As network data has become ubiquitous in the sciences, there has been growing interest in network models whose structure is driven by latent node-level variables in a (typically low-dimensional) latent geometric space. These "latent positions" are often estimated via embeddings, whereby the nodes of a network are mapped to points in Euclidean space so that "similar" nodes are mapped to nearby points. Under certain model assumptions, these embeddings are consistent estimates of the latent positions, but most such results require the embedding dimension to be chosen correctly. Methods for choosing the embedding dimension have been studied extensively, but little is known about the behavior of embeddings when the dimension is misspecified. In this work, we provide a theoretical description of the effects of dimension misspecification under the random dot product graph, a class of latent space network models that includes several widely-used network models, most notably the stochastic blockmodel, as special cases. We show that when the dimension is chosen too large, consistent estimation still holds, albeit at a slower rate than when the embedding dimension is chosen correctly. On the other hand, when the dimension is chosen too small, there is a fundamental estimation error lower bound that need not go to zero in the large-network limit. A range of synthetic data experiments support our theoretical results. Our main technical result, which may be of independent interest, is a generalization of earlier work in random matrix theory showing that all non-signal eigenvectors of a low-rank matrix subject to additive noise are delocalized.

In many applications, weighted networks are constructed based on time series data: each time series is associated to a vertex and edge weights are given by pairwise correlations. The result is a network whose edge dependency structure violates the assumptions of most common network models. Nonetheless, it is common to analyze these "correlation networks" using embedding methods derived from edge-independent network models, based on a belief that the edges are approximately independent. In this work, we put this modeling choice on firm theoretical ground. We show that when the time series are expressible in terms of a small number of Fourier basis elements (or in some other suitably-chosen basis), correlation networks correspond to latent space networks with dependent edge noise in which the vertex-level latent variables encode the basis coefficients. Further, we show that when time series are observed subject to noise, spectral embedding of the resulting noisy correlation network still recovers these true vertex-level latent representations under suitable assumptions. This characterization of embeddings as learning Fourier coefficients appears to be folklore in the signal processing community in the context of principal component analysis, but is, to the best of our knowledge, new to the statistical network analysis literature.

Peer effect estimation requires precise network measurement, yet most empirical networks are noisy, rendering standard estimators inconsistent. To address measurement error in networks, we propose a method to estimate peer effects in networks whose expected adjacency matrix is low-rank. Our key result shows that peer effects over a true unobserved network are asymptotically equivalent to peer effects over the expected adjacency matrix. This result reduces peer effect estimation in noisy networks to low-rank matrix estimation targeting the expected adjacency matrix. We develop our theory for weighted networks observed with additive noise, but simulations suggest approach can be applied more generally when there is a low-rank estimation method suited to a particular noise structure. We demonstrate via simulations that our approach applies to egocentric samples, aggregated relational data, and networks with missing edges, each requiring a different low-rank estimation method.

As network data has grown in popularity and ubiquity, collections of networks have become a common object of study, with appropriate data analysis typically requiring the identification of structural similarities and differences across networks and at different scales within them. We describe a statistically principled, scalable "omnibus embedding", in which multiple graphs on the same vertex set are jointly mapped into a single space with a distinct representation for each graph. This embedding streamlines graph comparison and comes with performance guarantees, including consistency and a central limit theorem, for the estimation of underlying network parameters. The joint embedding and accompanying central limit theorem provide solutions to several multiscale graph inference questions, such as the identification of graph-wide and vertex-specific differences across networks. We show that in simulated data, the omnibus embedding exhibits near-optimal estimation accuracy when the networks have the same generative structure, while still preserving discriminative power when the networks are different. We also analyze neuroscientific data collected from human subjects, and use the omnibus embedding to successfully identify specific brain regions associated with structural differences and markers of pathology.

We develop a method to decompose causal effects on a social network into an indirect effect mediated by the network, and a direct effect independent of the social network. To handle the complexity of network structures, we assume that latent social groups act as causal mediators. We develop principal components network regression models to differentiate the social effect from the non-social effect. Fitting the regression models is as simple as principal components analysis followed by ordinary least squares estimation. We prove asymptotic theory for regression coefficients from this procedure and show that it is widely applicable, allowing for a variety of distributions on the regression errors and network edges. We carefully characterize the counterfactual assumptions necessary to use the regression models for causal inference, and show that current approaches to causal network regression may result in over-control bias. The method is very general, so that it is applicable to many types of structured data beyond social networks, such as text, areal data, psychometrics, images and omics.

