arXiv subjects

Eric D. Kolaczyk

Publications and source records attributed to Eric D. Kolaczyk.

At least 37 records · Page 2Linked to original sources

arXiv2020

Sensor-based localization of epidemic sources on human mobility networks

We investigate the source detection problem in epidemiology, which is one of the most important issues for control of epidemics. Mathematically, we reformulate the problem as one of identifying the relevant component in a multivariate Gaussian mixture model. Focusing on the study of cholera and diseases with similar modes of transmission, we calibrate the parameters of our mixture model using human mobility networks within a stochastic, spatially explicit epidemiological model for waterborne disease. Furthermore, we adopt a Bayesian perspective, so that prior information on source location can be incorporated (e.g., reflecting the impact of local conditions). Posterior-based inference is performed, which permits estimates in the form of either individual locations or regions. Importantly, our estimator only requires first-arrival times of the epidemic by putative observers, typically located only at a small proportion of nodes. The proposed method is demonstrated within the context of the 2000-2002 cholera outbreak in the KwaZulu-Natal province of South Africa.

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arXiv2020

Estimation of the Epidemic Branching Factor in Noisy Contact Networks

Many fundamental concepts in network-based epidemic modeling depend on the branching factor, which captures a sense of dispersion in the network connectivity and quantifies the rate of spreading across the network. Moreover, contact network information generally is available only up to some level of error. We study the propagation of such errors to the estimation of the branching factor. Specifically, we characterize the impact of network noise on the bias and variance of the observed branching factor for arbitrary true networks, with examples in sparse, dense, homogeneous and inhomogeneous networks. In addition, we propose a method-of-moments estimator for the true branching factor. We illustrate the practical performance of our estimator through simulation studies and with contact networks observed in British secondary schools and a French hospital.

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arXiv2020

Estimation of subgraph density in noisy networks

While it is common practice in applied network analysis to report various standard network summary statistics, these numbers are rarely accompanied by uncertainty quantification. Yet any error inherent in the measurements underlying the construction of the network, or in the network construction procedure itself, necessarily must propagate to any summary statistics reported. Here we study the problem of estimating the density of an arbitrary subgraph, given a noisy version of some underlying network as data. Under a simple model of network error, we show that consistent estimation of such densities is impossible when the rates of error are unknown and only a single network is observed. Accordingly, we develop method-of-moment estimators of network subgraph densities and error rates for the case where a minimal number of network replicates are available. These estimators are shown to be asymptotically normal as the number of vertices increases to infinity. We also provide confidence intervals for quantifying the uncertainty in these estimates based on the asymptotic normality. To construct the confidence intervals, a new and non-standard bootstrap method is proposed to compute asymptotic variances, which is infeasible otherwise. We illustrate the proposed methods in the context of gene coexpression networks.

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arXiv2018

Approximation of the difference of two Poisson-like counts with Skellam

Poisson-like behavior for event count data is ubiquitous in nature. At the same time, differencing of such counts arises in the course of data processing in a variety of areas of application. As a result, the Skellam distribution -- defined as the distribution of the difference of two independent Poisson random variables -- is a natural candidate for approximating the difference of Poisson-like event counts. However, in many contexts strict independence, whether between counts or among events within counts, is not a tenable assumption. Here we characterize the accuracy in approximating the difference of Poisson-like counts by a Skellam random variable. Our results fully generalize existing, more limited results in this direction and, at the same time, our derivations are significantly more concise and elegant. We illustrate the potential impact of these results in the context of problems from network analysis and image processing, where various forms of weak dependence can be expected.

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arXiv2017

Dynamic Networks with Multi-scale Temporal Structure

We describe a novel method for modeling non-stationary multivariate time series, with time-varying conditional dependencies represented through dynamic networks. Our proposed approach combines traditional multi-scale modeling and network based neighborhood selection, aiming at capturing temporally local structure in the data while maintaining sparsity of the potential interactions. Our multi-scale framework is based on recursive dyadic partitioning, which recursively partitions the temporal axis into finer intervals and allows us to detect local network structural changes at varying temporal resolutions. The dynamic neighborhood selection is achieved through penalized likelihood estimation, where the penalty seeks to limit the number of neighbors used to model the data. We present theoretical and numerical results describing the performance of our method, which is motivated and illustrated using task-based magnetoencephalography (MEG) data in neuroscience.

