Testing k-monotonicity of a discrete distribution. Application to the estimation of the number of classes in a population

We develop here several goodness-of-fit tests for testing the k-monotonicity of a discrete density, based on the empirical distribution of the observations. Our tests are non-parametric, easy to implement and are proved to be asymptotically of the desired level and consistent. We propose an estimator of the degree of k-monotonicity of the distribution based on the non-parametric goodness-of-fit tests. We apply our work to the estimation of the total number of classes in a population. A large simulation study allows to assess the performances of our procedures.Comment: 32 pages, 8 figure

Differential Equation Models Derived from an Individual-Based Model Can Help to Understand Emergent Effects

We study a model of primacy effect on individual's attitude. Typically, when receiving a strong negative feature first, the individual keeps a negative attitude whatever the number of moderate positive features it receives afterwards. We consider a population of individuals, which receive the features from a media, and communicate with each other. We observe that interactions favour the primacy effect, compared with a population of isolated individuals. We derive a differential equation system ruling the evolution of probabilities that individuals retain different sets of features. The study of this aggregated model of the IBM shows that interaction can increase or decrease the number of individuals exhibiting a primacy effect. We verify on the IBM that the interactions can decrease the primacy effect in the conditions suggested by the study of the aggregated model. We finally discuss the interest of such a double-modelling approach (using a model of the individual based model) for this application.Primacy Effect, Information Filtering, Agent-Based Model, Aggregated Model, Collective Effects of Interactions, Double-Modelling

The Leviathan model: Absolute dominance, generalised distrust, small worlds and other patterns emerging from combining vanity with opinion propagation

We propose an opinion dynamics model that combines processes of vanity and opinion propagation. The interactions take place between randomly chosen pairs. During an interaction, the agents propagate their opinions about themselves and about other people they know. Moreover, each individual is subject to vanity: if her interlocutor seems to value her highly, then she increases her opinion about this interlocutor. On the contrary she tends to decrease her opinion about those who seem to undervalue her. The combination of these dynamics with the hypothesis that the opinion propagation is more efficient when coming from highly valued individuals, leads to different patterns when varying the parameters. For instance, for some parameters the positive opinion links between individuals generate a small world network. In one of the patterns, absolute dominance of one agent alternates with a state of generalised distrust, where all agents have a very low opinion of all the others (including themselves). We provide some explanations of the mechanisms behind these emergent behaviors and finally propose a discussion about their interestComment: Improved version after referees comment

Testing convex hypotheses on the mean of a Gaussian vector. Application to testing qualitative hypotheses on a regression function

In this paper we propose a general methodology, based on multiple testing, for testing that the mean of a Gaussian vector in R^n belongs to a convex set. We show that the test achieves its nominal level, and characterize a class of vectors over which the tests achieve a prescribed power. In the functional regression model this general methodology is applied to test some qualitative hypotheses on the regression function. For example, we test that the regression function is positive, increasing, convex, or more generally, satisfies a differential inequality. Uniform separation rates over classes of smooth functions are established and a comparison with other results in the literature is provided. A simulation study evaluates some of the procedures for testing monotonicity.Comment: Published at http://dx.doi.org/10.1214/009053604000000896 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Taking into Account the Variations of Neighbourhood Sizes in the Mean-Field Approximation of the Threshold Model on a Random Network

We compare the individual-based \'threshold model\' of innovation diffusion in the version which has been studied by Young (1998), with an aggregate model we derived from it. This model allows us to formalise and test hypotheses on the influence of individual characteristics upon global evolution. The classical threshold model supposes that an individual adopts a behaviour according to a trade-off between a social pressure and a personal interest. Our study considers only the case where all have the same threshold. We present an aggregated model, which takes into account variations of the neighbourhood sizes, whereas previous work assumed this size fixed (Edwards et al. 2003a). The comparison between the aggregated models (the first one assuming a neighbourhood size and the second one, a variable one) points out an improvement of the approximation in most of the value of parameter space. This proves that the average degree of connectivity (first aggregated model) is not sufficient for characterising the evolution, and that the node degree variability has an impact on the diffusion dynamics. Remaining differences between both models give us some clues about the specific ability of individual-based model to maintain a minority behaviour which becomes a majority by an addition of stochastic effects.Aggregate; Individual-Based Model; Innovation Diffusion; Mean Field Approximation; Model Comparison; Social Network Effect

A Universal Model of Commuting Networks

We test a recently proposed model of commuting networks on 80 case studies from different regions of the world (Europe and United-States) and with geographic units of different sizes (municipality, county, region). The model takes as input the number of commuters coming in and out of each geographic unit and generates the matrix of commuting flows betwen the geographic units. We show that the single parameter of the model, which rules the compromise between the influence of the distance and job opportunities, follows a universal law that depends only on the average surface of the geographic units. We verified that the law derived from a part of the case studies yields accurate results on other case studies. We also show that our model significantly outperforms the two other approaches proposing a universal commuting model (Balcan et al. (2009); Simini et al. (2012)), particularly when the geographic units are small (e.g. municipalities).Comment: 11 pages, 5 figure

A commuting network model: going to the bulk

The influence of commuting in socio-economic dynamics increases constantly. Analysing and modelling the networks formed by commuters to help decision-making regarding the land-use has become crucial. This paper presents a simple spatial interaction simulated model with only one parameter. The proposed algorithm considers each individual who wants to commute, starting from their living place to all their workplaces. It decides where the location of the workplace following the classical rule inspired from the gravity law consisting in a compromise between the job offers and the distance to the jobs. The further away the job offer is, the more important it must be in order to be considered. Inversely, only the quantity of offers is important for the decision when these offers are close. The paper also presents a comparative analysis of the structure of the commuting networks of the four European regions to which we apply our model. The model is calibrated and validated on these regions. Results from the analysis shows that the model is very efficient in reproducing most of the statistical properties of the network given by the data sources.Comment: submitted to JASS