A core problem in statistical network analysis is to develop network analogues of classical techniques. The problem of bootstrapping network data stands out as especially challenging, since typically one observes only a single network, rather than a sample. Here we propose two methods for obtaining bootstrap samples for networks drawn from latent space models. The first method generates bootstrap replicates of network statistics that can be represented as U-statistics in the latent positions, and avoids actually constructing new bootstrapped networks. The second method generates bootstrap replicates of whole networks, and thus can be used for bootstrapping any network function. Commonly studied network quantities that can be represented as U-statistics include many popular summaries, such as average degree and subgraph counts, but other equally popular summaries, such as the clustering coefficient, are not expressible as U-statistics and thus require the second bootstrap method. Under the assumption of a random dot product graph, a type of latent space network model, we show consistency of the proposed bootstrap methods. We give motivating examples throughout and demonstrate the effectiveness of our methods on synthetic data.

Spectral estimators are fundamental in lowrank matrix models and arise throughout machine learning and statistics, with applications including network analysis, matrix completion and PCA. These estimators aim to recover the leading eigenvalues and eigenvectors of an unknown signal matrix observed subject to noise. While extensive research has addressed the statistical accuracy of spectral estimators under a variety of conditions, most previous work has assumed that the signal eigenvectors are incoherent with respect to the standard basis. This assumption typically arises because of suboptimal dependence on coherence in one or more concentration inequalities. Using a new matrix concentration result that may be of independent interest, we establish estimation error bounds for eigenvector and eigenvalue recovery whose dependence on coherence significantly improves upon prior work. Our results imply that coherence-free bounds can be achieved when the standard deviation of the noise is comparable to its Orlicz 1-norm (i.e., its subexponential norm). This matches known minimax lower bounds under Gaussian noise up to logarithmic factors.

We study the problem of modeling multiple symmetric, weighted networks defined on a common set of nodes, where networks arise from different groups or conditions. We propose a model in which each network is expressed as the sum of a shared low-rank structure and a node-sparse matrix that captures the differences between conditions. This formulation is motivated by practical scenarios, such as in connectomics, where most nodes share a global connectivity structure while only a few exhibit condition-specific deviations. We develop a multi-stage estimation procedure that combines a spectral initialization step, semidefinite programming for support recovery, and a debiased refinement step for low-rank estimation. We establish minimax-optimal guarantees for recovering the shared low-rank component under the row-wise ℓ_2,∞ norm and elementwise ℓ_∞ norm, as well as for detecting node-level perturbations under various signal-to-noise regimes. We demonstrate that the availability of multiple networks can significantly enhance estimation accuracy compared to single-network settings. Additionally, we show that commonly-used methods such as group Lasso may provably fail to recover the sparse structure in this setting, a result which might be of independent interest.

In many application domains, networks are observed with node-level features. In such settings, a common problem is to assess whether or not nodal covariates are correlated with the network structure itself. Here, we present four novel methods for addressing this problem. Two of these are based on a linear model relating node-level covariates to latent node-level variables that drive network structure. The other two are based on applying canonical correlation analysis to the node features and network structure, avoiding the linear modeling assumptions. We provide theoretical guarantees for all four methods when the observed network is generated according to a low-rank latent space model endowed with node-level covariates, which we allow to be high-dimensional. Our methods are computationally cheaper and require fewer modeling assumptions than previous approaches to network dependency testing. We demonstrate and compare the performance of our novel methods on both simulated and real-world data.

Spatial networks are widely used in various fields to represent and analyze interactions or relationships between locations or spatially distributed entities. Existing studies have proposed methods for hub identification and community detection in spatial networks, but relatively few have focused on quantifying the strength or density of connections shared within a community of hubs across space and time. Borrowing from network science, there is a relevant concept known as the rich club phenomenon, which describes the tendency of rich nodes to form densely interconnected subnetworks. Although there are established methods to quantify topological, weighted, and temporal rich clubs individually, there is limited research on measuring the rich club effect in spatially weighted temporal networks, which could be particularly useful for studying dynamic spatial interaction networks. To address this gap, we introduce the spatially weighted temporal rich club (WTRC), a metric that quantifies the strength and consistency of connections between rich nodes in a spatiotemporal network. Additionally, we present a unified rich club framework that distinguishes the WTRC effect from other rich club effects, providing a way to measure topological, weighted, and temporal rich club effects together. Through two case studies of human mobility networks at different spatial scales, we demonstrate how the WTRC is able to identify significant weighted temporal rich club effects, whereas the unweighted equivalent in the same network either fails to detect a rich club effect or inaccurately estimates its significance. In each case study, we explore the spatial layout and temporal variations revealed by the WTRC analysis, showcasing its value in studying spatiotemporal interaction networks. This research offers new insights into the study of spatiotemporal networks, with critical implications for applications such as transportation, redistricting, and epidemiology.

