A perturbative approach to non-linearities in the information carried by a two layer neural network

We evaluate the mutual information between the input and the output of a two layer network in the case of a noisy and non-linear analogue channel. In the case where the non-linearity is small with respect to the variability in the noise, we derive an exact expression for the contribution to the mutual information given by the non-linear term in first order of perturbation theory. Finally we show how the calculation can be simplified by means of a diagrammatic expansion. Our results suggest that the use of perturbation theories applied to neural systems might give an insight on the contribution of non-linearities to the information transmission and in general to the neuronal dynamics.Comment: Accepted as a preprint of ICTP, Triest

Stability of the replica symmetric solution for the information conveyed by by a neural network

The information that a pattern of firing in the output layer of a feedforward network of threshold-linear neurons conveys about the network's inputs is considered. A replica-symmetric solution is found to be stable for all but small amounts of noise. The region of instability depends on the contribution of the threshold and the sparseness: for distributed pattern distributions, the unstable region extends to higher noise variances than for very sparse distributions, for which it is almost nonexistant.Comment: 19 pages, LaTeX, 5 figures. Also available at http://www.mrc-bbc.ox.ac.uk/~schultz/papers.html . Submitted to Phys. Rev. E Minor change

Effective Free Energy for Individual Dynamics

Physics and economics are two disciplines that share the common challenge of linking microscopic and macroscopic behaviors. However, while physics is based on collective dynamics, economics is based on individual choices. This conceptual difference is one of the main obstacles one has to overcome in order to characterize analytically economic models. In this paper, we build both on statistical mechanics and the game theory notion of Potential Function to introduce a rigorous generalization of the physicist's free energy, which includes individual dynamics. Our approach paves the way to analytical treatments of a wide range of socio-economic models and might bring new insights into them. As first examples, we derive solutions for a congestion model and a residential segregation model.Comment: 8 pages, 2 figures, presented at the ECCS'10 conferenc

Multifractal Properties of Price Fluctuations of Stocks and Commodities

We analyze daily prices of 29 commodities and 2449 stocks, each over a period of $\approx 15$ years. We find that the price fluctuations for commodities have a significantly broader multifractal spectrum than for stocks. We also propose that multifractal properties of both stocks and commodities can be attributed mainly to the broad probability distribution of price fluctuations and secondarily to their temporal organization. Furthermore, we propose that, for commodities, stronger higher order correlations in price fluctuations result in broader multifractal spectra.Comment: Published in Euro Physics Letters (14 pages, 5 figures

Synchronization interfaces and overlapping communities in complex networks

We show that a complex network of phase oscillators may display interfaces between domains (clusters) of synchronized oscillations. The emergence and dynamics of these interfaces are studied in the general framework of interacting phase oscillators composed of either dynamical domains (influenced by different forcing processes), or structural domains (modular networks). The obtained results allow to give a functional definition of overlapping structures in modular networks, and suggest a practical method to identify them. As a result, our algorithm could detect information on both single overlapping nodes and overlapping clusters.Comment: 5 pages, 4 figure

Precision radial velocities of double-lined spectroscopic binaries with an iodine absorption cell

A spectroscopic technique employing an iodine absorption cell (I_2) to superimpose a reference spectrum onto a stellar spectrum is currently the most widely adopted approach to obtain precision radial velocities of solar-type stars. It has been used to detect ~80 extrasolar planets out of ~130 know. Yet in its original version, it only allows us to measure precise radial velocities of single stars. In this paper, we present a novel method employing an I_2 absorption cell that enables us to accurately determine radial velocities of both components of double-lined binaries. Our preliminary results based on the data from the Keck I telescope and HIRES spectrograph demonstrate that 20-30 m/s radial velocity precision can be routinely obtained for "early" type binaries (F3-F8). For later type binaries, the precision reaches ~10 m/s. We discuss applications of the technique to stellar astronomy and searches for extrasolar planets in binary systems. In particular, we combine the interferometric data collected with the Palomar Testbed Interferometer with our preliminary precision velocities of the spectroscopic double-lined binary HD 4676 to demonstrate that with such a combination one can routinely obtain masses of the binary components accurate at least at the level of 1.0%.Comment: Accepted for publication in The Astrophysical Journa

The mutual information of a stochastic binary channel: validity of the Replica Symmetry Ansatz

We calculate the mutual information (MI) of a two-layered neural network with noiseless, continuous inputs and binary, stochastic outputs under several assumptions on the synaptic efficiencies. The interesting regime corresponds to the limit where the number of both input and output units is large but their ratio is kept fixed at a value $\alpha$. We first present a solution for the MI using the replica technique with a replica symmetric (RS) ansatz. Then we find an exact solution for this quantity valid in a neighborhood of $\alpha = 0$. An analysis of this solution shows that the system must have a phase transition at some finite value of $\alpha$. This transition shows a singularity in the third derivative of the MI. As the RS solution turns out to be infinitely differentiable, it could be regarded as a smooth approximation to the MI. This is checked numerically in the validity domain of the exact solution.Comment: Latex, 29 pages, 2 Encapsulated Post Script figures. To appear in Journal of Physics

Modelling the individual and collective dynamics of the propensity to offend

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.Fil: Nadal, J. P.. Centre National de la Recherche Scientifique; Francia. École des Hautes Études en Sciences Sociales; Francia. Université Pierre et Marie Curie; FranciaFil: Gordon, M. B.. Université de Grenoble I; FranciaFil: Iglesias, J. R.. Universidade Federal do Rio Grande do Sul; BrasilFil: Semeshenko, Viktoriya. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Económicas. Departamento de Economía; Argentin