arXiv subjects

Carter T. Butts

Publications and source records attributed to Carter T. Butts.

At least 37 records · Page 2Linked to original sources

arXiv2020

Kernel-based Approximate Bayesian Inference for Exponential Family Random Graph Models

Bayesian inference for exponential family random graph models (ERGMs) is a doubly-intractable problem because of the intractability of both the likelihood and posterior normalizing factor. Auxiliary variable based Markov Chain Monte Carlo (MCMC) methods for this problem are asymptotically exact but computationally demanding, and are difficult to extend to modified ERGM families. In this work, we propose a kernel-based approximate Bayesian computation algorithm for fitting ERGMs. By employing an adaptive importance sampling technique, we greatly improve the efficiency of the sampling step. Though approximate, our easily parallelizable approach is yields comparable accuracy to state-of-the-art methods with substantial improvements in compute time on multi-core hardware. Our approach also flexibly accommodates both algorithmic enhancements (including improved learning algorithms for estimating conditional expectations) and extensions to non-standard cases such as inference from non-sufficient statistics. We demonstrate the performance of this approach on two well-known network data sets, comparing its accuracy and efficiency with results obtained using the approximate exchange algorithm. Our tests show a wallclock time advantage of up to 50% with five cores, and the ability to fit models in 1/5th the time at 30 cores; further speed enhancements are possible when more cores are available.

stat.CO↗

arXiv2020

Spatial Heterogeneity Can Lead to Substantial Local Variations in COVID-19 Timing and Severity

Standard epidemiological models for COVID-19 employ variants of compartment (SIR) models at local scales, implicitly assuming spatially uniform local mixing. Here, we examine the effect of employing more geographically detailed diffusion models based on known spatial features of interpersonal networks, most particularly the presence of a long-tailed but monotone decline in the probability of interaction with distance, on disease diffusion. Based on simulations of unrestricted COVID-19 diffusion in 19 U.S cities, we conclude that heterogeneity in population distribution can have large impacts on local pandemic timing and severity, even when aggregate behavior at larger scales mirrors a classic SIR-like pattern. Impacts observed include severe local outbreaks with long lag time relative to the aggregate infection curve, and the presence of numerous areas whose disease trajectories correlate poorly with those of neighboring areas. A simple catchment model for hospital demand illustrates potential implications for health care utilization, with substantial disparities in the timing and extremity of impacts even without distancing interventions. Likewise, analysis of social exposure to others who are morbid or deceased shows considerable variation in how the epidemic can appear to individuals on the ground, potentially affecting risk assessment and compliance with mitigation measures. These results demonstrate the potential for spatial network structure to generate highly non-uniform diffusion behavior even at the scale of cities, and suggest the importance of incorporating such structure when designing models to inform healthcare planning, predict community outcomes, or identify potential disparities.

physics.soc-ph↗

arXiv2020

Finite Mixtures of ERGMs for Modeling Ensembles of Networks

Ensembles of networks arise in many scientific fields, but there are few statistical tools for inferring their generative processes, particularly in the presence of both dyadic dependence and cross-graph heterogeneity. To fill in this gap, we propose characterizing network ensembles via finite mixtures of exponential family random graph models, a framework for parametric statistical modeling of graphs that has been successful in explicitly modeling the complex stochastic processes that govern the structure of edges in a network. Our proposed modeling framework can also be used for applications such as model-based clustering of ensembles of networks and density estimation for complex graph distributions. We develop a Metropolis-within-Gibbs algorithm to conduct fully Bayesian inference and adapt a version of deviance information criterion for missing data models to choose the number of latent heterogeneous generative mechanisms. Simulation studies show that the proposed procedure can recover the true number of latent heterogeneous generative processes and corresponding parameters. We demonstrate the utility of the proposed approach using an ensemble of political co-voting networks among U.S. Senators.

stat.ME↗

arXiv2020

Phase Transitions in the Edge/Concurrent Vertex Model

Although it is well-known that some exponential family random graph model (ERGM) families exhibit phase transitions (in which small parameter changes lead to qualitative changes in graph structure), the behavior of other models is still poorly understood. Recently, Krivitsky and Morris have reported a previously unobserved phase transition in the edge/concurrent vertex family (a simple starting point for models of sexual contact networks). Here, we examine this phase transition, showing it to be a first order transition with respect to an order parameter associated with the fraction of concurrent vertices. This transition stems from weak cooperativity in the recruitment of vertices to the concurrent phase, which may not be a desirable property in some applications.

