We describe our work in the collection and analysis of massive data describing the connections between participants to online social networks. Alternative approaches to social network data collection are defined and evaluated in practice, against the popular Facebook Web site. Thanks to our ad-hoc, privacy-compliant crawlers, two large samples, comprising millions of connections, have been collected; the data is anonymous and organized as an undirected graph. We describe a set of tools that we developed to analyze specific properties of such social-network graphs, i.e., among others, degree distribution, centrality measures, scaling laws and distribution of friendship.
In this paper we describe in details several results, some of them already presented in a preliminary work @cite_14 based on a significantly smaller sample acquired with a naive technique.
@cite_14
2762725970 2119885577 2962865631 1599875939
In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly overdamped) Langevin diffusion and establish guarantees on its error measured in the Wasserstein-2 distance. Our guarantees improve or extend the state-of-the-art results in three directions. First, we provide an upper bound on the error of the first-order Langevin Monte Carlo (LMC) algorithm with optimized varying step-size. This result has the advantage of being horizon free (we do not need to know in advance the target precision) and to improve by a logarithmic factor the corresponding result for the constant step-size. Second, we study the case where accurate evaluations of the gradient of the log-density are unavailable, but one can have access to approximations of the aforementioned gradient. In such a situation, we consider both deterministic and stochastic approximations of the gradient and provide an upper bound on the sampling error of the first-order LMC that quantifies the impact of the gradient evaluation inaccuracies. Third, we establish upper bounds for two versions of the second-order LMC, which leverage the Hessian of the log-density. We nonasymptotic guarantees on the sampling error of these second-order LMCs. These guarantees reveal that the second-order LMC algorithms improve on the first-order LMC in ill-conditioned settings. We introduce fast algorithms for selecting a random sample of n records without replacement from a pool of N records, where the value of N is unknown beforehand. The main result of the paper is the design and analysis of Algorithm Z; it does the sampling in one pass using constant space and in O ( n (1 + log( N n ))) expected time, which is optimum, up to a constant factor. Several optimizations are studied that collectively improve the speed of the naive version of the algorithm by an order of magnitude. We give an efficient Pascal-like implementation that incorporates these modifications and that is suitable for general use. Theoretical and empirical results indicate that Algorithm Z outperforms current methods by a significant margin. Abstract In this paper, we consider the problem of recovering a compactly supported multivariate function from a collection of pointwise samples of its Fourier transform taken nonuniformly. We do this by using the concept of weighted Fourier frames. A seminal result of Beurling shows that sampling points give rise to a classical Fourier frame provided they are relatively separated and of sufficient density. However, this result does not allow for arbitrary clustering of sampling points, as is often the case in practice. Whilst keeping the density condition sharp and dimension independent, our first result removes the separation condition and shows that density alone suffices. However, this result does not lead to estimates for the frame bounds. A known result of Grochenig provides explicit estimates, but only subject to a density condition that deteriorates linearly with dimension. In our second result we improve these bounds by reducing the dimension dependence. In particular, we provide explicit frame bounds which are dimensionless for functions having compact support contained in a sphere. Next, we demonstrate how our two main results give new insight into a reconstruction algorithm—based on the existing generalized sampling framework—that allows for stable and quasi-optimal reconstruction in any particular basis from a finite collection of samples. Finally, we construct sufficiently dense sampling schemes that are often used in practice—jittered, radial and spiral sampling schemes—and provide several examples illustrating the effectiveness of our approach when tested on these schemes. This paper proposes a unified framework to quantify local and global inferential uncertainty for high dimensional nonparanormal graphical models. In particular, we consider the problems of testing the presence of a single edge and constructing a uniform confidence subgraph. Due to the presence of unknown marginal transformations, we propose a pseudo likelihood based inferential approach. In sharp contrast to the existing high dimensional score test method, our method is free of tuning parameters given an initial estimator, and extends the scope of the existing likelihood based inferential framework. Furthermore, we propose a U-statistic multiplier bootstrap method to construct the confidence subgraph. We show that the constructed subgraph is contained in the true graph with probability greater than a given nominal level. Compared with existing methods for constructing confidence subgraphs, our method does not rely on Gaussian or sub-Gaussian assumptions. The theoretical properties of the proposed inferential methods are verified by thorough numerical experiments and real data analysis.