arXiv subjects

Daniel L. Sussman

Publications and source records attributed to Daniel L. Sussman.

At least 19 recordsLinked to original sources

arXiv2025

Attractor-Based Coevolving Dot Product Random Graph Model

We introduce the attractor-based coevolving dot product random graph model (ABCDPRGM) to analyze time-series network data manifesting polarizing or flocking behavior. Graphs are generated based on latent positions under the random dot product graph regime. We assign group membership to each node. When evolving through time, the latent position of each node will change based on its current position and two attractors, which are defined to be the centers of the latent positions of all of its neighbors who share its group membership or who have different group membership than it. Parameters are assigned to the attractors to quantify the amount of influence that the attractors have on the trajectory of the latent position of each node. We developed estimators for the parameters, demonstrated their consistency, and established convergence rates under specific assumptions. Through the ABCDPRGM, we provided a novel framework for quantifying and understanding the underlying forces influencing the polarizing or flocking behaviors in dynamic network data.

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arXiv2024

Gotta match 'em all: Solution diversification in graph matching matched filters

We present a novel approach for finding multiple noisily embedded template graphs in a very large background graph. Our method builds upon the graph-matching-matched-filter technique proposed in Sussman et al., with the discovery of multiple diverse matchings being achieved by iteratively penalizing a suitable node-pair similarity matrix in the matched filter algorithm. In addition, we propose algorithmic speed-ups that greatly enhance the scalability of our matched-filter approach. We present theoretical justification of our methodology in the setting of correlated Erdos-Renyi graphs, showing its ability to sequentially discover multiple templates under mild model conditions. We additionally demonstrate our method's utility via extensive experiments both using simulated models and real-world dataset, include human brain connectomes and a large transactional knowledge base.

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arXiv2022

Unbiased estimation for additive exposure models

Causal inference methods have been applied in various fields where researchers want to estimate treatment effects. In traditional causal inference settings, one assumes that the outcome of a unit does not depend on treatments of other units. However, as causal inference methods are extended to more applications, there is a greater need for estimators of general causal effects. We use an exposure mapping framework [Aronow and Samii, 2017] to map the relationship between the treatment allocation and the potential outcomes. Under the exposure model, we propose linear unbiased estimators (LUEs) for general causal effects under the assumption that treatment effects are additive. Additivity provides statistical advantages, where contrasts in exposures are now equivalent, and so the set of estimators considered grows. We identify a subset of LUEs that forms an affine basis for LUEs, and we characterize optimal LUEs with minimum integrated variance through defining conditions on the support of the estimator. We show, through simulations that our proposed estimators are fairly robust to violations of the additivity assumption, and in general, there is benefit in leveraging information from all exposures.

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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.

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arXiv2021

Bias-Variance Tradeoffs in Joint Spectral Embeddings

Joint spectral embeddings facilitate analysis of multiple network data by simultaneously mapping vertices in each network to points in Euclidean space where statistical inference is then performed. In this work, we consider one such joint embedding technique, the omnibus embedding of arXiv:1705.09355 , which has been successfully used for community detection, anomaly detection, and hypothesis testing tasks. To date the theoretical properties of this method have only been established under the strong assumption that the networks are conditionally i.i.d. random dot product graphs. Herein, we take a first step in characterizing the theoretical properties of the omnibus embedding in the presence of heterogeneous network data. Under a latent position model, we show the omnibus embedding implicitly regularizes its latent position estimates which induces a finite-sample bias-variance tradeoff for latent position estimation. We establish an explicit bias expression, derive a uniform concentration bound on the residual, and prove a central limit theorem characterizing the distributional properties of these estimates. These explicit bias and variance expressions enable us to state sufficient conditions for exact recovery in community detection tasks and develop a pivotal test statistic to determine whether two graphs share the same set of latent positions; demonstrating that accurate inference is achievable despite the estimator's inconsistency. These results are demonstrated in several experimental settings where statistical procedures utilizing the omnibus embedding are competitive, and oftentimes preferable, to comparable embedding techniques. These observations accentuate the viability of the omnibus embedding for multiple graph inference beyond the homogeneous network setting.

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arXiv2021

Multiplex graph matching matched filters

We consider the problem of detecting a noisy induced multiplex template network in a larger multiplex background network. Our approach, which extends the framework of Sussman et al. (2019) to the multiplex setting, leverages a multiplex analogue of the classical graph matching problem to use the template as a matched filter for efficiently searching the background for candidate template matches. The effectiveness of our approach is demonstrated both theoretically and empirically, with particular attention paid to the potential benefits of considering multiple channels.