cs.SI↗

arXiv2019

A Dynamic Process Reference Model for Sparse Networks with Reciprocity

Many social and other networks exhibit stable size scaling relationships, such that features such as mean degree or reciprocation rates change slowly or are approximately constant as the number of vertices increases. Statistical network models built on top of simple Bernoulli baseline (or reference) measures often behave unrealistically in this respect, leading to the development of sparse reference models that preserve features such as mean degree scaling. In this paper, we generalize recent work on the micro-foundations of such reference models to the case of sparse directed graphs with non-vanishing reciprocity, providing a dynamic process interpretation of the emergence of stable macroscopic behavior.

stat.ME↗

arXiv2019

Exponential-Family Models of Random Graphs: Inference in Finite-, Super-, and Infinite Population Scenarios

Exponential-family Random Graph Models (ERGMs) constitute a large statistical framework for modeling sparse and dense random graphs, short- and long-tailed degree distributions, covariates, and a wide range of complex dependencies. Special cases of ERGMs are generalized linear models (GLMs), Bernoulli random graphs, $β$-models, $p_1$-models, and models related to Markov random fields in spatial statistics and other areas of statistics. While widely used in practice, questions have been raised about the theoretical properties of ERGMs. These include concerns that some ERGMs are near-degenerate and that many ERGMs are non-projective. To address them, careful attention must be paid to model specifications and their underlying assumptions, and in which inferential settings models are employed. As we discuss, near-degeneracy can affect simplistic ERGMs lacking structure, but well-posed ERGMs with additional structure can be well-behaved. Likewise, lack of projectivity can affect non-likelihood-based inference, but likelihood-based inference does not require projectivity. Here, we review well-posed ERGMs along with likelihood-based inference. We first clarify the core statistical notions of "sample" and "population" in the ERGM framework, and separate the process that generates the population graph from the observation process. We then review likelihood-based inference in finite-, super-, and infinite-population scenarios. We conclude with consistency results, and an application to human brain networks

stat.ME↗

arXiv2019

Local Graph Stability in Exponential Family Random Graph Models

Exponential family Random Graph Models (ERGMs) can be viewed as expressing a probability distribution on graphs arising from the action of competing social forces that make ties more or less likely, depending on the state of the rest of the graph. Such forces often lead to a complex pattern of dependence among edges, with non-trivial large-scale structures emerging from relatively simple local mechanisms. While this provides a powerful tool for probing macro-micro connections, much remains to be understood about how local forces shape global outcomes. One simple question of this type is that of the conditions needed for social forces to stabilize a particular structure. We refer to this property as local stability and seek a general means of identifying the set of parameters under which a target graph is locally stable with respect to a set of alternatives. Here, we provide a complete characterization of the region of the parameter space inducing local stability, showing it to be the interior of a convex cone whose faces can be derived from the change-scores of the sufficient statistics vis-a-vis the alternative structures. As we show, local stability is a necessary but not sufficient condition for more general notions of stability, the latter of which can be explored more efficiently by using the ``stable cone'' within the parameter space as a starting point. In addition, we show how local stability can be used to determine whether a fitted model implies that an observed structure would be expected to arise primarily from the action of social forces, versus by merit of the model permitting a large number of high probability structures, of which the observed structure is one. We also use our approach to identify the dyads within a given structure that are the least stable, and hence predicted to have the highest probability of changing over time.

cs.DM↗

arXiv2019

Selection of Exponential-Family Random Graph Models via Held-Out Predictive Evaluation (HOPE)

