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We present a novel approach for computing a variant of eigenvector centrality for multilayer networks with inter-layer constraints on node importance. Specifically, we consider a multilayer network defined by multiple edge-weighted, potentially directed, graphs over the same set of nodes with each graph representing one layer of the network and no inter-layer edges. As in the standard eigenvector centrality construction, the importance of each node in a given layer is based on the weighted sum of the importance of adjacent nodes in that same layer. Unlike standard eigenvector centrality, we assume that the adjacency relationship and the importance of adjacent nodes may be based on distinct layers. Importantly, this type of centrality constraint is only partially supported by existing frameworks for multilayer eigenvector centrality that use edges between nodes in different layers to capture inter-layer dependencies. For our model, constrained, layer-specific eigenvector centrality values are defined by a system of independent eigenvalue problems and dependent pseudo-eigenvalue problems, whose solution can be efficiently realized using an interleaved power iteration algorithm.

We describe a family of iterative algorithms that involve the repeated execution of discrete and inverse discrete Fourier transforms. One interesting member of this family is motivated by the discrete Fourier transform uncertainty principle and involves the application of a sparsification operation to both the time domain and frequency domain data with convergence obtained when time domain sparsity hits a stable pattern. This sparsification variant has practical utility for signal denoising, in particular the recovery of a periodic spike signal in the presence of Gaussian noise. General convergence properties and denoising performance are demonstrated using simulation studies. We are not aware of prior work on such iterative Fourier transformation algorithms and have written this paper in part to solicit feedback from others in the field who may be familiar with similar techniques.

Additional file 1: Contains supplementary notes (Notes 1–3), supplementary figures (Figures S1-S11), supplementary Table 1 and 2. Figure S1. Inter-omics Procrustes biplots comparing PCoA ordinations of untargeted metabolite profiles and taxonomic relative abundances for 6 weeks (left panels) (n = 158) and 12 months (right panels) (n = 262). Top panels present analyses based on ordinations from Euclidean distances of genus level abundances after centered log ratio transformation and Euclidean distances of arcsine square root transformed metabolite relative abundances. Bottom panel presents analyses based on generalized Unifrac distance of amplicon sequence variant (ASV) relative abundances and Euclidean distances of arcsine square root transformed metabolite relative abundances. Figure S2. Pairwise Spearman correlation of metabolite bins and genus-level taxonomic abundances for 6-weeks (panel A, N = 158) and 12-months (panel B, N = 282) infants. Left panel displays the overall correlation pattern, where non-significant correlations are not colored (false discovery rate (FDR) controlled q-value 0.05) are not colored. Table S1. Metabolites selected for targeted analysis and their potential biological functions. Table S2. Primers used for bacterial 16S rRNA gene sequencing.

Additional file 1.

Additional file 1: This file contains information on the TCGA cohorts and pathway collections used in this study and supplemental results for both the simulation and real data analyses.

We develop new mathematical tools for the study of the double copy and colour-kinematics duality for tree-level scattering amplitudes using the properties of Lie polynomials. We show that the $S$-map that was defined to simplify super-Yang--Mills multiparticle superfields is in fact a new Lie bracket on the dual space of Lie polynomials. We introduce {\it Lie polynomial currents} based on Berends-Giele recursion for biadjoint scalar tree amplitudes that take values in Lie polynomials. Field theory amplitudes are obtained from the Lie polynomial amplitudes by numerators characterized as homomorphisms from the free Lie algebra to kinematic data. Examples are presented for the biadjoint scalar, Yang--Mills theory and the nonlinear sigma model. That these theories satisfy the Bern-Carrasco-Johansson amplitude relations follows from the identities we prove for the Lie polynomial amplitudes and the existence of BCJ numerators. A KLT map from Lie polynomials to their dual is obtained by nesting the S-map Lie bracket; the matrix elements of this map yield a recently proposed `generalized KLT matrix', and this reduces to the usual KLT matrix when its entries are restricted to a basis. Using this, we give an algebraic proof for the cancellation of double poles in the KLT formula for gravity amplitudes. We finish with some remarks on numerators and colour-kinematics duality from this perspective., Comment: 48 pp

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