency between their inputs and their outputs in terms of a small number of parameters. To validate this idea, a procedure is demonstrated whereby the brain of an organism with arbitrary input and output connectivity can deduce the dimensionality of the rigid group of the space underlying its input-output relationship, that is the dimension of what the organism will call physical space.
(© 2003 The MIT Press)

Identifying genes from up-down properties of microarray expression series

Karen Willbrand, Francois Radvanyi, Jean-Pierre Nadal, Jean-Paul Thiery, and Thomas M. A. Fink

Motivation: We consider any collection of microarrays that can be ordered to form a progression, as a function of time, or severity of disease, or dose of a stimulant, for example. By plotting the expression level of each gene as a function of time, or severity, or dose, we form an expression series, or curve, for each gene. While most of these curves will exhibit random fluctuations, some will contain pattern, and it is these genes which are most likely associated with the independent variable.
Results: We introduce a method of identifying pattern and hence genes in microarray expression curves without knowing what kind of pattern to look for. Key to our approach is the sequence of ups and downs formed by pairs of consecutive data points in each curve. As a benchmark, we blindly identified yeast cell cycles genes without selecting for periodic or any other anticipated behaviour.
(Copyright © 2005 Oxford Journals)

Optimal Information Storage and the Distribution of Synaptic Weights: Perceptron versus Purkinje Cell

Nicolas Brunel, Vincent Hakim, Philippe Isope, Jean-Pierre Nadal and Boris Barbour

It is widely believed that synaptic modifications under lie learning and memory. However, few studies have examined what can be deduced about the learning process from the distribution of synaptic weights. We analyze the perceptron, a prototypical feedforward neural network, and obtain the optimal synaptic weight distribution for a perceptron with excitatory synapses. It contains more than 50% silent synapses, and this fraction increases with storage reliability: silent synapses are therefore a necessary byproduct of optimizing learning and reliability. Exploiting the classical analogy between the perceptron and the cerebellar Purkinje cell, we fitted the optimal weight distribution to that measured for granule cell-Purkinje cell synapses. The two distributions agreed well, suggesting that the Purkinje cell can learn up to 5 kilobytes of information in the form of 40,000 input-output associations.
(Copyright © 2004 by Cell Press)

Pre-attentive segmentation of oriented textures

Ingrid Machecler and Jean-Pierre Nadal

The interpretation of geophysical data, such as images of subsurface rocks (seismic data, borehole scans), requires one in particular to perform an elaborate segmentation analysis on strongly textured, anisotropic, and not necessarily brightness-contrasted images. In this paper we explore the possibility of deriving new segmentation algorithms from recent advances in the neural modelling of pre-attentive segmentation in human vision. More specifically we consider a neural model proposed by Zhaoping Li. First, we reproduce some specific results obtained by Zhaoping Li on simple artificial and real images sharing some textural characteristics with geophysical data. Next, from the analysis of the model behaviour, we propose an image processing workflow depending on the textural characteristics and on the type of segmentation (contour enhancement or texture edge detection) one is interested in. With this algorithm one gets promising results: from the computation of a single attribute one extracts the oriented textured feature boundaries without prior classification.
(Copyright © Institute of Physics and IOP Publishing Limited 2004)

Heterogeneity and feedback in an agent-based market model

François Ghoulmié, Rama Cont and Jean-Pierre Nadal

We propose an agent-based model of a single-asset financial market, described in terms of a small number of parameters, which generates price returns with statistical properties similar to the stylized facts observed in financial time series. Our agent-based model generically leads to the absence of autocorrelation in returns, self-sustaining excess volatility, mean-reverting volatility, volatility clustering and endogenous bursts of market activity non-attributable to external noise. The parsimonious structure of the model allows the identification of feedback and heterogeneity as the key mechanisms leading to these effects.
(Copyright © Institute of Physics and IOP Publishing Limited 2005)

Seller's dilemma due to social interactions between customers

Mirta B. Gordon, Jean-Pierre Nadal, Denis Phan and Jean Vannimenus

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 strong enough, there is a regime where, if the mean willingness to pay increases, or if the production costs decrease, the optimal solution for the monopolist jumps from a solution with a high price and a small number of buyers, to a solution with a low price and a large number of buyers. Depending on the path of prices adjustments by the monopolist, simulations show hysteretic effects on the fraction of buyers.
(Copyright © 2005 Elsevier B.V.)

