arXiv subjects

Eric D. Kolaczyk

Publications and source records attributed to Eric D. Kolaczyk.

At least 19 recordsLinked to original sources

arXiv2026

Inference for subgraph densities in noisy dynamic networks

In this work we develop statistical methodology to estimate and perform inference on subgraph densities using time-indexed, or dynamic network sequences. These estimates explicitly adjust for observation errors for the network edges, and have good theoretical properties as the size of the network grows. By specifying a stochastically evolving hidden Markov network model, we address two important directions for further investigation identified by Chang et al. (2022): robustness to non-identical network replicates, and efficient aggregation of multiple available network snapshots. These new methods vastly expand the analysis of noisy networks to new data settings, as network replicates are commonly observed dynamically. The methodology is also extended to consider joint inference for subgraph densities at multiple time points, to facilitate formal statistical comparison of dynamic network snapshots.

stat.ME↗

arXiv2026

Hierarchical Clustering of Networks via Hierarchical Distance Matrices

Clustering populations of networks while recovering their latent hierarchical organization is a fundamental yet largely unexplored problem in network analysis. To formalize this, we introduce the Hierarchical Distance Matrix, a specific class of population-level distance matrices that encodes latent hierarchical organization through recursively nested distance separation, accommodating unbalanced tree depths. Building on this framework, we propose a fully data-driven top-down procedure: network hierarchical clustering based on two-sample testing (NHC-TST). The algorithm recursively splits networks via spectral clustering and uses a graph-based two-sample stopping rule. The procedure adaptively determines the branching structure without requiring prior knowledge of the number of clusters or tree depth. Theoretically, we establish exact recovery of the population-level hierarchical structure and statistical consistency in the empirical procedure. Simulation studies demonstrate highly accurate recovery of both cluster memberships and hierarchical relationships across a wide range of settings. Applied to a global migration dataset, NHC-TST uncovers interpretable multi-resolution temporal structures that are not revealed by conventional flat clustering approaches.

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arXiv2026

Isotonic Conformal Prediction

A point prediction that is well calibrated on average can still be systematically biased conditional on its own value, undermining its use in downstream decision-making. We consider two objectives for reliable uncertainty quantification: self-calibration, requiring a point prediction to be unbiased conditional on its own value, and prediction-conditional validity, requiring a prediction interval to attain nominal coverage conditional on the prediction. Self-Calibrating Conformal Prediction (SC-CP) attains both objectives exactly in finite samples, but requires refitting its calibrator for every candidate outcome, which is computationally prohibitive for continuous outcomes. We propose Isotonic Conformal Prediction (ICP), a framework that decouples calibration from prediction-set construction by fitting a single isotonic recalibration map and constructing prediction intervals within strata of similar recalibrated predictions. Within this framework we develop two procedures. Split Isotonic Conformal Prediction (SICP) attains prediction-conditional validity in finite samples and self-calibration asymptotically, at the computational cost of split conformal prediction. Transductive Isotonic Conformal Prediction (TICP) attains both objectives exactly in finite samples through a per-test-point inner loop that avoids refitting the isotonic calibrator. On synthetic heteroscedastic regression problems and a real-world healthcare-utilization dataset, both procedures match the coverage of SC-CP at substantially lower computational cost.

stat.ML↗

arXiv2026

Consistent Identification of Top-$K$ Nodes in Noisy Networks

Identifying the most influential nodes in a network, typically using centrality measures, is a central task in applied network analysis. However, real-world networks are often constructed from noisy or incomplete data, which can distort rankings and lead to errors in identifying the true top-$k$ nodes. In this paper, we study how network noise affects the recovery of the true top-$k$ node set based on degree centrality. Specifically, we consider a noisy network observation in which edges are randomly added or removed according to a probabilistic noise model, and analyze the resulting empirical top-$k$ set. We show that top-$k$ recovery under network noise is governed by the relationship between the degree gap and the noise magnitude, which separates recoverable and unrecoverable regimes. To quantify ranking stability, we derive upper and lower bounds on the expected discrepancy between the empirical and true top-$k$ sets in a general framework and for specific network models. We also extend the analysis to eigenvector centrality, showing that similar noise-gap tradeoffs arise in spectral rankings. Simulation studies support our theoretical findings and illustrate the practical impact of network noise across a range of settings.