Statistical models for networks with complex dependencies pose particular challenges for model selection and evaluation. In particular, many well-established statistical tools for selecting between models assume conditional independence of observations and/or conventional asymptotics, and their theoretical foundations are not always applicable in a network modeling context. While simulation-based approaches to model adequacy assessment are now widely used, there remains a need for procedures that quantify a model's performance in a manner suitable for selecting among competing models. Here, we propose to address this issue by developing a predictive evaluation strategy for exponential family random graph models that is analogous to cross-validation. Our approach builds on the held-out predictive evaluation (HOPE) scheme introduced by Wang et al. (2016) to assess imputation performance. We systematically hold out parts of the observed network to: evaluate how well the model is able to predict the held-out data; identify where the model performs poorly based on which data are held-out, indicating e.g. potential weaknesses; and calculate general summaries of predictive performance that can be used for model selection. As such, HOPE can assist researchers in improving models by indicating where a model performs poorly, and by quantitatively comparing predictive performance across competing models. The proposed method is applied to model selection problem of two well-known data sets, and the results are compared to those obtained via nominal AIC and BIC scores.

stat.ME↗

arXiv2019

Incorporating Structural Stigma into Network Analysis

A rich literature has explored the modeling of homophily and other forms of nonuniform mixing associated with individual-level covariates within the exponential family random graph (ERGM) framework. Such differential mixing does not fully explain phenomena such as stigma, however, which involve the active maintenance of social boundaries by ostracism of persons with out-group ties. Here, we introduce a new statistic that allows for such effects to be captured, making it possible to probe for the potential presence of boundary maintenance above and beyond simple differences in nomination rates. We demonstrate this statistic in the context of gender segregation in a school classroom.

cs.SI↗

arXiv2018

A Dynamic Process Interpretation of the Sparse ERGM Reference Model

Exponential family random graph models (ERGMs) can be understood in terms of a set of structural biases that act on an underlying reference distribution. This distribution determines many aspects of the behavior and interpretation of the ERGM families incorporating it. One important innovation in this area has been the development of an ERGM reference model that produces realistic behavior when generalized to sparse networks of varying size. Here, we show that this model can be derived from a latent dynamic process in which tie formation takes place within small local settings between which individuals move. This derivation provides one possible micro-process interpretation of the sparse ERGM reference model, and sheds light on the conditions under which constant mean degree scaling can emerge.

math.ST↗

arXiv2018

Spectral Graph Forge: Graph Generation Targeting Modularity

Community structure is an important property that captures inhomogeneities common in large networks, and modularity is one of the most widely used metrics for such community structure. In this paper, we introduce a principled methodology, the Spectral Graph Forge, for generating random graphs that preserves community structure from a real network of interest, in terms of modularity. Our approach leverages the fact that the spectral structure of matrix representations of a graph encodes global information about community structure. The Spectral Graph Forge uses a low-rank approximation of the modularity matrix to generate synthetic graphs that match a target modularity within user-selectable degree of accuracy, while allowing other aspects of structure to vary. We show that the Spectral Graph Forge outperforms state-of-the-art techniques in terms of accuracy in targeting the modularity and randomness of the realizations, while also preserving other local structural properties and node attributes. We discuss extensions of the Spectral Graph Forge to target other properties beyond modularity, and its applications to anonymization.

cs.SI↗

arXiv2017

A Perfect Sampling Method for Exponential Family Random Graph Models

Generation of deviates from random graph models with non-trivial edge dependence is an increasingly important problem. Here, we introduce a method which allows perfect sampling from random graph models in exponential family form ("exponential family random graph" models), using a variant of Coupling From The Past. We illustrate the use of the method via an application to the Markov graphs, a family that has been the subject of considerable research. We also show how the method can be applied to a variant of the biased net models, which are not exponentially parameterized.

stat.CO↗

arXiv2017

Baseline Mixture Models for Social Networks

Continuous mixtures of distributions are widely employed in the statistical literature as models for phenomena with highly divergent outcomes; in particular, many familiar heavy-tailed distributions arise naturally as mixtures of light-tailed distributions (e.g., Gaussians), and play an important role in applications as diverse as modeling of extreme values and robust inference. In the case of social networks, continuous mixtures of graph distributions can likewise be employed to model social processes with heterogeneous outcomes, or as robust priors for network inference. Here, we introduce some simple families of network models based on continuous mixtures of baseline distributions. While analytically and computationally tractable, these models allow more flexible modeling of cross-graph heterogeneity than is possible with conventional baseline (e.g., Bernoulli or $U|man$ distributions). We illustrate the utility of these baseline mixture models with application to problems of multiple-network ERGMs, network evolution, and efficient network inference. Our results underscore the potential ubiquity of network processes with nontrivial mixture behavior in natural settings, and raise some potentially disturbing questions regarding the adequacy of current network data collection practices.