Spectral methods are widely used to estimate eigenvectors of a low-rank signal matrix subject to noise. These methods use the leading eigenspace of an observed matrix to estimate this low-rank signal. Typically, the entrywise estimation error of these methods depends on the coherence of the low-rank signal matrix with respect to the standard basis. In this work, we present a novel method for eigenvector estimation that avoids this dependence on coherence. Assuming a rank-one signal matrix, under mild technical conditions, the entrywise estimation error of our method provably has no dependence on the coherence under Gaussian noise (i.e., in the spiked Wigner model), and achieves the optimal estimation rate up to logarithmic factors. Simulations demonstrate that our method performs well under non-Gaussian noise and that an extension of our method to the case of a rank-$r$ signal matrix has little to no dependence on the coherence. In addition, we derive new metric entropy bounds for rank-$r$ singular subspaces under $\ell_{2,\infty}$ distance, which may be of independent interest. We use these new bounds to improve the best known lower bound for rank-$r$ eigenspace estimation under $\ell_{2,\infty}$ distance.

The linear-in-means model is widely used to study peer influence in social networks. We consider estimation in the linear-in-means model when a randomized treatment is applied to nodes in a network. We show that even when peer effects are identified, they may not be estimable at standard rates, due to near-perfect collinearity. We prove a minimax lower bound on estimation error and show that estimation becomes more difficult as networks grow denser. In sufficiently dense networks, consistent estimation of peer effects is impossible. To address this challenge, we investigate network-dependent treatment assignment. Using random dot product graphs, we show that treatments depending on network structure can prevent asymptotic collinearity when there is sufficient degree heterogeneity. However, such dependence is not a panacea, as different dependence structures must be individually evaluated for estimability. These results suggest caution when using the linear-in-means model to estimate peer effects and highlight the importance of explicitly modeling the relationship between treatments and network structure.

Recent advances in Bayesian models for random partitions have led to the formulation and exploration of Exchangeable Sequences of Clusters (ESC) models. Under ESC models, it is the cluster sizes that are exchangeable, rather than the observations themselves. This property is particularly useful for obtaining microclustering behavior, whereby cluster sizes grow sublinearly in the number of observations, as is common in applications such as record linkage, sparse networks and genomics. Unfortunately, the exchangeable clusters property comes at the cost of projectivity. As a consequence, in contrast to more traditional Dirichlet Process or Pitman-Yor process mixture models, samples a priori from ESC models cannot be easily obtained in a sequential fashion and instead require the use of rejection or importance sampling. In this work, drawing on connections between ESC models and discrete renewal theory, we obtain closed-form expressions for certain ESC models and develop faster methods for generating samples a priori from these models compared with the existing state of the art. In the process, we establish analytical expressions for the distribution of the number of clusters under ESC models, which was unknown prior to this work.

Latent space models play an important role in the modeling and analysis of network data. Under these models, each node has an associated latent point in some (typically low-dimensional) geometric space, and network formation is driven by this unobserved geometric structure. The random dot product graph (RDPG) and its generalization (GRDPG) are latent space models under which this latent geometry is taken to be Euclidean. These latent vectors can be efficiently and accurately estimated using well-studied spectral embeddings. In this paper, we develop a minimax lower bound for estimating the latent positions in the RDPG and the GRDPG models under the two-to-infinity norm, and show that a particular spectral embedding method achieves this lower bound. We also derive a minimax lower bound for the related task of subspace estimation under the two-to-infinity norm that holds in general for low-rank plus noise network models, of which the RDPG and GRDPG are special cases. The lower bounds are achieved by a novel construction based on Hadamard matrices.

Many statistical settings call for estimating a population parameter, most typically the population mean, based on a sample of matrices. The most natural estimate of the population mean is the arithmetic mean, but there are many other matrix means that may behave differently, especially in high dimensions. Here we consider the matrix harmonic mean as an alternative to the arithmetic matrix mean. We show that in certain high-dimensional regimes, the harmonic mean yields an improvement over the arithmetic mean in estimation error as measured by the operator norm. Counter-intuitively, studying the asymptotic behavior of these two matrix means in a spiked covariance estimation problem, we find that this improvement in operator norm error does not imply better recovery of the leading eigenvector. We also show that a Rao-Blackwellized version of the harmonic mean is equivalent to a linear shrinkage estimator studied previously in the high-dimensional covariance estimation literature, while applying a similar Rao-Blackwellization to regularized sample covariance matrices yields a novel nonlinear shrinkage estimator. Simulations complement the theoretical results, illustrating the conditions under which the harmonic matrix mean yields an empirically better estimate.

