arXiv subjects

Gesine Reinert

Publications and source records attributed to Gesine Reinert.

At least 55 records · Page 3Linked to original sources

arXiv2022

DIGRAC: Digraph Clustering Based on Flow Imbalance

Node clustering is a powerful tool in the analysis of networks. We introduce a graph neural network framework, named DIGRAC, to obtain node embeddings for directed networks in a self-supervised manner, including a novel probabilistic imbalance loss, which can be used for network clustering. Here, we propose \textit{directed flow imbalance} measures, which are tightly related to directionality, to reveal clusters in the network even when there is no density difference between clusters. In contrast to standard approaches in the literature, in this paper, directionality is not treated as a nuisance, but rather contains the main signal. DIGRAC optimizes directed flow imbalance for clustering without requiring label supervision, unlike existing graph neural network methods, and can naturally incorporate node features, unlike existing spectral methods. Extensive experimental results on synthetic data, in the form of directed stochastic block models, and real-world data at different scales, demonstrate that our method, based on flow imbalance, attains state-of-the-art results on directed graph clustering when compared against 10 state-of-the-art methods from the literature, for a wide range of noise and sparsity levels, graph structures, and topologies, and even outperforms supervised methods.

stat.ML↗

arXiv2022

On RKHS Choices for Assessing Graph Generators via Kernel Stein Statistics

Score-based kernelised Stein discrepancy (KSD) tests have emerged as a powerful tool for the goodness of fit tests, especially in high dimensions; however, the test performance may depend on the choice of kernels in an underlying reproducing kernel Hilbert space (RKHS). Here we assess the effect of RKHS choice for KSD tests of random networks models, developed for exponential random graph models (ERGMs) in Xu and Reinert (2021)and for synthetic graph generators in Xu and Reinert (2022). We investigate the power performance and the computational runtime of the test in different scenarios, including both dense and sparse graph regimes. Experimental results on kernel performance for model assessment tasks are shown and discussed on synthetic and real-world network applications.

stat.ML↗

arXiv2022

GNNRank: Learning Global Rankings from Pairwise Comparisons via Directed Graph Neural Networks

Recovering global rankings from pairwise comparisons has wide applications from time synchronization to sports team ranking. Pairwise comparisons corresponding to matches in a competition can be construed as edges in a directed graph (digraph), whose nodes represent e.g. competitors with an unknown rank. In this paper, we introduce neural networks into the ranking recovery problem by proposing the so-called GNNRank, a trainable GNN-based framework with digraph embedding. Moreover, new objectives are devised to encode ranking upsets/violations. The framework involves a ranking score estimation approach, and adds an inductive bias by unfolding the Fiedler vector computation of the graph constructed from a learnable similarity matrix. Experimental results on extensive data sets show that our methods attain competitive and often superior performance against baselines, as well as showing promising transfer ability. Codes and preprocessed data are at: \url{https://github.com/SherylHYX/GNNRank}.

cs.LG↗

arXiv2022

Bounds for the chi-square approximation of Friedman's statistic by Stein's method

Friedman's chi-square test is a non-parametric statistical test for $r\geq2$ treatments across $n\ge1$ trials to assess the null hypothesis that there is no treatment effect. We use Stein's method with an exchangeable pair coupling to derive an explicit bound on the distance between the distribution of Friedman's statistic and its limiting chi-square distribution, measured using smooth test functions. Our bound is of the optimal order $n^{-1}$, and also has an optimal dependence on the parameter $r$, in that the bound tends to zero if and only if $r/n\rightarrow0$. From this bound, we deduce a Kolmogorov distance bound that decays to zero under the weaker condition $r^{1/2}/n\rightarrow0$.

math.ST↗

arXiv2022

Stein's Method Meets Computational Statistics: A Review of Some Recent Developments

