Randomized matrix sparsification has proven to be a fruitful technique for producing faster algorithms in applications ranging from graph partitioning to semidefinite programming. In the decade or so of research into this technique, the focus has been—with few exceptions—on ensuring the quality of approximation in the spectral and Frobenius norms. For certain graph algorithms, however, the ∞→1 norm may be a more natural measure of performance. This paper addresses the problem of approximating a real matrix A by a sparse random matrix X with respect to several norms. It provides the first results on approximation error in the ∞→1 and ∞→2 norms, and it uses a result of Lata la to study approximation error in the spectral norm. These bounds hold for a reasonable family of random sparsification schemes, those which ensure that the entries of X are independent and average to the corresponding entries of A. Optimality of the ∞→1 and ∞→2 error estimates is established. Concentration results for the three norms hold when the entries of X are uniformly bounded. The spectral error bound is used to predict the performance of several sparsification and quantization schemes that have appeared in the literature; the results are competitive with the performance guarantees given by earlier scheme-specific analyses.
In @cite_5 , Spielman and Srivastava improve upon this sampling scheme, instead keeping an edge with probability proportional to its ---a measure of how likely it is to appear in a random spanning tree of the graph. They provide an algorithm which produces a sparsifier with @math edges, where @math is the number of vertices in the graph. They obtain this result by reducing the problem to the behavior of projection matrices @math and @math associated with the original graph and the sparsifier, and appealing to a spectral norm concentration result.
@cite_5
2014255080 2114244473 2124450885 2950224979
We consider a variation of the spectral sparsification problem where we are required to keep a subgraph of the original graph. Formally, given a union of two weighted graphs G and W and an integer k, we are asked to find a k-edge weighted graph Wk such that G+Wk is a good spectral sparsifer of G+W. We will refer to this problem as the subgraph (spectral) sparsification. We present a nontrivial condition on G and W such that a good sparsifier exists and give a polynomial-time algorithm to find the sparsifer. As an application of our technique, we show that for each positive integer k, every n-vertex weighted graph has an (n-1+k)-edge spectral sparsifier with relative condition number at most n k log n, O(log log n) where O() hides lower order terms. Our bound nearly settles a question left open by Spielman and Teng about ultrasparsifiers. We also present another application of our technique to spectral optimization in which the goal is to maximize the algebraic connectivity of a graph (e.g. turn it into an expander) with a limited number of edges. A statistical approach to a posteriori blockmodeling for graphs is proposed. The model assumes that the vertices of the graph are partitioned into two unknown blocks and that the probability of an edge between two vertices depends only on the blocks to which they belong. Statistical procedures are derived for estimating the probabilities of edges and for predicting the block structure from observations of the edge pattern only. ML estimators can be computed using the EM algorithm, but this strategy is practical only for small graphs. A Bayesian estimator, based on Gibbs sampling, is proposed. This estimator is practical also for large graphs. When ML estimators are used, the block structure can be predicted based on predictive likelihood. When Gibbs sampling is used, the block structure can be predicted from posterior predictive probabilities. A side result is that when the number of vertices tends to infinity while the probabilities remain constant, the block structure can be recovered correctly with probability tending to 1. (Less) Sampling is a standard approach in big-graph analytics; the goal is to efficiently estimate the graph properties by consulting a sample of the whole population. A perfect sample is assumed to mirror every property of the whole population. Unfortunately, such a perfect sample is hard to collect in complex populations such as graphs (e.g. web graphs, social networks), where an underlying network connects the units of the population. Therefore, a good sample will be representative in the sense that graph properties of interest can be estimated with a known degree of accuracy. While previous work focused particularly on sampling schemes to estimate certain graph properties (e.g. triangle count), much less is known for the case when we need to estimate various graph properties with the same sampling scheme. In this paper, we pro- pose a generic stream sampling framework for big-graph analytics, called Graph Sample and Hold (gSH), which samples from massive graphs sequentially in a single pass, one edge at a time, while maintaining a small state in memory. We use a Horvitz-Thompson construction in conjunction with a scheme that samples arriving edges without adjacencies to previously sampled edges with probability p and holds edges with adjacencies with probability q. Our sample and hold framework facilitates the accurate estimation of subgraph patterns by enabling the dependence of the sampling process to vary based on previous history. Within our framework, we show how to produce statistically unbiased estimators for various graph properties from the sample. Given that the graph analytics will run on a sample instead of the whole population, the runtime complexity is kept under control. Moreover, given that the estimators are unbiased, the approximation error is also kept under control. Finally, we test the performance of the proposed framework (gSH) on various types of graphs, showing that from a sample with -- 40K edges, it produces estimates with relative errors In this work we show that for every @math , such that for all @math where the parameters of the model do not depend on @math . They also provide a rare example where one can prove a polynomial time mixing of Gibbs sampler in a situation where the actual mixing time is slower than @math . Our proof exploits in novel ways the local treelike structure of Erd o s-R 'enyi random graphs, comparison and block dynamics arguments and a recent result of Weitz. Our results extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. In particular, they apply to any graph for which every vertex @math of the graph has a neighborhood @math of radius @math in which the induced sub-graph is a tree union at most @math edges and where for each simple path in @math the sum of the vertex degrees along the path is @math . Moreover, our result apply also in the case of arbitrary external fields and provide the first FPRAS for sampling the Ising distribution in this case. We finally present a non Markov Chain algorithm for sampling the distribution which is effective for a wider range of parameters. In particular, for @math it applies for all external fields and @math , where @math is the critical point for decay of correlation for the Ising model on @math .