In increasingly many settings, data sets consist of multiple samples from a population of networks, with vertices aligned across these networks. For example, brain connectivity networks in neuroscience consist of measures of interaction between brain regions that have been aligned to a common template. We consider the setting where the observed networks have a shared expectation, but may differ in the noise structure on their edges. Our approach exploits the shared mean structure to denoise edge-level measurements of the observed networks and estimate the underlying population-level parameters. We also explore the extent to which edge-level errors influence estimation and downstream inference. We establish a finite-sample concentration inequality for the low-rank eigenvalue truncation of a random weighted adjacency matrix that may be of independent interest. The proposed approach is illustrated on synthetic networks and on data from an fMRI study of schizophrenia.

We consider the setting where many networks are observed on a common node set, and each observation comprises edge weights of a network, covariates observed at each node, and an overall response. The goal is to use the edge weights and node covariates to predict the response while identifying an interpretable set of predictive features. Our motivating application is neuroimaging, where edge weights encode functional connectivity measured between brain regions, node covariates encode task activations at each brain region, and the response is disease status or score on a behavioral task. We propose an approach that constructs feature groups based on assumed community structure (naturally occurring in neuroimaging applications). We propose two feature grouping schemes that incorporate both edge weights and node covariates, and we derive algorithms for optimization using an overlapping group LASSO penalty. Empirical results on synthetic data show that our method, relative to competing approaches, has similar or improved prediction error along with superior support recovery, enabling a more interpretable and potentially more accurate understanding of the underlying process. We also apply the method to neuroimaging data from the Human Connectome Project. Our approach is widely applicable in neuroimaging where interpretability is highly desired.

We present a low-constant approximation for the metric k-median problem on insertiononly streams using O(epsilon(-3)k log n) space. In particular, we present a streaming (O(epsilon(-3)k log n), 2+epsilon)-bicriterion solution that reports cluster weights. Running the offline approximation algorithm due to Byrka et al. (2015) on this bicriterion solution yields a (17.66 + epsilon)-approximation (Guha et al., 2003; Charikar et al., 2003; Braverman et al., 2011). Our result matches the best-known space requirements for streaming k-median clustering while significantly improving the approximation accuracy. We also provide a lower bound, showing that any polylog(n)-space streaming algorithm that maintains an (alpha, beta)-bicriterion must have beta >= 2. Our technique breaks the stream into segments defined by jumps in the optimal clustering cost, which increases monotonically as the stream progresses. By storing an accurate summary of recent segments of the stream and a lower-space summary of older segments, our algorithm maintains a (O(epsilon(-3)k log n), 2 + epsilon)-bicriterion solution for the entirety of the stream. In addition to our main result, we introduce a novel construction that we call a candidate set. This is a collection of points that, with high probability, contains k points that yield a near-optimal k-median cost. We present an algorithm called monotone faraway sampling (MFS) for constructing a candidate set in a single pass over a data stream. We show that using this candidate set in tandem with a coreset speeds up the search for a solution set of k cluster centers upon termination of the data stream. While coresets of smaller asymptotic size are known, comparative simplicity of MFS makes it appealing as a practical technique. (c) 2021 Elsevier B.V. All rights reserved.

We view ridge regression through the lens of eigenvalue shrinkage, and consider its influence on two modern problems in high-dimensional statistical inference: covariance estimation and community d...

Vertex nomination is a lightly-supervised network information retrieval task in which vertices of interest in one graph are used to query a second graph to discover vertices of interest in the second graph. Similar to other information retrieval tasks, the output of a vertex nomination scheme is a ranked list of the vertices in the second graph, with the heretofore unknown vertices of interest ideally concentrating at the top of the list. Vertex nomination schemes provide a useful suite of tools for efficiently mining complex networks for pertinent information. In this paper, we explore, both theoretically and practically, the dual roles of content (i.e., edge and vertex attributes) and context (i.e., network topology) in vertex nomination. We provide necessary and sufficient conditions under which vertex nomination schemes that leverage both content and context outperform schemes that leverage only content or context separately. While the joint utility of both content and context has been demonstrated empirically in the literature, the framework presented in this paper provides a novel theoretical basis for understanding the potential complementary roles of network features and topology.