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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

Maximum Likelihood Estimation and Graph Matching in Errorfully Observed Networks

Given a pair of graphs with the same number of vertices, the inexact graph matching problem consists in finding a correspondence between the vertices of these graphs that minimizes the total number of induced edge disagreements. We study this problem from a statistical framework in which one of the graphs is an errorfully observed copy of the other. We introduce a corrupting channel model, and show that in this model framework, the solution to the graph matching problem is a maximum likelihood estimator. Necessary and sufficient conditions for consistency of this MLE are presented, as well as a relaxed notion of consistency in which a negligible fraction of the vertices need not be matched correctly. The results are used to study matchability in several families of random graphs, including edge independent models, random regular graphs and small-world networks. We also use these results to introduce measures of matching feasibility, and experimentally validate the results on simulated and real-world networks.

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arXiv2019

Matched Filters for Noisy Induced Subgraph Detection

The problem of finding the vertex correspondence between two noisy graphs with different number of vertices where the smaller graph is still large has many applications in social networks, neuroscience, and computer vision. We propose a solution to this problem via a graph matching matched filter: centering and padding the smaller adjacency matrix and applying graph matching methods to align it to the larger network. The centering and padding schemes can be incorporated into any algorithm that matches using adjacency matrices. Under a statistical model for correlated pairs of graphs, which yields a noisy copy of the small graph within the larger graph, the resulting optimization problem can be guaranteed to recover the true vertex correspondence between the networks. However, there are currently no efficient algorithms for solving this problem. To illustrate the possibilities and challenges of such problems, we use an algorithm that can exploit a partially known correspondence and show via varied simulations and applications to {\it Drosophila} and human connectomes that this approach can achieve good performance.

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arXiv2019

Matchability of heterogeneous networks pairs

We consider the problem of graph matchability in non-identically distributed networks. In a general class of edge-independent networks, we demonstrate that graph matchability can be lost with high probability when matching the networks directly. We further demonstrate that under mild model assumptions, matchability is almost perfectly recovered by centering the networks using Universal Singular Value Thresholding before matching. These theoretical results are then demonstrated in both real and synthetic simulation settings. We also recover analogous core-matchability results in a very general core-junk network model, wherein some vertices do not correspond between the graph pair.

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arXiv2018

Connectome Smoothing via Low-rank Approximations

In statistical connectomics, the quantitative study of brain networks, estimating the mean of a population of graphs based on a sample is a core problem. Often, this problem is especially difficult because the sample or cohort size is relatively small, sometimes even a single subject. While using the element-wise sample mean of the adjacency matrices is a common approach, this method does not exploit any underlying structural properties of the graphs. We propose using a low-rank method which incorporates tools for dimension selection and diagonal augmentation to smooth the estimates and improve performance over the naive methodology for small sample sizes. Theoretical results for the stochastic blockmodel show that this method offers major improvements when there are many vertices. Similarly, we demonstrate that the low-rank methods outperform the standard sample mean for a variety of independent edge distributions as well as human connectome data derived from magnetic resonance imaging, especially when sample sizes are small. Moreover, the low-rank methods yield "eigen-connectomes", which correlate with the lobe-structure of the human brain and superstructures of the mouse brain. These results indicate that low-rank methods are an important part of the tool box for researchers studying populations of graphs in general, and statistical connectomics in particular.

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arXiv2018

Tractable Graph Matching via Soft Seeding

The graph matching problem aims to discover a latent correspondence between the vertex sets of two observed graphs. This problem has proven to be quite challenging, with few satisfying methods that are computationally tractable and widely applicable. The FAQ algorithm has proven to have good performance on benchmark problems and works with a indefinite relaxation of the problem. Due to the indefinite relaxation, FAQ is not guaranteed to find the global maximum. However, if prior information is known about the true correspondence, this can be leveraged to initialize the algorithm near the truth. We show that given certain properties of this initial matrix, with high probability the FAQ algorithm will converge in two steps to the truth under a flexible model for pairs of random graphs. Importantly, this result implies that there will be no local optima near the global optima, providing a method to assess performance.

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arXiv2017

Statistical inference on random dot product graphs: a survey

The random dot product graph (RDPG) is an independent-edge random graph that is analytically tractable and, simultaneously, either encompasses or can successfully approximate a wide range of random graphs, from relatively simple stochastic block models to complex latent position graphs. In this survey paper, we describe a comprehensive paradigm for statistical inference on random dot product graphs, a paradigm centered on spectral embeddings of adjacency and Laplacian matrices. We examine the analogues, in graph inference, of several canonical tenets of classical Euclidean inference: in particular, we summarize a body of existing results on the consistency and asymptotic normality of the adjacency and Laplacian spectral embeddings, and the role these spectral embeddings can play in the construction of single- and multi-sample hypothesis tests for graph data. We investigate several real-world applications, including community detection and classification in large social networks and the determination of functional and biologically relevant network properties from an exploratory data analysis of the Drosophila connectome. We outline requisite background and current open problems in spectral graph inference.

