Most studies of networks have only looked at small subsets of the true network. Here, we discuss the sampling properties of a network?s degree distribution under the most parsimonious sampling scheme. Only if the degree distributions of the network and randomly sampled subnets belong to the same family of probability distributions is it possible to extrapolate from subnet data to properties of the global network. We show that this condition is indeed satisfied for some important classes of networks, notably classical random graphs and exponential random graphs. For scale-free degree distributions, however, this is not the case. Thus, inferences about the scale-free nature of a network may have to be treated with some caution. The work presented here has important implications for the analysis of molecular networks as well as for graph theory and the theory of networks in general.

Abstract: This letter introduces a new representation of discrete signals based on the mathematical notions of functionals and continuous dual spaces. A new and more general sampling theorem is also suggested. Next, the problems of interpolating and resampling discrete signals are addressed; and a general solution using functional interpolation-which is applicable to many different settings-is proposed. Families of resampling filters dubbed de Boor-Ron filters that use de Boor-Ron interpolation are introduced, and their numerical realization is discussed. Some applications of this research are suggested. Index Terms: de Boor–Ron filters, image interpolation, multiresolution, multivariate interpolation, resampling, samplingtheorem, super-resolution, wavelets.

improving motif finding combining expression and chip data