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arXiv2017

Hypothesis Testing For Network Data in Functional Neuroimaging

In recent years, it has become common practice in neuroscience to use networks to summarize relational information in a set of measurements, typically assumed to be reflective of either functional or structural relationships between regions of interest in the brain. One of the most basic tasks of interest in the analysis of such data is the testing of hypotheses, in answer to questions such as "Is there a difference between the networks of these two groups of subjects?" In the classical setting, where the unit of interest is a scalar or a vector, such questions are answered through the use of familiar two-sample testing strategies. Networks, however, are not Euclidean objects, and hence classical methods do not directly apply. We address this challenge by drawing on concepts and techniques from geometry, and high-dimensional statistical inference. Our work is based on a precise geometric characterization of the space of graph Laplacian matrices and a nonparametric notion of averaging due to Fréchet. We motivate and illustrate our resulting methodologies for testing in the context of networks derived from functional neuroimaging data on human subjects from the 1000 Functional Connectomes Project. In particular, we show that this global test is more statistical powerful, than a mass-univariate approach. In addition, we have also provided a method for visualizing the individual contribution of each edge to the overall test statistic.

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arXiv2016

On the Propagation of Low-Rate Measurement Error to Subgraph Counts in Large Networks

Our work in this paper is inspired by a statistical observation that is both elementary and broadly relevant to network analysis in practice -- that the uncertainty in approximating some true network graph $G=(V,E)$ by some estimated graph $\hat{G}=(V,\hat{E})$ manifests as errors in the status of (non)edges that must necessarily propagate to any estimates of network summaries $η(G)$ we seek. Motivated by the common practice of using plug-in estimates $η(\hat{G})$ as proxies for $η(G)$, our focus is on the problem of characterizing the distribution of the discrepancy $D=η(\hat{G}) - η(G)$, in the case where $η(\cdot)$ is a subgraph count. Specifically, we study the fundamental case where the statistic of interest is $|E|$, the number of edges in $G$. Our primary contribution in this paper is to show that in the empirically relevant setting of large graphs with low-rate measurement errors, the distribution of $D_E=|\hat{E}| - |E|$ is well-characterized by a Skellam distribution, when the errors are independent or weakly dependent. Under an assumption of independent errors, we are able to further show conditions under which this characterization is strictly better than that of an appropriate normal distribution. These results derive from our formulation of a general result, quantifying the accuracy with which the difference of two sums of dependent Bernoulli random variables may be approximated by the difference of two independent Poisson random variables, i.e., by a Skellam distribution. This general result is developed through the use of Stein's method, and may be of some general interest. We finish with a discussion of possible extension of our work to subgraph counts $η(G)$ of higher order.

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arXiv2016

Detection of multiple perturbations in multi-omics biological networks

Cellular mechanism-of-action is of fundamental concern in many biological studies. It is of particular interest for identifying the cause of disease and learning the way in which treatments act against disease. However, pinpointing such mechanisms is difficult, due to the fact that small perturbations to the cell can have wide-ranging downstream effects. Given a snapshot of cellular activity, it can be challenging to tell where a disturbance originated. The presence of an ever-greater variety of high-throughput biological data offers an opportunity to examine cellular behavior from multiple angles, but also presents the statistical challenge of how to effectively analyze data from multiple sources. In this setting, we propose a method for mechanism-of-action inference by extending network filtering to multi-attribute data. We first estimate a joint Gaussian graphical model across multiple data types using penalized regression and filter for network effects. We then apply a set of likelihood ratio tests to identify the most likely site of the original perturbation. In addition, we propose a conditional testing procedure to allow for detection of multiple perturbations. We demonstrate this methodology on paired gene expression and methylation data from The Cancer Genome Atlas (TCGA).

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arXiv2015

On the Question of Effective Sample Size in Network Modeling: An Asymptotic Inquiry

The modeling and analysis of networks and network data has seen an explosion of interest in recent years and represents an exciting direction for potential growth in statistics. Despite the already substantial amount of work done in this area to date by researchers from various disciplines, however, there remain many questions of a decidedly foundational nature - natural analogues of standard questions already posed and addressed in more classical areas of statistics - that have yet to even be posed, much less addressed. Here we raise and consider one such question in connection with network modeling. Specifically, we ask, "Given an observed network, what is the sample size?" Using simple, illustrative examples from the class of exponential random graph models, we show that the answer to this question can very much depend on basic properties of the networks expected under the model, as the number of vertices $n_V$ in the network grows. In particular, adopting the (asymptotic) scaling of the variance of the maximum likelihood parameter estimates as a notion of effective sample size ($n_{\mathrm{eff}}$), we show that when modeling the overall propensity to have ties and the propensity to reciprocate ties, whether the networks are sparse or not under the model (i.e., having a constant or an increasing number of ties per vertex, respectively) is sufficient to yield an order of magnitude difference in $n_{\mathrm{eff}}$, from $O(n_V)$ to $O(n^2_V)$. In addition, we report simulation study results that suggest similar properties for models for triadic (friend-of-a-friend) effects. We then explore some practical implications of this result, using both simulation and data on food-sharing from Lamalera, Indonesia.

