Random networks with sublinear preferential attachment: The giant component

We study a dynamical random network model in which at every construction step a new vertex is introduced and attached to every existing vertex independently with a probability proportional to a concave function f of its current degree. We give a criterion for the existence of a giant component, which is both necessary and sufficient, and which becomes explicit when f is linear. Otherwise it allows the derivation of explicit necessary and sufficient conditions, which are often fairly close. We give an explicit criterion to decide whether the giant component is robust under random removal of edges. We also determine asymptotically the size of the giant component and the empirical distribution of component sizes in terms of the survival probability and size distribution of a multitype branching random walk associated with f.Comment: Published in at http://dx.doi.org/10.1214/11-AOP697 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Minimization of multi-penalty functionals by alternating iterative thresholding and optimal parameter choices

Inspired by several recent developments in regularization theory, optimization, and signal processing, we present and analyze a numerical approach to multi-penalty regularization in spaces of sparsely represented functions. The sparsity prior is motivated by the largely expected geometrical/structured features of high-dimensional data, which may not be well-represented in the framework of typically more isotropic Hilbert spaces. In this paper, we are particularly interested in regularizers which are able to correctly model and separate the multiple components of additively mixed signals. This situation is rather common as pure signals may be corrupted by additive noise. To this end, we consider a regularization functional composed by a data-fidelity term, where signal and noise are additively mixed, a non-smooth and non-convex sparsity promoting term, and a penalty term to model the noise. We propose and analyze the convergence of an iterative alternating algorithm based on simple iterative thresholding steps to perform the minimization of the functional. By means of this algorithm, we explore the effect of choosing different regularization parameters and penalization norms in terms of the quality of recovering the pure signal and separating it from additive noise. For a given fixed noise level numerical experiments confirm a significant improvement in performance compared to standard one-parameter regularization methods. By using high-dimensional data analysis methods such as Principal Component Analysis, we are able to show the correct geometrical clustering of regularized solutions around the expected solution. Eventually, for the compressive sensing problems considered in our experiments we provide a guideline for a choice of regularization norms and parameters.Comment: 32 page

Emergence of condensation in Kingman's model of selection and mutation

We describe the onset of condensation in the simple model for the balance between selection and mutation given by Kingman in terms of a scaling limit theorem. Loosely speaking, this shows that the wave moving towards genes of maximal fitness has the shape of a gamma distribution. We conjecture that this wave shape is a universal phenomenon that can also be found in a variety of more complex models, well beyond the genetics context, and provide some further evidence for this

Subsistence Agriculture in Central and Eastern Europe: Determinants and Perspectives

Subsistence agriculture in Central and Eastern Europe (CEE) has only recently gained interest from agricultural economists. Their origin, their future and even their definition is still not well elaborated. This paper tries to throw light on the issue of subsistence farming in CEE. It first discusses the theoretical and empirical background of subsistence agriculture. This part is followed by a typology of subsistence farming as found in CEE. Analysis considers several hypotheses on the cause of subsistence agriculture, among them the structure of land ownership, market imperfections and lack of alternative income sources or low opportunity costs of labour respectively. Of all these hypotheses, only the latter can be proofed empirically, which is done by a nonlinear regression analysis. The paper concludes that this gives reason to argue that rather economic problems than specific problems related to the agricultural structure in CEE determine the degree of subsistence farming. Consequently, structural and social policies rather than agricultural policies like market intervention are to be considered.Subsistence agriculture, transition, Central and Eastern Europe, Food Security and Poverty,