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arXiv2025

High-Dimensional Privacy-Utility Dynamics of Noisy Stochastic Gradient Descent on Least Squares

The interplay between optimization and privacy has become a central theme in privacy-preserving machine learning. Noisy stochastic gradient descent (SGD) has emerged as a cornerstone algorithm, particularly in large-scale settings. These variants of gradient methods inject carefully calibrated noise into each update to achieve differential privacy, the gold standard notion of rigorous privacy guarantees. Prior work primarily provides various bounds on statistical risk and privacy loss for noisy SGD, yet the \textit{exact} behavior of the process remains unclear, particularly in high-dimensional settings. This work leverages a diffusion approach to analyze noisy SGD precisely, providing a continuous-time perspective that captures both statistical risk evolution and privacy loss dynamics in high dimensions. Moreover, we study a variant of noisy SGD that does not require explicit knowledge of gradient sensitivity, unlike existing work that assumes or enforces sensitivity through gradient clipping. Specifically, we focus on the least squares problem with $\ell_2$ regularization.

cs.LG↗

arXiv2025

Minority representation and fairness in network ranking: An application to school contact diary data

Considerations of bias, fairness and representation are a prerequisite of responsible modern statistics. In statistical network analysis, observed networks are often incomplete or systematically biased, which can lead to systematic underrepresentation of protected groups, and affect any downstream ranking or decision based on the observed network. In this paper, we study a high school contact network constructed from self-reported contact diaries and introduce a formal measure of minority representation, defined as the proportion of minority nodes among the top-ranked individuals. We model systematic bias through group-dependent missing edge mechanisms and develop statistical methods to estimate and test for such bias. When bias is detected, we propose a re-ranking procedure based on an asymptotic approximation that improves group representation. Applying the framework to the high school contact network reveals systematic underreporting of cross-group contacts consistent with recall bias. These findings highlight the importance of modeling and correcting systematic bias in social networks with heterogeneous groups.

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arXiv2024

Stochastic gradient descent-based inference for dynamic network models with attractors

In Coevolving Latent Space Networks with Attractors (CLSNA) models, nodes in a latent space represent social actors, and edges indicate their dynamic interactions. Attractors are added at the latent level to capture the notion of attractive and repulsive forces between nodes, borrowing from dynamical systems theory. However, CLSNA reliance on MCMC estimation makes scaling difficult, and the requirement for nodes to be present throughout the study period limit practical applications. We address these issues by (i) introducing a Stochastic gradient descent (SGD) parameter estimation method, (ii) developing a novel approach for uncertainty quantification using SGD, and (iii) extending the model to allow nodes to join and leave over time. Simulation results show that our extensions result in little loss of accuracy compared to MCMC, but can scale to much larger networks. We apply our approach to the longitudinal social networks of members of US Congress on the social media platform X. Accounting for node dynamics overcomes selection bias in the network and uncovers uniquely and increasingly repulsive forces within the Republican Party.

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arXiv2024

Differentially Private Linear Regression with Linked Data

There has been increasing demand for establishing privacy-preserving methodologies for modern statistics and machine learning. Differential privacy, a mathematical notion from computer science, is a rising tool offering robust privacy guarantees. Recent work focuses primarily on developing differentially private versions of individual statistical and machine learning tasks, with nontrivial upstream pre-processing typically not incorporated. An important example is when record linkage is done prior to downstream modeling. Record linkage refers to the statistical task of linking two or more data sets of the same group of entities without a unique identifier. This probabilistic procedure brings additional uncertainty to the subsequent task. In this paper, we present two differentially private algorithms for linear regression with linked data. In particular, we propose a noisy gradient method and a sufficient statistics perturbation approach for the estimation of regression coefficients. We investigate the privacy-accuracy tradeoff by providing finite-sample error bounds for the estimators, which allows us to understand the relative contributions of linkage error, estimation error, and the cost of privacy. The variances of the estimators are also discussed. We demonstrate the performance of the proposed algorithms through simulations and an application to synthetic data.

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arXiv2024

Autoregressive networks with dependent edges

We propose an autoregressive framework for modelling dynamic networks with dependent edges. It encompasses models that accommodate, for example, transitivity, degree heterogenenity, and other stylized features often observed in real network data. By assuming the edges of networks at each time are independent conditionally on their lagged values, the models, which exhibit a close connection with temporal ERGMs, facilitate both simulation and the maximum likelihood estimation in a straightforward manner. Due to the possibly large number of parameters in the models, the natural MLEs may suffer from slow convergence rates. An improved estimator for each component parameter is proposed based on an iteration employing projection, which mitigates the impact of the other parameters (Chang et al., 2021; Chang et al., 2023). Leveraging a martingale difference structure, the asymptotic distribution of the improved estimator is derived without the assumption of stationarity. The limiting distribution is not normal in general, although it reduces to normal when the underlying process satisfies some mixing conditions. Illustration with a transitivity model was carried out in both simulation and a real network data set.