Multiple equilibria in a monopoly market with heterogeneous agents and externalities

Jean-Pierre Nadal, Denis Phan, Mirta B. Gordon and Jean Vannimenus

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 consider two special cases of heterogeneity in the individuals' decision rules, corresponding in the literature to the Random Utility Models of Thurstone, and of McFadden and Manski. In the first one the heterogeneity fluctuates with time, leading to a standard model in Physics: the Ising model at finite temperature (known as annealed disorder) in a uniform external field. In the second approach the heterogeneity among agents is fixed; in Physics this is a particular case of quenched disorder models known as random field Ising model, at zero temperature. We study analytically the equilibrium properties of the market in the limiting case where each agent is influenced by all the others (the mean field limit), and we illustrate some dynamic properties of these models making use of numerical simulations in an Agent based Computational Economics approach.
(Copyright © 2005 Taylor & Francis)

The acquisition of allophonic rules: statistical learning with linguistic constraints

Sharon Peperkamp, Rozenn Le Calvez, Jean-Pierre Nadal and Emmanuel Dupoux

Phonological rules relate surface phonetic word forms to abstract underlying forms that are stored in the lexicon. Infants must thus acquire these rules in order to infer the abstract representation of words. We implement a statistical learning algorithm for the acquisition of one type of rule, namely allophony, which introduces context-sensitive phonetic variants of phonemes. This algorithm is based on the observation that different realizations of a single phoneme typically do not appear in the same contexts (ideally, they have complementary distributions). In particular, it measures the discrepancies in context probabilities for each pair of phonetic segments. In Experiment 1, we test the algorithm.s performances on a pseudo-language and show that it is robust to statistical noise due to sampling and coding errors, and to non-systematic rule application. In Experiment 2, we show that a natural corpus of semiphonetically transcribed child-directed speech in French presents a very large number of near-complementary distributions that do not correspond to existing allophonic rules. These spurious allophonic rules can be eliminated by a linguistically motivated filtering mechanism based on a phonetic representation of segments. We discuss the role of a priori linguistic knowledge in the statistical learning of phonology.
(Copyright © 2005 Elsevier B.V)

Discrete Choices under Social Influence: Generic Properties

Mirta B. Gordon, Jean-Pierre Nadal, Denis Phan and Viktoriya Semeshenko

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 Idiosyncratic Willingnesses to Pay (IWP).
Positive additive externalities yield a family of inverse demand curves that include the classical downward sloping ones but also new ones with non constant convexity. When $j$, the ratio of the social influene strength to the standard deviation of the IWP distribution, is small enough, the inverse demand is a classical monotonic (decreasing) function of the adoption rate. Even if the IWP distribution is mono-modal, there is a critical value of $j$ above which the inverse demand is non monotonic, decreasing for small and high adoption rates, but increasing within some intermediate range. Depending on the price there are thus either one or two equilibria.
Beyond this first result, we exhibit the {\em generic} properties of the boundaries limiting the regions where the system presents different types of equilibria (unique or multiple). These properties are shown to depend {\em only} on qualitative features of the IWP distribution: modality (number of maxima), smoothness and type of support (compact or infinite). The main results are summarized as {\em phase diagrams} in the space of the model parameters, on which the regions of multiple equilibria are precisely delimited.

