Lower Bounds on Expansions of Graph Powers

Given a lazy regular graph G, we prove that the expansion of G^t is at least sqrt(t) times the expansion of G. This bound is tight and can be generalized to small set expansion. We show some applications of this result

Cheeger Inequalities for Vertex Expansion and Reweighted Eigenvalues

The classical Cheeger's inequality relates the edge conductance $\phi$ of a graph and the second smallest eigenvalue $\lambda_2$ of the Laplacian matrix. Recently, Olesker-Taylor and Zanetti discovered a Cheeger-type inequality $\psi^2 / \log |V| \lesssim \lambda_2^* \lesssim \psi$ connecting the vertex expansion $\psi$ of a graph $G=(V,E)$ and the maximum reweighted second smallest eigenvalue $\lambda_2^*$ of the Laplacian matrix. In this work, we first improve their result to $\psi^2 / \log d \lesssim \lambda_2^* \lesssim \psi$ where $d$ is the maximum degree in $G$, which is optimal assuming the small-set expansion conjecture. Also, the improved result holds for weighted vertex expansion, answering an open question by Olesker-Taylor and Zanetti. Building on this connection, we then develop a new spectral theory for vertex expansion. We discover that several interesting generalizations of Cheeger inequalities relating edge conductances and eigenvalues have a close analog in relating vertex expansions and reweighted eigenvalues. These include an analog of Trevisan's result on bipartiteness, an analog of higher order Cheeger's inequality, and an analog of improved Cheeger's inequality. Finally, inspired by this connection, we present negative evidence to the $0/1$-polytope edge expansion conjecture by Mihail and Vazirani. We construct $0/1$-polytopes whose graphs have very poor vertex expansion. This implies that the fastest mixing time to the uniform distribution on the vertices of these $0/1$-polytopes is almost linear in the graph size. This does not provide a counterexample to the conjecture, but this is in contrast with known positive results which proved poly-logarithmic mixing time to the uniform distribution on the vertices of subclasses of $0/1$-polytopes.Comment: 65 pages, 1 figure. Minor change

Improved Cheeger's Inequality: Analysis of Spectral Partitioning Algorithms through Higher Order Spectral Gap

Let \phi(G) be the minimum conductance of an undirected graph G, and let 0=\lambda_1 <= \lambda_2 <=... <= \lambda_n <= 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2, \phi(G) = O(k) \lambda_2 / \sqrt{\lambda_k}, and this performance guarantee is achieved by the spectral partitioning algorithm. This improves Cheeger's inequality, and the bound is optimal up to a constant factor for any k. Our result shows that the spectral partitioning algorithm is a constant factor approximation algorithm for finding a sparse cut if \lambda_k$ is a constant for some constant k. This provides some theoretical justification to its empirical performance in image segmentation and clustering problems. We extend the analysis to other graph partitioning problems, including multi-way partition, balanced separator, and maximum cut

Lupus nephritis in Chinese children--a territory-wide cohort study in Hong Kong

We report a multicenter study of Chinese children in Hong Kong with systemic lupus erythematosus (SLE) nephritis. Children were included if: they fulfilled the ACR criteria, had significant proteinuria or casturia, were Chinese and younger than 19 years and had been diagnosed with SLE between January 1990 and December 2003. Investigators in each center retrieved data on clinical features, biopsy reports, treatment and outcome of these patients. There were 128 patients (eight boys, 120 girls; mean age: 11.9+/-2.8 years). About 50% presented with multisystem illness and 40% with nephritic/nephrotic symptoms. Negative anti-dsDNA antibodies were found in 6% of the patients. Renal biopsy revealed WHO Class II, III, IV and V nephritis in 13 (10%), 22 (17%), 69 (54%) and 13 (10%) patients, respectively. The clinical severity of the nephritis did not accurately predict renal biopsy findings. The follow-up period ranged from 1 to 16.5 years (mean+/-SD: 5.76+/-3.61 years). During the study five patients died (two from lupus flare, one from cardiomyopathy, two from infections). Four patients had endstage renal failure (ESRF) (one died during a lupus flare). All deaths and end-stage renal failure occurred in the Class IV nephritis group. Chronic organ damage was infrequent in the survivors. The actuarial patient survival rates at 5, 10 and 15 years of age were 95.3, 91.8, and 91.8%, respectively. For Class IV nephritis patients, the survival rates without ESRF at 5, 10, and 15 years were 91.5, 82.3 and 76%, respectively. The survival and chronic morbidity rates of the Chinese SLE children in the present study are comparable to those of other published studies.postprin