math.ST↗

arXiv2024

Differentially Private Confidence Intervals for Proportions under Stratified Random Sampling

Confidence intervals are a fundamental tool for quantifying the uncertainty of parameters of interest. With the increase of data privacy awareness, developing a private version of confidence intervals has gained growing attention from both statisticians and computer scientists. Differential privacy is a state-of-the-art framework for analyzing privacy loss when releasing statistics computed from sensitive data. Recent work has been done around differentially private confidence intervals, yet to the best of our knowledge, rigorous methodologies on differentially private confidence intervals in the context of survey sampling have not been studied. In this paper, we propose three differentially private algorithms for constructing confidence intervals for proportions under stratified random sampling. We articulate two variants of differential privacy that make sense for data from stratified sampling designs, analyzing each of our algorithms within one of these two variants. We establish analytical privacy guarantees and asymptotic properties of the estimators. In addition, we conduct simulation studies to evaluate the proposed private confidence intervals, and two applications to the 1940 Census data are provided.

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arXiv2024

Edge differentially private estimation in the $β$-model via jittering and method of moments

A standing challenge in data privacy is the trade-off between the level of privacy and the efficiency of statistical inference. Here we conduct an in-depth study of this trade-off for parameter estimation in the $β$-model (Chatterjee, Diaconis and Sly, 2011) for edge differentially private network data released via jittering (Karwa, Krivitsky and Slavković, 2017). Unlike most previous approaches based on maximum likelihood estimation for this network model, we proceed via method-of-moments. This choice facilitates our exploration of a substantially broader range of privacy levels - corresponding to stricter privacy - than has been to date. Over this new range we discover our proposed estimator for the parameters exhibits an interesting phase transition, with both its convergence rate and asymptotic variance following one of three different regimes of behavior depending on the level of privacy. Because identification of the operable regime is difficult if not impossible in practice, we devise a novel adaptive bootstrap procedure to construct uniform inference across different phases. In fact, leveraging this bootstrap we are able to provide for simultaneous inference of all parameters in the $β$-model (i.e., equal to the number of nodes), which, to our best knowledge, is the first result of its kind. Numerical experiments confirm the competitive and reliable finite sample performance of the proposed inference methods, next to a comparable maximum likelihood method, as well as significant advantages in terms of computational speed and memory.

math.ST↗

arXiv2022

Causal Inference under Network Interference with Noise

Increasingly, there is a marked interest in estimating causal effects under network interference due to the fact that interference manifests naturally in networked experiments. However, network information generally is available only up to some level of error. We study the propagation of such errors to estimators of average causal effects under network interference. Specifically, assuming a four-level exposure model and Bernoulli random assignment of treatment, we characterize the impact of network noise on the bias and variance of standard estimators in homogeneous and inhomogeneous networks. In addition, we propose method-of-moments estimators for bias reduction where a minimal number of network replicates are available. We show our estimators are asymptotically normal and provide confidence intervals for quantifying the uncertainty in these estimates. We illustrate the practical performance of our estimators through simulation studies in British secondary school contact networks.

stat.ME↗

arXiv2022

Disentangling positive and negative partisanship in social media interactions using a coevolving latent space network with attractors model

We develop a broadly applicable class of coevolving latent space network with attractors (CLSNA) models, where nodes represent individual social actors assumed to lie in an unknown latent space, edges represent the presence of a specified interaction between actors, and attractors are added in the latent level to capture the notion of attractive and repulsive forces. We apply the CLSNA models to understand the dynamics of partisan polarization on social media, where we expect Republicans and Democrats to increasingly interact with their own party and disengage with the opposing party. Using longitudinal social networks from the social media platforms Twitter and Reddit, we investigate the relative contributions of positive (attractive) and negative (repulsive) forces among political elites and the public, respectively. Our goals are to disentangle the positive and negative forces within and between parties and explore if and how they change over time. Our analysis confirms the existence of partisan polarization in social media interactions among both political elites and the public. Moreover, while positive partisanship is the driving force of interactions across the full periods of study for both the public and Democratic elites, negative partisanship has come to dominate Republican elites' interactions since the run-up to the 2016 presidential election.

stat.AP↗

arXiv2022

Quantifying Uncertainty for Temporal Motif Estimation in Graph Streams under Sampling

