laboratoire de physique statistique
laboratoire de physique statistique


A MODEL OF RIOTS DYNAMICS: SHOCKS, DIFFUSION AND THRESHOLDS - Berestycki, Henri and Nadal, Jean-Pierre and Rodiguez, Nancy

Abstract : We introduce and analyze several variants of a system of differential equations which model the dynamics of social outbursts, such as riots. The systems involve the coupling of an explicit variable representing the intensity of rioting activity and an underlying (implicit) field of social tension. Our models include the effects of exogenous and endogenous factors as well as various propagation mechanisms. From numerical and mathematical analysis of these models we show that the assumptions made on how different locations influence one another and how the tension in the system disperses play a major role on the qualitative behavior of bursts of social unrest. Furthermore, we analyze here various properties of these systems, such as the existence of traveling wave solutions, and formulate some new open mathematical problems which arise from our work.
Entanglement Between Demand and Supply in Markets with Bandwagon Goods - Gordon, Mirta B. and Nadal, Jean-Pierre and Denis Phan and Semeshenko, Viktoriya

Abstract : 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. This leads to a dilemma for the seller: should he sell at a high price, targeting a small number of buyers, or at low price targeting a large number of buyers? In this paper we show that the interaction between demand and supply is even more complex than expected, leading to what we call the curse of coordination: the pricing strategy for the seller which aimed at maximizing his profit corresponds to posting a price which, not only assumes that the customers will coordinate, but also lies very near the critical price value at which such high demand no more exists. This is obtained by the detailed mathematical analysis of a particular model formally related to the Random Field Ising Model and to a model introduced in social sciences by T.C. Schelling in the 70's.
Modeling urban housing market dynamics: Can the socio-spatial segregation preserve some social diversity? - Gauvin, Laetitia and Vignes, Annick and Nadal, Jean-Pierre

Abstract : Addressing issues of social diversity, we introduce a model of housing transactions between agents who are heterogeneous in their willingness to pay. A key assumption is that agents' preferences for a location depend on both an intrinsic attractiveness and on the social characteristics of the neighborhood. The stationary space distribution of income is analytically and numerically characterized. The main results are that socio-spatial segregation occurs if - and only if - the social influence is strong enough, but even so, some social diversity is preserved at most locations. Comparison with data on the Paris housing market shows that the results reproduce general trends of price distribution and spatial income segregation. (C) 2013 Elsevier B.V. All rights reserved.
Perception of categories: From coding efficiency to reaction times - Bonnasse-Gahot, Laurent and Nadal, Jean-Pierre
BRAIN RESEARCH 143447-61 (2012) 

Abstract : Reaction-times in perceptual tasks are the subject of many experimental and theoretical studies. With the neural decision making process as main focus, most of these works concern discrete (typically binary) choice tasks, implying the identification of the stimulus as an exemplar of a category. Here we address issues specific to the perception of categories (e.g. vowels, familiar faces,...), making a clear distinction between identifying a category (an element of a discrete set) and estimating a continuous parameter (such as a direction). We exhibit a link between optimal Bayesian decoding and coding efficiency, the latter being measured by the mutual information between the discrete category set and the neural activity. We characterize the properties of the best estimator of the likelihood of the category, when this estimator takes its inputs from a large population of stimulus-specific coding cells. Adopting the diffusion-to-bound approach to model the decisional process, this allows to relate analytically the bias and variance of the diffusion process underlying decision making to macroscopic quantities that are behaviorally measurable. A major consequence is the existence of a quantitative link between reaction times and discrimination accuracy. The resulting analytical expression of mean reaction times during an identification task accounts for empirical facts, both qualitatively (e.g. more time is needed to identify a category from a stimulus at the boundary compared to a stimulus lying within a category), and quantitatively (working on published experimental data on phoneme identification tasks). (C) 2011 Elsevier B.V. All rights reserved.
Storage of Correlated Patterns in Standard and Bistable Purkinje Cell Models - Clopath, Claudia and Nadal, Jean-Pierre and Brunel, Nicolas