Meta-analysis Followed by Replication Identifies Loci in or near CDKN1B, TET3, CD80, DRAM1, and ARID5B as Associated with Systemic Lupus Erythematosus in Asians

Systemic lupus erythematosus (SLE) is a prototype autoimmune disease with a strong genetic involvement and ethnic differences. Susceptibility genes identified so far only explain a small portion of the genetic heritability of SLE, suggesting that many more loci are yet to be uncovered for this disease. In this study, we performed a meta-analysis of genome-wide association studies on SLE in Chinese Han populations and followed up the findings by replication in four additional Asian cohorts with a total of 5,365 cases and 10,054 corresponding controls. We identified genetic variants in or near CDKN1B, TET3, CD80, DRAM1, and ARID5B as associated with the disease. These findings point to potential roles of cell-cycle regulation, autophagy, and DNA demethylation in SLE pathogenesis. For the region involving TET3 and that involving CDKN1B, multiple independent SNPs were identified, highlighting a phenomenon that might partially explain the missing heritability of complex diseases

Antimicrobial resistance among migrants in Europe: a systematic review and meta-analysis

BACKGROUND: Rates of antimicrobial resistance (AMR) are rising globally and there is concern that increased migration is contributing to the burden of antibiotic resistance in Europe. However, the effect of migration on the burden of AMR in Europe has not yet been comprehensively examined. Therefore, we did a systematic review and meta-analysis to identify and synthesise data for AMR carriage or infection in migrants to Europe to examine differences in patterns of AMR across migrant groups and in different settings. METHODS: For this systematic review and meta-analysis, we searched MEDLINE, Embase, PubMed, and Scopus with no language restrictions from Jan 1, 2000, to Jan 18, 2017, for primary data from observational studies reporting antibacterial resistance in common bacterial pathogens among migrants to 21 European Union-15 and European Economic Area countries. To be eligible for inclusion, studies had to report data on carriage or infection with laboratory-confirmed antibiotic-resistant organisms in migrant populations. We extracted data from eligible studies and assessed quality using piloted, standardised forms. We did not examine drug resistance in tuberculosis and excluded articles solely reporting on this parameter. We also excluded articles in which migrant status was determined by ethnicity, country of birth of participants' parents, or was not defined, and articles in which data were not disaggregated by migrant status. Outcomes were carriage of or infection with antibiotic-resistant organisms. We used random-effects models to calculate the pooled prevalence of each outcome. The study protocol is registered with PROSPERO, number CRD42016043681. FINDINGS: We identified 2274 articles, of which 23 observational studies reporting on antibiotic resistance in 2319 migrants were included. The pooled prevalence of any AMR carriage or AMR infection in migrants was 25·4% (95% CI 19·1-31·8; I2 =98%), including meticillin-resistant Staphylococcus aureus (7·8%, 4·8-10·7; I2 =92%) and antibiotic-resistant Gram-negative bacteria (27·2%, 17·6-36·8; I2 =94%). The pooled prevalence of any AMR carriage or infection was higher in refugees and asylum seekers (33·0%, 18·3-47·6; I2 =98%) than in other migrant groups (6·6%, 1·8-11·3; I2 =92%). The pooled prevalence of antibiotic-resistant organisms was slightly higher in high-migrant community settings (33·1%, 11·1-55·1; I2 =96%) than in migrants in hospitals (24·3%, 16·1-32·6; I2 =98%). We did not find evidence of high rates of transmission of AMR from migrant to host populations. INTERPRETATION: Migrants are exposed to conditions favouring the emergence of drug resistance during transit and in host countries in Europe. Increased antibiotic resistance among refugees and asylum seekers and in high-migrant community settings (such as refugee camps and detention facilities) highlights the need for improved living conditions, access to health care, and initiatives to facilitate detection of and appropriate high-quality treatment for antibiotic-resistant infections during transit and in host countries. Protocols for the prevention and control of infection and for antibiotic surveillance need to be integrated in all aspects of health care, which should be accessible for all migrant groups, and should target determinants of AMR before, during, and after migration. FUNDING: UK National Institute for Health Research Imperial Biomedical Research Centre, Imperial College Healthcare Charity, the Wellcome Trust, and UK National Institute for Health Research Health Protection Research Unit in Healthcare-associated Infections and Antimictobial Resistance at Imperial College London