Stein's method compares probability distributions through the study of a class of linear operators called Stein operators. While mainly studied in probability and used to underpin theoretical statistics, Stein's method has led to significant advances in computational statistics in recent years. The goal of this survey is to bring together some of these recent developments and, in doing so, to stimulate further research into the successful field of Stein's method and statistics. The topics we discuss include tools to benchmark and compare sampling methods such as approximate Markov chain Monte Carlo, deterministic alternatives to sampling methods, control variate techniques, parameter estimation and goodness-of-fit testing.

stat.ME↗

arXiv2022

Multivariate Central Limit Theorems for Random Clique Complexes

Motivated by open problems in applied and computational algebraic topology, we establish multivariate normal approximation theorems for three random vectors which arise organically in the study of random clique complexes. These are: (1) the vector of critical simplex counts attained by a lexicographical Morse matching, (2) the vector of simplex counts in the link of a fixed simplex, and (3) the vector of total simplex counts. The first of these random vectors forms a cornerstone of modern homology algorithms, while the second one provides a natural generalisation for the notion of vertex degree, and the third one may be viewed from the perspective of U-statistics. To obtain distributional approximations for these random vectors, we extend the notion of dissociated sums to a multivariate setting and prove a new central limit theorem for such sums using Stein's method.

math.PR↗

arXiv2022

A Kernelised Stein Statistic for Assessing Implicit Generative Models

Synthetic data generation has become a key ingredient for training machine learning procedures, addressing tasks such as data augmentation, analysing privacy-sensitive data, or visualising representative samples. Assessing the quality of such synthetic data generators hence has to be addressed. As (deep) generative models for synthetic data often do not admit explicit probability distributions, classical statistical procedures for assessing model goodness-of-fit may not be applicable. In this paper, we propose a principled procedure to assess the quality of a synthetic data generator. The procedure is a kernelised Stein discrepancy (KSD)-type test which is based on a non-parametric Stein operator for the synthetic data generator of interest. This operator is estimated from samples which are obtained from the synthetic data generator and hence can be applied even when the model is only implicit. In contrast to classical testing, the sample size from the synthetic data generator can be as large as desired, while the size of the observed data, which the generator aims to emulate is fixed. Experimental results on synthetic distributions and trained generative models on synthetic and real datasets illustrate that the method shows improved power performance compared to existing approaches.

stat.ML↗

arXiv2022

Ranking of Communities in Multiplex Spatiotemporal Models of Brain Dynamics

As a relatively new field, network neuroscience has tended to focus on aggregate behaviours of the brain averaged over many successive experiments or over long recordings in order to construct robust brain models. These models are limited in their ability to explain dynamic state changes in the brain which occurs spontaneously as a result of normal brain function. Hidden Markov Models (HMMs) trained on neuroimaging time series data have since arisen as a method to produce dynamical models that are easy to train but can be difficult to fully parametrise or analyse. We propose an interpretation of these neural HMMs as multiplex brain state graph models we term Hidden Markov Graph Models (HMGMs). This interpretation allows for dynamic brain activity to be analysed using the full repertoire of network analysis techniques. Furthermore, we propose a general method for selecting HMM hyperparameters in the absence of external data, based on the principle of maximum entropy, and use this to select the number of layers in the multiplex model. We produce a new tool for determining important communities of brain regions using a spatiotemporal random walk-based procedure that takes advantage of the underlying Markov structure of the model. Our analysis of real multi-subject fMRI data provides new results that corroborate the modular processing hypothesis of the brain at rest as well as contributing new evidence of functional overlap between and within dynamic brain state communities. Our analysis pipeline provides a way to characterise dynamic network activity of the brain under novel behaviours or conditions.

q-bio.NC↗

arXiv2022

Relaxing the Gaussian assumption in Shrinkage and SURE in high dimension

Shrinkage estimation is a fundamental tool of modern statistics, pioneered by Charles Stein upon his discovery of the famous paradox involving the multivariate Gaussian. A large portion of the subsequent literature only considers the efficiency of shrinkage, and that of an associated procedure known as Stein's Unbiased Risk Estimate, or SURE, in the Gaussian setting of that original work. We investigate what extensions to the domain of validity of shrinkage and SURE can be made away from the Gaussian through the use of tools developed in the probabilistic area now known as Stein's method. We show that shrinkage is efficient away from the Gaussian under very mild conditions on the distribution of the noise. SURE is also proved to be adaptive under similar assumptions, and in particular in a way that retains the classical asymptotics of Pinsker's theorem. Notably, shrinkage and SURE are shown to be efficient under mild distributional assumptions, and particularly for general isotropic log-concave measures.