Collective states in social systems with interacting learning agents

Viktoriya Semeshenko, Mirta B. Gordon and Jean-Pierre Nadal

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 that the equilibrium reached by the system depends on the nature of the information agents use to estimate their expected utilities.
(Copyright © 2008 Elsevier B.V. )

Crime and punishment: the economic burden of impunity

Mirta B. Gordon, J. Roberto Iglesias, Viktoriya Semeshenko and Jean-Pierre Nadal

Crime is an economically relevant activity. It may represent a mechanism of wealth distribution but also a social and economic burden because of the interference with regular legal activities and the cost of the law enforcement system. Sometimes it may be less costly for the society to allow for some level of criminality. However, a drawback of such a policy is that it may lead to a high increase of criminal activity, that may become hard to reduce later on. Here we investigate the level of law enforcement required to keep crime within acceptable limits. A sharp phase transition is observed as a function of the probability of punishment. We also analyze other consequences of criminality as the growth of the economy, the inequality in the wealth distribution (the Gini coefficient) and other relevant quantities under different scenarios of criminal activity and probabilities of apprehension.
(Copyright © EDP Sciences, Societàà italiana di Fisica, Springer-Verlag 2009)

What can we learn from synaptic weight distributions?

Boris Barbour, Nicolas Brunel, Vincent Hakim and Jean-Pierre Nadal

Much research effort into synaptic plasticity has been motivated by the idea that modifications of synaptic weights (or strengths or efficacies) underlie learning and memory. Here, we examine the possibility of exploiting the statistics of experimentally measured synaptic weights to deduce information about the learning process. Analysing distributions of synaptic weights requires a theoretical framework to interpret the experimental measurements, but the results can be unexpectedly powerful, yielding strong constraints on possible learning theories as well as information that is difficult to obtain by other means, such as the information storage capacity of a cell. We review the available experimental and theoretical techniques as well as important open issues.
(Copyright © 2007 Elsevier B.V)

Neural Coding of Categories:
Information Efficiency and Optimal Population Codes

Pricing Strategies of Goods with Externalities

Mirta B. Gordon, Jean-Pierre Nadal, Denis Phan and Viktoriya Semeshenko

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 them at a higher price. This arises for numerous mono-modal distributions of idiosyncratic willingness to pay if the social influence is strong enough. The seller's optimal strategy switches from one regime to the other at parameter values where the demand has two different Nash equilibria; but the strategy of posting low prices to attract large fractions of buyers may fail due to a lack of coordination.

Phase diagram of a Schelling segregation model

Cycles of cooperation and free-riding in social systems

Yiping P. Ma, Sebastian Gonçalves, Sylvain Mignot, Jean-Pierre Nadal and Mirta B. Gordon

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 utility of joining the community is proportional to its size. We assume an idiosyncratic willingness to join the community ; cooperation bears a fixed cost while free-riding bears a (moral) idiosyncratic cost proportional to the fraction of cooperators. We show that the system presents two types of equilibria: fixed points (Nash equilibria) with a mixture of cooperators and free-riders and cycles where the size of the community, as well as the proportion of cooperators and free-riders, vary periodically.

Disentangling collective trends from local dynamics

Marc Barthelemy, Jean-Pierre Nadal and Henri Berestycki

A single social phenomenon (such as crime, unemployment or birth rate) can be observed through temporal series corresponding to units at different levels (cities, regions, countries...). Units at a given local level may follow a collective trend imposed by external conditions, but also may display fluctuations of purely local origin. The local behavior is usually computed as the difference between the local data and a global average (e.g. a national average), a view point which can be very misleading. In this article, we propose a method for separating the local dynamics from the global trend in a collection of correlated time series. We take an independent component analysis approach in which we do not assume a small average local contribution in contrast with previously proposed methods. We first test our method on financial time series for which various data analysis tools have already been used. For the S&P500 stocks, our method is able to identify two classes of stocks with marked different behaviors: the `followers' (stocks driven by the collective trend), and the `leaders' (stocks for which local fluctuations dominate). Furthermore, as a byproduct contributing to its validation, the method also allows to classify stocks in several groups consistent with industrials sectors. We then consider crime rate series, a domain where the separation between global and local policies is still a major subject of debate. We apply our method to the states in the US and the regions in France. In the case of the US data, we observe large fluctuations in the transition period of mid-70's during which crime rates increased significantly, whereas since the 80's, the state crime rates are governed by external factors, and the importance of local specificities being decreasing.

Schelling segregation in an open city: a kinetically constrained Blume-Emery-Griffiths spin-1 system

Perception of categories: from coding efficiency to reaction times

Modeling urban housing market dynamics: can the socio-spatial segregation preserve some social diversity?