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arXiv2015

Estimating network degree distributions under sampling: An inverse problem, with applications to monitoring social media networks

Networks are a popular tool for representing elements in a system and their interconnectedness. Many observed networks can be viewed as only samples of some true underlying network. Such is frequently the case, for example, in the monitoring and study of massive, online social networks. We study the problem of how to estimate the degree distribution - an object of fundamental interest - of a true underlying network from its sampled network. In particular, we show that this problem can be formulated as an inverse problem. Playing a key role in this formulation is a matrix relating the expectation of our sampled degree distribution to the true underlying degree distribution. Under many network sampling designs, this matrix can be defined entirely in terms of the design and is found to be ill-conditioned. As a result, our inverse problem frequently is ill-posed. Accordingly, we offer a constrained, penalized weighted least-squares approach to solving this problem. A Monte Carlo variant of Stein's unbiased risk estimation (SURE) is used to select the penalization parameter. We explore the behavior of our resulting estimator of network degree distribution in simulation, using a variety of combinations of network models and sampling regimes. In addition, we demonstrate the ability of our method to accurately reconstruct the degree distributions of various sub-communities within online social networks corresponding to Friendster, Orkut and LiveJournal. Overall, our results show that the true degree distributions from both homogeneous and inhomogeneous networks can be recovered with substantially greater accuracy than reflected in the empirical degree distribution resulting from the original sampling.

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arXiv2014

Perturbation Detection Through Modeling of Gene Expression on a Latent Biological Pathway Network: A Bayesian hierarchical approach

Cellular response to a perturbation is the result of a dynamic system of biological variables linked in a complex network. A major challenge in drug and disease studies is identifying the key factors of a biological network that are essential in determining the cell's fate. Here our goal is the identification of perturbed pathways from high-throughput gene expression data. We develop a three-level hierarchical model, where (i) the first level captures the relationship between gene expression and biological pathways using confirmatory factor analysis, (ii) the second level models the behavior within an underlying network of pathways induced by an unknown perturbation using a conditional autoregressive model, and (iii) the third level is a spike-and-slab prior on the perturbations. We then identify perturbations through posterior-based variable selection. We illustrate our approach using gene transcription drug perturbation profiles from the DREAM7 drug sensitivity predication challenge data set. Our proposed method identified regulatory pathways that are known to play a causative role and that were not readily resolved using gene set enrichment analysis or exploratory factor models. Simulation results are presented assessing the performance of this model relative to a network-free variant and its robustness to inaccuracies in biological databases.

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arXiv2014

Percolation under Noise: Detecting Explosive Percolation Using the Second Largest Component

We consider the problem of distinguishing classical (Erdős-Rényi) percolation from explosive (Achlioptas) percolation, under noise. A statistical model of percolation is constructed allowing for the birth and death of edges as well as the presence of noise in the observations. This graph-valued stochastic process is composed of a latent and an observed non-stationary process, where the observed graph process is corrupted by Type I and Type II errors. This produces a hidden Markov graph model. We show that for certain choices of parameters controlling the noise, the classical (ER) percolation is visually indistinguishable from the explosive (Achlioptas) percolation model. In this setting, we compare two different criteria for discriminating between these two percolation models, based on a quantile difference (QD) of the first component's size and on the maximal size of the second largest component. We show through data simulations that this second criterion outperforms the QD of the first component's size, in terms of discriminatory power. The maximal size of the second component therefore provides a useful statistic for distinguishing between the ER and Achlioptas models of percolation, under physically motivated conditions for the birth and death of edges, and under noise. The potential application of the proposed criteria for percolation detection in clinical neuroscience is also discussed.

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arXiv2013

Exponential-type Inequalities Involving Ratios of the Modified Bessel Function of the First Kind and their Applications

The modified Bessel function of the first kind, $I_ν(x)$, arises in numerous areas of study, such as physics, signal processing, probability, statistics, etc. As such, there has been much interest in recent years in deducing properties of functionals involving $I_ν(x)$, in particular, of the ratio ${I_{ν+1}(x)}/{I_ν(x)}$, when $ν,x\geq 0$. In this paper we establish sharp upper and lower bounds on $H(ν,x)=\sum_{k=1}^{\infty} {I_{ν+k}(x)}/{I_ν(x)}$ for $ν,x\geq 0$ that appears as the complementary cumulative hazard function for a Skellam$(λ,λ)$ probability distribution in the statistical analysis of networks. Our technique relies on bounding existing estimates of ${I_{ν+1}(x)}/{I_ν(x)}$ from above and below by quantities with nicer algebraic properties, namely exponentials, to better evaluate the sum, while optimizing their rates in the regime when $ν+1\leq x$ in order to maintain their precision. We demonstrate the relevance of our results through applications, providing an improvement for the well-known asymptotic $\exp(-x)I_ν(x)\sim {1}/{\sqrt{2πx}}$ as $x\rightarrow \infty$, upper and lower bounding $\mathbb{P}\left[W=ν\right]$ for $W\sim Skellam(λ_1,λ_2)$, and deriving a novel concentration inequality on the $Skellam(λ,λ)$ probability distribution from above and below.