math.ST↗

arXiv2022

SSSNET: Semi-Supervised Signed Network Clustering

Node embeddings are a powerful tool in the analysis of networks; yet, their full potential for the important task of node clustering has not been fully exploited. In particular, most state-of-the-art methods generating node embeddings of signed networks focus on link sign prediction, and those that pertain to node clustering are usually not graph neural network (GNN) methods. Here, we introduce a novel probabilistic balanced normalized cut loss for training nodes in a GNN framework for semi-supervised signed network clustering, called SSSNET. The method is end-to-end in combining embedding generation and clustering without an intermediate step; it has node clustering as main focus, with an emphasis on polarization effects arising in networks. The main novelty of our approach is a new take on the role of social balance theory for signed network embeddings. The standard heuristic for justifying the criteria for the embeddings hinges on the assumption that "an enemy's enemy is a friend". Here, instead, a neutral stance is assumed on whether or not the enemy of an enemy is a friend. Experimental results on various data sets, including a synthetic signed stochastic block model, a polarized version of it, and real-world data at different scales, demonstrate that SSSNET can achieve comparable or better results than state-of-the-art spectral clustering methods, for a wide range of noise and sparsity levels. SSSNET complements existing methods through the possibility of including exogenous information, in the form of node-level features or labels.

cs.SI↗

arXiv2022

Lead-lag detection and network clustering for multivariate time series with an application to the US equity market

In multivariate time series systems, it has been observed that certain groups of variables partially lead the evolution of the system, while other variables follow this evolution with a time delay; the result is a lead-lag structure amongst the time series variables. In this paper, we propose a method for the detection of lead-lag clusters of time series in multivariate systems. We demonstrate that the web of pairwise lead-lag relationships between time series can be helpfully construed as a directed network, for which there exist suitable algorithms for the detection of pairs of lead-lag clusters with high pairwise imbalance. Within our framework, we consider a number of choices for the pairwise lead-lag metric and directed network clustering components. Our framework is validated on both a synthetic generative model for multivariate lead-lag time series systems and daily real-world US equity prices data. We showcase that our method is able to detect statistically significant lead-lag clusters in the US equity market. We study the nature of these clusters in the context of the empirical finance literature on lead-lag relations and demonstrate how these can be used for the construction of predictive financial signals.

stat.ML↗

arXiv2021

Estimating the correlation in network disturbance models

The Network Disturbance Model of Doreian (1989) expresses the dependency between observations taken at the vertices of a network by modelling the correlation between neighbouring vertices, using a single correlation parameter $ρ$. It has been observed that estimation of $ρ$ in dense graphs, using the method of Maximum Likelihood, leads to results that can be both biased and very unstable. In this paper, we sketch why this is the case, showing that the variability cannot be avoided, no matter how large the network. We also propose a more intuitive estimator of $ρ$, which shows little bias. The related Network Effects Model is briefly discussed.

math.ST↗

arXiv2021

A Stein Goodness of fit Test for Exponential Random Graph Models

We propose and analyse a novel nonparametric goodness of fit testing procedure for exchangeable exponential random graph models (ERGMs) when a single network realisation is observed. The test determines how likely it is that the observation is generated from a target unnormalised ERGM density. Our test statistics are derived from a kernel Stein discrepancy, a divergence constructed via Steins method using functions in a reproducing kernel Hilbert space, combined with a discrete Stein operator for ERGMs. The test is a Monte Carlo test based on simulated networks from the target ERGM. We show theoretical properties for the testing procedure for a class of ERGMs. Simulation studies and real network applications are presented.

