We prove that every graph has a spectral sparsifier with a number of edges linear in its number of vertices. As linearsized spectral sparsifiers of complete graphs are expanders, our sparsifiers of arbitrary graphs can be viewed as generalizations of expander graphs.In particular, we prove that for every d > 1 and every undirected, weighted graph G = (V, E, w) on n vertices, there exists a weighted graph H = (V, F,w) with at most d(n − 1) edges such that for every x ∈ R V ,LG and LH are the Laplacian matrices of G and H, respectively. Thus, H approximates G spectrally at least as well as a Ramanujan expander with dn/2 edges approximates the complete graph. We give an elementary deterministic polynomial time algorithm for constructing H.
We prove that for every d > 1 and every undirected, weighted graph G = (V, E), there exists a weighted graph H with at most ⌈d |V |⌉ edges such that for everywhereLG and LH are the Laplacian matrices of G and H, respectively.
The growing prevalence of deadly microbes with resistance to previously life-saving drug therapies is a dire threat to human health. Detection of low abundance pathogen sequences remains a challenge for metagenomic Next Generation Sequencing (NGS). We introduce FLASH (Finding Low Abundance Sequences by Hybridization), a next-generation CRISPR/Cas9 diagnostic method that takes advantage of the efficiency, specificity and flexibility of Cas9 to enrich for a programmed set of sequences. FLASH-NGS achieves up to 5 orders of magnitude of enrichment and sub-attomolar gene detection with minimal background. We provide an open-source software tool (FLASHit) for guide RNA design. Here we applied it to detection of antimicrobial resistance genes in respiratory fluid and dried blood spots, but FLASH-NGS is applicable to all areas that rely on multiplex PCR.
Mosquitoes are major infectious disease-carrying vectors. Assessment of current and future risks associated with the mosquito population requires knowledge of the full repertoire of pathogens they carry, including novel viruses, as well as their blood meal sources. Unbiased metatranscriptomic sequencing of individual mosquitoes offers a straightforward, rapid, and quantitative means to acquire this information. Here, we profile 148 diverse wild-caught mosquitoes collected in California and detect sequences from eukaryotes, prokaryotes, 24 known and 46 novel viral species. Importantly, sequencing individuals greatly enhanced the value of the biological information obtained. It allowed us to (a) speciate host mosquito, (b) compute the prevalence of each microbe and recognize a high frequency of viral co-infections, (c) associate animal pathogens with specific blood meal sources, and (d) apply simple co-occurrence methods to recover previously undetected components of highly prevalent segmented viruses. In the context of emerging diseases, where knowledge about vectors, pathogens, and reservoirs is lacking, the approaches described here can provide actionable information for public health surveillance and intervention decisions.
We introduce a new notion of graph sparsification based on spectral similarity of graph Laplacians: spectral sparsification requires that the Laplacian quadratic form of the sparsifier approximate that of the original. This is equivalent to saying that the Laplacian of the sparsifier is a good preconditioner for the Laplacian of the original.We prove that every graph has a spectral sparsifier of nearly-linear size. Moreover, we present an algorithm that produces spectral sparsifiers in time O (m log c m), where m is the number of edges in the original graph and c is some absolute constant. This construction is a key component of a nearly-linear time algorithm for solving linear equations in diagonallydominant matrices.Our sparsification algorithm makes use of a nearly-linear time algorithm for graph partitioning that satisfies a strong guarantee: if the partition it outputs is very unbalanced, then the larger part is contained in a subgraph of high conductance. * This paper is the second in a sequence of three papers expanding on material that appeared first under the title "Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems" [ST04]. The first paper, "A Local Clustering Algorithm for Massive Graphs and its Application to Nearly-Linear Time Graph Partitioning" [ST08a] contains graph partitioning algorithms that are used to construct the sparsifiers in this paper. The third paper, "Nearly-Linear Time Algorithms for Preconditioning and Solving Symmetric, Diagonally Dominant Linear Systems" [ST08b] contains the results on solving linear equations and approximating eigenvalues and eigenvectors.
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