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arXiv2012

Weighted Frechet Means as Convex Combinations in Metric Spaces: Properties and Generalized Median Inequalities

In this short note, we study the properties of the weighted Frechet mean as a convex combination operator on an arbitrary metric space, (Y,d). We show that this binary operator is commutative, non-associative, idempotent, invariant to multiplication by a constant weight and possesses an identity element. We also treat the properties of the weighted cumulative Frechet mean. These tools allow us to derive several types of median inequalities for abstract metric spaces that hold for both negative and positive Alexandrov spaces. In particular, we show through an example that these bounds cannot be improved upon in general metric spaces. For weighted Frechet means, however, such inequalities can solely be derived for weights equal or greater than one. This latter limitation highlights the inherent difficulties associated with working with abstract-valued random variables.

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arXiv2012

Inference and Characterization of Multi-Attribute Networks with Application to Computational Biology

Our work is motivated by and illustrated with application of association networks in computational biology, specifically in the context of gene/protein regulatory networks. Association networks represent systems of interacting elements, where a link between two different elements indicates a sufficient level of similarity between element attributes. While in reality relational ties between elements can be expected to be based on similarity across multiple attributes, the vast majority of work to date on association networks involves ties defined with respect to only a single attribute. We propose an approach for the inference of multi-attribute association networks from measurements on continuous attribute variables, using canonical correlation and a hypothesis-testing strategy. Within this context, we then study the impact of partial information on multi-attribute network inference and characterization, when only a subset of attributes is available. We consider in detail the case of two attributes, wherein we examine through a combination of analytical and numerical techniques the implications of the choice and number of node attributes on the ability to detect network links and, more generally, to estimate higher-level network summary statistics, such as node degree, clustering coefficients, and measures of centrality. Illustration and applications throughout the paper are developed using gene and protein expression measurements on human cancer cell lines from the NCI-60 database.

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arXiv2012

A Compressed PCA Subspace Method for Anomaly Detection in High-Dimensional Data

Random projection is widely used as a method of dimension reduction. In recent years, its combination with standard techniques of regression and classification has been explored. Here we examine its use with principal component analysis (PCA) and subspace detection methods. Specifically, we show that, under appropriate conditions, with high probability the magnitude of the residuals of a PCA analysis of randomly projected data behaves comparably to that of the residuals of a similar PCA analysis of the original data. Our results indicate the feasibility of applying subspace-based anomaly detection algorithms to randomly projected data, when the data are high-dimensional but have a covariance of an appropriately compressed nature. We illustrate in the context of computer network traffic anomaly detection.

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arXiv2010

Target Detection via Network Filtering

A method of `network filtering' has been proposed recently to detect the effects of certain external perturbations on the interacting members in a network. However, with large networks, the goal of detection seems a priori difficult to achieve, especially since the number of observations available often is much smaller than the number of variables describing the effects of the underlying network. Under the assumption that the network possesses a certain sparsity property, we provide a formal characterization of the accuracy with which the external effects can be detected, using a network filtering system that combines Lasso regression in a sparse simultaneous equation model with simple residual analysis. We explore the implications of the technical conditions underlying our characterization, in the context of various network topologies, and we illustrate our method using simulated data.

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arXiv2009

Network inference - with confidence - from multivariate time series

Networks - collections of interacting elements or nodes - abound in the natural and manmade worlds. For many networks, complex spatiotemporal dynamics stem from patterns of physical interactions unknown to us. To infer these interactions, it is common to include edges between those nodes whose time series exhibit sufficient functional connectivity, typically defined as a measure of coupling exceeding a pre-determined threshold. However, when uncertainty exists in the original network measurements, uncertainty in the inferred network is likely, and hence a statistical propagation-of-error is needed. In this manuscript, we describe a principled and systematic procedure for the inference of functional connectivity networks from multivariate time series data. Our procedure yields as output both the inferred network and a quantification of uncertainty of the most fundamental interest: uncertainty in the number of edges. To illustrate this approach, we apply our procedure to simulated data and electrocorticogram data recorded from a human subject during an epileptic seizure. We demonstrate that the procedure is accurate and robust in both the determination of edges and the reporting of uncertainty associated with that determination.

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