Dynamic networks, a.k.a. graph streams, consist of a set of vertices and a collection of timestamped interaction events (i.e., temporal edges) between vertices. Temporal motifs are defined as classes of (small) isomorphic induced subgraphs on graph streams, considering both edge ordering and duration. As with motifs in static networks, temporal motifs are the fundamental building blocks for temporal structures in dynamic networks. Several methods have been designed to count the occurrences of temporal motifs in graph streams, with recent work focusing on estimating the count under various sampling schemes along with concentration properties. However, little attention has been given to the problem of uncertainty quantification and the asymptotic statistical properties for such count estimators. In this work, we establish the consistency and the asymptotic normality of a certain Horvitz-Thompson type of estimator in an edge sampling framework for deterministic graph streams, which can be used to construct confidence intervals and conduct hypothesis testing for the temporal motif count under sampling. We also establish similar results under an analogous stochastic model. Our results are relevant to a wide range of applications in social, communication, biological, and brain networks, for tasks involving pattern discovery.

stat.ME↗

arXiv2021

Network Recovery from Unlabeled Noisy Samples

There is a growing literature on the statistical analysis of multiple networks in which the network is the fundamental data object. However, most of this work requires networks on a shared set of labeled vertices. In this work, we consider the question of recovering a parent network based on noisy unlabeled samples. We identify a specific regime in the noisy network literature for recovery that is asymptotically unbiased and computationally tractable based on a three-stage recovery procedure: first, we align the networks via a sequential pairwise graph matching procedure; next, we compute the sample average of the aligned networks; finally, we obtain an estimate of the parent by thresholding the sample average. Previous work on multiple unlabeled networks is only possible for trivial networks due to the complexity of brute-force computations.

stat.ME↗

arXiv2021

Inferring the Type of Phase Transitions Undergone in Epileptic Seizures Using Random Graph Hidden Markov Models for Percolation in Noisy Dynamic Networks

In clinical neuroscience, epileptic seizures have been associated with the sudden emergence of coupled activity across the brain. The resulting functional networks - in which edges indicate strong enough coupling between brain regions - are consistent with the notion of percolation, which is a phenomenon in complex networks corresponding to the sudden emergence of a giant connected component. Traditionally, work has concentrated on noise-free percolation with a monotonic process of network growth, but real-world networks are more complex. We develop a class of random graph hidden Markov models (RG-HMMs) for characterizing percolation regimes in noisy, dynamically evolving networks in the presence of edge birth and edge death, as well as noise. This class is used to understand the type of phase transitions undergone in a seizure, and in particular, distinguishing between different percolation regimes in epileptic seizures. We develop a hypothesis testing framework for inferring putative percolation mechanisms. As a necessary precursor, we present an EM algorithm for estimating parameters from a sequence of noisy networks only observed at a longitudinal subsampling of time points. Our results suggest that different types of percolation can occur in human seizures. The type inferred may suggest tailored treatment strategies and provide new insights into the fundamental science of epilepsy.

stat.AP↗

arXiv2021

Bayesian classification, anomaly detection, and survival analysis using network inputs with application to the microbiome

While the study of a single network is well-established, technological advances now allow for the collection of multiple networks with relative ease. Increasingly, anywhere from several to thousands of networks can be created from brain imaging, gene co-expression data, or microbiome measurements. And these networks, in turn, are being looked to as potentially powerful features to be used in modeling. However, with networks being non-Euclidean in nature, how best to incorporate them into standard modeling tasks is not obvious. In this paper, we propose a Bayesian modeling framework that provides a unified approach to binary classification, anomaly detection, and survival analysis with network inputs. We encode the networks in the kernel of a Gaussian process prior via their pairwise differences and we discuss several choices of provably positive definite kernel that can be plugged into our models. Although our methods are widely applicable, we are motivated here in particular by microbiome research (where network analysis is emerging as the standard approach for capturing the interconnectedness of microbial taxa across both time and space) and its potential for reducing preterm delivery and improving personalization of prenatal care.

stat.AP↗

arXiv2020

A spectral-based framework for hypothesis testing in populations of networks

In this paper, we propose a new spectral-based approach to hypothesis testing for populations of networks. The primary goal is to develop a test to determine whether two given samples of networks come from the same random model or distribution. Our test statistic is based on the trace of the third order for a centered and scaled adjacency matrix, which we prove converges to the standard normal distribution as the number of nodes tends to infinity. The asymptotic power guarantee of the test is also provided. The proper interplay between the number of networks and the number of nodes for each network is explored in characterizing the theoretical properties of the proposed testing statistics. Our tests are applicable to both binary and weighted networks, operate under a very general framework where the networks are allowed to be large and sparse, and can be extended to multiple-sample testing. We provide an extensive simulation study to demonstrate the superior performance of our test over existing methods and apply our test to three real datasets.

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