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arXiv2017

Elements of estimation theory for causal effects in the presence of network interference

Randomized experiments in which the treatment of a unit can affect the outcomes of other units are becoming increasingly common in healthcare, economics, and in the social and information sciences. From a causal inference perspective, the typical assumption of no interference becomes untenable in such experiments. In many problems, however, the patterns of interference may be informed by the observation of network connections among the units of analysis. Here, we develop elements of optimal estimation theory for causal effects leveraging an observed network, by assuming that the potential outcomes of an individual depend only on the individual's treatment and on the treatment of the neighbors. We propose a collection of exclusion restrictions on the potential outcomes, and show how subsets of these restrictions lead to various parameterizations. Considering the class of linear unbiased estimators of the average direct treatment effect, we derive conditions on the design that lead to the existence of unbiased estimators, and offer analytical insights on the weights that lead to minimum integrated variance estimators. We illustrate the improved performance of these estimators when compared to more standard biased and unbiased estimators, using simulations.

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arXiv2016

Empirical Bayes Estimation for the Stochastic Blockmodel

Inference for the stochastic blockmodel is currently of burgeoning interest in the statistical community, as well as in various application domains as diverse as social networks, citation networks, brain connectivity networks (connectomics), etc. Recent theoretical developments have shown that spectral embedding of graphs yields tractable distributional results; in particular, a random dot product latent position graph formulation of the stochastic blockmodel informs a mixture of normal distributions for the adjacency spectral embedding. We employ this new theory to provide an empirical Bayes methodology for estimation of block memberships of vertices in a random graph drawn from the stochastic blockmodel, and demonstrate its practical utility. The posterior inference is conducted using a Metropolis-within-Gibbs algorithm. The theory and methods are illustrated through Monte Carlo simulation studies, both within the stochastic blockmodel and beyond, and experimental results on a Wikipedia data set are presented.

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arXiv2015

A nonparametric two-sample hypothesis testing problem for random dot product graphs

We consider the problem of testing whether two finite-dimensional random dot product graphs have generating latent positions that are independently drawn from the same distribution, or distributions that are related via scaling or projection. We propose a test statistic that is a kernel-based function of the adjacency spectral embedding for each graph. We obtain a limiting distribution for our test statistic under the null and we show that our test procedure is consistent across a broad range of alternatives.

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arXiv2015

Analyzing statistical and computational tradeoffs of estimation procedures

The recent explosion in the amount and dimensionality of data has exacerbated the need of trading off computational and statistical efficiency carefully, so that inference is both tractable and meaningful. We propose a framework that provides an explicit opportunity for practitioners to specify how much statistical risk they are willing to accept for a given computational cost, and leads to a theoretical risk-computation frontier for any given inference problem. We illustrate the tradeoff between risk and computation and illustrate the frontier in three distinct settings. First, we derive analytic forms for the risk of estimating parameters in the classical setting of estimating the mean and variance for normally distributed data and for the more general setting of parameters of an exponential family. The second example concentrates on computationally constrained Hodges-Lehmann estimators. We conclude with an evaluation of risk associated with early termination of iterative matrix inversion algorithms in the context of linear regression.

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arXiv2015

A semiparametric two-sample hypothesis testing problem for random dot product graphs

Two-sample hypothesis testing for random graphs arises naturally in neuroscience, social networks, and machine learning. In this paper, we consider a semiparametric problem of two-sample hypothesis testing for a class of latent position random graphs. We formulate a notion of consistency in this context and propose a valid test for the hypothesis that two finite-dimensional random dot product graphs on a common vertex set have the same generating latent positions or have generating latent positions that are scaled or diagonal transformations of one another. Our test statistic is a function of a spectral decomposition of the adjacency matrix for each graph and our test procedure is consistent across a broad range of alternatives. We apply our test procedure to real biological data: in a test-retest data set of neural connectome graphs, we are able to distinguish between scans from different subjects; and in the {\em C.elegans} connectome, we are able to distinguish between chemical and electrical networks. The latter example is a concrete demonstration that our test can have power even for small sample sizes. We conclude by discussing the relationship between our test procedure and generalized likelihood ratio tests.

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