stat.ME↗

arXiv2020

Ruin probabilities for risk processes in a bipartite network

This paper studies risk balancing features in an insurance market by evaluating ruin probabilities for single and multiple components of a multivariate compound Poisson risk process. The dependence of the components of the process is induced by a random bipartite network. In analogy with the non-network scenario, a network ruin parameter is introduced. This random parameter, which depends on the bipartite network, is crucial for the ruin probabilities. Under certain conditions on the network and for light-tailed claim size distributions we obtain Lundberg bounds and, for exponential claim size distributions, exact results for the ruin probabilities. For large sparse networks, the network ruin parameter is approximated by a function of independent Poisson variables. T

math.PR↗

arXiv2019

Core-Periphery Structure in Directed Networks

While studies of meso-scale structures in networks often focus on community structure, core--periphery structures can reveal new insights. This structure typically consists of a well-connected core and a periphery that is well connected to the core but sparsely connected internally. Most studies of core--periphery structure focus on undirected networks. We propose a generalisation of core-periphery structure to directed networks. Our approach yields a family of core-periphery block model formulations in which core and periphery sets are edge-direction dependent. We mainly focus on a particular core--periphery structure consisting of two core sets and two periphery sets which we motivate empirically. To detect this directed core-periphery structure we propose four different methods, with different trade-offs between computational complexity and accuracy. We assess these methods on three benchmarks and compare to four standard methods. On simulated data, the proposed methods match or outperform the standard methods. Applying our methods to three empirical networks -- a political blogs networks, a faculty hiring network, and a trade network -- illustrates that this directed core--periphery structure can offer novel insights about the underlying dataset.

cs.SI↗

arXiv2019

Stein operators, kernels and discrepancies for multivariate continuous distributions

In this paper we present a general framework for Stein's method for multivariate continuous distributions. The approach gives a collection of Stein characterisations, among which we highlight score-Stein operators and kernel Stein operators. Applications include copulas and distance between posterior distributions. We give a general construction for Stein kernels for elliptical distributions and discuss Stein kernels in generality, highlighting connections with Fisher information and mass transport. Finally, a goodness-of-fit test based on Stein discrepancies is given.

math.PR↗

arXiv2019

Topological Methods for Characterising Spatial Networks: A Case Study in Tumour Vasculature

Understanding how the spatial structure of blood vessel networks relates to their function in healthy and abnormal biological tissues could improve diagnosis and treatment for diseases such as cancer. New imaging techniques can generate multiple, high-resolution images of the same tissue region, and show how vessel networks evolve during disease onset and treatment. Such experimental advances have created an exciting opportunity for discovering new links between vessel structure and disease through the development of mathematical tools that can analyse these rich datasets. Here we explain how topological data analysis (TDA) can be used to study vessel network structures. TDA is a growing field in the mathematical and computational sciences, that consists of algorithmic methods for identifying global and multi-scale structures in high-dimensional data sets that may be noisy and incomplete. TDA has identified the effect of ageing on vessel networks in the brain and more recently proposed to study blood flow and stenosis. Here we present preliminary work which shows how TDA of spatial network structure can be used to characterise tumour vasculature.

q-bio.QM↗

arXiv2019

First order covariance inequalities via Stein's method

We propose probabilistic representations for inverse Stein operators (i.e. solutions to Stein equations) under general conditions; in particular we deduce new simple expressions for the Stein kernel. These representations allow to deduce uniform and non-uniform Stein factors (i.e. bounds on solutions to Stein equations) and lead to new covariance identities expressing the covariance between arbitrary functionals of an arbitrary {univariate} target in terms of a weighted covariance of the derivatives of the functionals. Our weights are explicit, easily computable in most cases, and expressed in terms of objects familiar within the context of Stein's method. Applications of the Cauchy-Schwarz inequality to these weighted covariance identities lead to sharp upper and lower covariance bounds and, in particular, weighted Poincaré inequalities. Many examples are given and, in particular, classical variance bounds due to Klaassen, Brascamp and Lieb or Otto and Menz are corollaries. Connections with more recent literature are also detailed.

math.PR↗