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Human complex diseases are affected by both genetic and environmental factors. When multiple environmental risk factors are present, the interaction effect between a gene and the environmental mixture can be larger than the addition of individual interactions, resulting in the so-called synergistic gene–environment (GxE) interactions. Existing literature has shown the power of synergistic gene-environment interaction analysis with cross-sectional traits. In this work, we propose a functional varying index coefficient model for longitudinal traits together with multiple longitudinal environmental risk factors and assess how the genetic effects on a longitudinal disease trait are nonlinearly modified by a mixture of environmental influences. We derive an estimation procedure for the nonparametric functional varying index coefficients under the quadratic inference function and penalized spline framework. We evaluate some theoretical properties such as estimation consistency and asymptotic normality of the estimates. We further propose a hypothesis testing procedure to assess the significance of the synergistic GxE effect. The performance of the estimation and testing procedure is evaluated through Monte Carlo simulation studies. Finally, the utility of the method is illustrated by a real dataset from a pain sensitivity study in which SNP effects are nonlinearly modulated by a mixture of drug dosages and other environmental variables to affect patients’ blood pressure and heart rate. 

Label Propagation is not only a well-known machine learning algorithm for classification, but it is also an effective method for discovering communities and connected components in networks. We propose a new Direction-Optimizing Label Propagation Algorithm (DOLPA) framework that enhances the performance of the standard Label Propagation Algorithm (LPA), increases its scalability, and extends its versatility and application scope. As a central feature, the DOLPA framework relies on the use of frontiers and alternates between label push and label pull operations to attain high performance. It is formulated in such a way that the same basic algorithm can be used for finding communities or connected components in graphs by only changing the objective function used. Additionally, DOLPA has parameters for tuning the processing order of vertices in a graph to reduce the number of edges visited and improve the quality of solution obtained. We present the design and implementation of the enhanced algorithm as well as our shared-memory parallelization of it using OpenMP. We also present an extensive experimental evaluation of our implementations using the LFR benchmark and real-world networks drawn from various domains. Compared with an implementation of LPA for community detection available in a widely used network analysis software, we achieve at most five times the F-Score while maintaining similar runtime for graphs with overlapping communities. We also compare DOLPA against an implementation of the Louvain method for community detection using the same LFR-graphs and show that DOLPA achieves about three times the F-Score at just 10% of the runtime. For connected component decomposition, our algorithm achieves orders of magnitude speedups over the basic LP-based algorithm on large diameter graphs, up to 13.2 × speedup over the Shiloach-Vishkin algorithm, and up to 1.6 × speedup over Afforest on an Intel Xeon processor using 40 threads.