Abstract : The cerebellum has long been considered to undergo supervised learning, with climbing fibers acting as a `teaching' or `error' signal. Purkinje cells (PCs), the sole output of the cerebellar cortex, have been considered as analogs of perceptrons storing input/output associations. In support of this hypothesis, a recent study found that the distribution of synaptic weights of a perceptron at maximal capacity is in striking agreement with experimental data in adult rats. However, the calculation was performed using random uncorrelated inputs and outputs. This is a clearly unrealistic assumption since sensory inputs and motor outputs carry a substantial degree of temporal correlations. In this paper, we consider a binary output neuron with a large number of inputs, which is required to store associations between temporally correlated sequences of binary inputs and outputs, modelled as Markov chains. Storage capacity is found to increase with both input and output correlations, and diverges in the limit where both go to unity. We also investigate the capacity of a bistable output unit, since PCs have been shown to be bistable in some experimental conditions. Bistability is shown to enhance storage capacity whenever the output correlation is stronger than the input correlation. Distribution of synaptic weights at maximal capacity is shown to be independent on correlations, and is also unaffected by the presence of bistability.
Disentangling collective trends from local dynamics - Barthelemy, Marc and Nadal, Jean-Pierre and Berestycki, Henri

Abstract : A single social phenomenon (such as crime, unemployment, or birthrate) can be observed through temporal series corresponding to units at different levels (i. e., cities, regions, and 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 viewpoint that can be very misleading. We propose here 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 synthetic series generated by correlated random walkers. We then consider crime rate series (in the United States and France) and the evolution of obesity rate in the United States, which are two important examples of societal measures. For the crime rates in the United States, we observe large fluctuations in the transition period of mid-70s during which crime rates increased significantly, whereas since the 80s, the state crime rates are governed by external factors and the importance of local specificities being decreasing. In the case of obesity, our method shows that external factors dominate the evolution of obesity since 2000, and that different states can have different dynamical behavior even if their obesity prevalence is similar.
Schelling segregation in an open city: A kinetically constrained Blume-Emery-Griffiths spin-1 system - Gauvin, Laetitia and Nadal, Jean-Pierre and Vannimenus, Jean

Abstract : In the 70s Schelling introduced a multiagent model to describe the segregation dynamics that may occur with individuals having only weak preferences for ``similar'' neighbors. Recently variants of this model have been discussed, in particular, with emphasis on the links with statistical physics models. Whereas these models consider a fixed number of agents moving on a lattice, here, we present a version allowing for exchanges with an external reservoir of agents. The density of agents is controlled by a parameter which can be viewed as measuring the attractiveness of the city lattice. This model is directly related to the zero-temperature dynamics of the Blume-Emery-Griffiths spin-1 model, with kinetic constraints. With a varying vacancy density, the dynamics with agents making deterministic decisions leads to a variety of ``phases'' whose main features are the characteristics of the interfaces between clusters of agents of different types. The domains of existence of each type of interface are obtained analytically as well as numerically. These interfaces may completely isolate the agents leading to another type of segregation as compared to what is observed in the original Schelling model, and we discuss its possible socioeconomic correlates.
MATHEMATICS AND COMPLEXITY IN LIFE AND HUMAN SCIENCES - Bellomo, N. and Berestycki, H. and Brezzi, F. and Nadal, J. -P.
Modelling the individual and collective dynamics of the propensity to offend - Nadal, J-P and Gordon, M. B. and Iglesias, J. R. and Semeshenko, V.

Abstract : 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 outcome and of the crime setting (the context which makes a crime more or less likely to occur, such as the presence or not of a guardian). Within this framework we explore some consequences of the working hypothesis that punishment has a deterrent effect, assuming that, after a criminal act, an agent's honesty index may increase if he is caught and decrease otherwise. We provide both analytical and numerical results. We show that in the space of parameters characterizing the probability of punishment, there are two `phases': one corresponding to a population with a low crime rate and the other to a population with a large crime rate. We speculate on the possible existence of a self-organized state in which, due to the society reaction against crime activities, the population dynamics would be stabilized on the critical line, leading to a wide distribution of propensities to offend in the population. In view of empirical works on the causes of the recent evolution of crime rates in developed countries, we discuss how changes of socio-economic conditions may affect the model parameters, and hence the crime rate in the population. We suggest possible extensions of the model that will allow us to take into account more realistic features.
Self-organised critical hot spots of criminal activity - Berestycki, H. and Nadal, J-P

Abstract : In this paper(1) we introduce a family of models to describe the spatio-temporal dynamics of criminal activity. It is argued here that with a minimal set of mechanisms corresponding to elements that are basic in the study of crime, one can observe the formation of hot spots. By analysing the simplest versions of our model, we exhibit a self-organised critical state of illegal activities that we propose to call a warm spot or a tepid milieu(2) depending on the context. It is characterised by a positive level of illegal or uncivil activity that maintains itself without exploding, in contrast with genuine hot spots where localised high level or peaks are being formed. Within our framework, we further investigate optimal policy issues under the constraint of limited resources in law enforcement and deterrence. We also introduce extensions of our model that take into account repeated victimisation effects, local and long range interactions, and briefly discuss some of the resulting effects such as hysteresis phenomena.
Crime and punishment: the economic burden of impunity - Gordon, M. B. and Iglesias, J. R. and Semeshenko, V. and Nadal, J. P.

Abstract : 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.
Phase diagram of a Schelling segregation model - Gauvin, L. and Vannimenus, J. and Nadal, J. -P.

Abstract : The collective behavior in a variant of Schelling's segregation model is characterized with methods borrowed from statistical physics, in a context where their relevance was not conspicuous. A measure of segregation based on cluster geometry is defined and several quantities analogous to those used to describe physical lattice models at equilibrium are introduced. This physical approach allows to distinguish quantitatively several regimes and to characterize the transitions between them, leading to the building of a phase diagram. Some of the transitions evoke empirical sudden ethnic turnovers. We also establish links with `spin-1' models in physics. Our approach provides generic tools to analyze the dynamics of other socio-economic systems.
DISCRETE CHOICES UNDER SOCIAL INFLUENCE: GENERIC PROPERTIES - Gordon, Mirta B. and Nadal, Jean-Pierre and Phan, Denis and Semeshenko, Viktoriya

Abstract : 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). When j, the ratio of the social influence 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. However, even if the IWP distribution is mono-modal, there is a critical value of j above which the inverse demand is non-monotonic. Thus, depending on the price, there are either one or several equilibria. Beyond this first result, we exhibit the 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 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 phase diagrams in the space of the model parameters, on which the regions of multiple equilibria are precisely delimited. We also discuss the links between the model and the random field Ising model studied in the physics literature.
MATHEMATICS AND COMPLEXITY IN LIFE AND HUMAN SCIENCES - Bellomo, N. and Berestycki, H. and Brezzi, F. and Nadal, J-P.
Cycles of cooperation and free-riding in social systems - Ma, Y. P. and Goncalves, S. and Mignot, S. and Nadal, J. -P. and Gordon, M. B.

Abstract : Basic evidences on non-profit making and other forms of benevolent-based organizations reveal a rough partition of members between some pure consumers of the public good (free-riders) and 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.
Collective states in social systems with interacting learning agents - Semeshenko, Viktoriya and Gordon, Mirta B. and Nadal, Jean-Pierre

Abstract : 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 model is close to Random Field Ising Models, where the random field corresponds to the idiosyncratic willingness to pay. We show that the equilibrium reached depends on the nature of the information agents use to estimate their expected utilities. It may be different from the systems' Nash equilibria. (C) 2008 Elsevier B.V. All rights reserved.
Neural coding of categories: information efficiency and optimal population codes - Bonnasse-Gahot, Laurent and Nadal, Jean-Pierre

Abstract : This paper deals with the analytical study of coding a discrete set of categories by a large assembly of neurons. We consider population coding schemes, which can also be seen as instances of exemplar models proposed in the literature to account for phenomena in the psychophysics of categorization. We quantify the coding efficiency by the mutual information between the set of categories and the neural code, and we characterize the properties of the most efficient codes, considering different regimes corresponding essentially to different signal-to-noise ratio. One main outcome is to find that, in a high signal-to-noise ratio limit, the Fisher information at the population level should be the greatest between categories, which is achieved by having many cells with the stimulus-discriminating parts (steepest slope) of their tuning curves placed in the transition regions between categories in stimulus space. We show that these properties are in good agreement with both psychophysical data and with the neurophysiology of the inferotemporal cortex in the monkey, a cortex area known to be specifically involved in classification tasks.
What can we learn from synaptic weight distributions? - Barbour, Boris and Brunel, Nicolas and Hakim, Vincent and Nadal, Jean-Pierre

Abstract : 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.