stat.ME↗

arXiv2017

A Relational Event Approach to Modeling Behavioral Dynamics

This chapter provides an introduction to the analysis of relational event data (i.e., actions, interactions, or other events involving multiple actors that occur over time) within the R/statnet platform. We begin by reviewing the basics of relational event modeling, with an emphasis on models with piecewise constant hazards. We then discuss estimation for dyadic and more general relational event models using the relevent package, with an emphasis on hands-on applications of the methods and interpretation of results. Statnet is a collection of packages for the R statistical computing system that supports the representation, manipulation, visualization, modeling, simulation, and analysis of relational data. Statnet packages are contributed by a team of volunteer developers, and are made freely available under the GNU Public License. These packages are written for the R statistical computing environment, and can be used with any computing platform that supports R (including Windows, Linux, and Mac)."

stat.ME↗

arXiv2017

A note on the role of projectivity in likelihood-based inference for random graph models

There is widespread confusion about the role of projectivity in likelihood-based inference for random graph models. The confusion is rooted in claims that projectivity, a form of marginalizability, may be necessary for likelihood-based inference and consistency of maximum likelihood estimators. We show that likelihood-based superpopulation inference is not affected by lack of projectivity and that projectivity is not a necessary condition for consistency of maximum likelihood estimators.

math.ST↗

arXiv2017

Construction of Directed 2K Graphs

We study the problem of constructing synthetic graphs that resemble real-world directed graphs in terms of their degree correlations. We define the problem of directed 2K construction (D2K) that takes as input the directed degree sequence (DDS) and a joint degree and attribute matrix (JDAM) so as to capture degree correlation specifically in directed graphs. We provide necessary and sufficient conditions to decide whether a target D2K is realizable, and we design an efficient algorithm that creates realizations with that target D2K. We evaluate our algorithm in creating synthetic graphs that target real-world directed graphs (such as Twitter) and we show that it brings significant benefits compared to state-of-the-art approaches.

cs.SI↗

arXiv2016

Are you going to the party: depends, who else is coming? [Learning hidden group dynamics via conditional latent tree models]

Scalable probabilistic modeling and prediction in high dimensional multivariate time-series is a challenging problem, particularly for systems with hidden sources of dependence and/or homogeneity. Examples of such problems include dynamic social networks with co-evolving nodes and edges and dynamic student learning in online courses. Here, we address these problems through the discovery of hierarchical latent groups. We introduce a family of Conditional Latent Tree Models (CLTM), in which tree-structured latent variables incorporate the unknown groups. The latent tree itself is conditioned on observed covariates such as seasonality, historical activity, and node attributes. We propose a statistically efficient framework for learning both the hierarchical tree structure and the parameters of the CLTM. We demonstrate competitive performance in multiple real world datasets from different domains. These include a dataset on students' attempts at answering questions in a psychology MOOC, Twitter users participating in an emergency management discussion and interacting with one another, and windsurfers interacting on a beach in Southern California. In addition, our modeling framework provides valuable and interpretable information about the hidden group structures and their effect on the evolution of the time series.

cs.SI↗

arXiv2015

Estimating Subgraph Frequencies with or without Attributes from Egocentrically Sampled Data

In this paper we show how to efficiently produce unbiased estimates of subgraph frequencies from a probability sample of egocentric networks (i.e., focal nodes, their neighbors, and the induced subgraphs of ties among their neighbors). A key feature of our proposed method that differentiates it from prior methods is the use of egocentric data. Because of this, our method is suitable for estimation in large unknown graphs, is easily parallelizable, handles privacy sensitive network data (e.g. egonets with no neighbor labels), and supports counting of large subgraphs (e.g. maximal clique of size 205 in Section 6) by building on top of existing exact subgraph counting algorithms that may not support sampling. It gracefully handles a variety of sampling designs such as uniform or weighted independence or random walk sampling. Our method can be used for subgraphs that are: (i) undirected or directed; (ii) induced or non-induced; (iii) maximal or non-maximal; and (iv) potentially annotated with attributes. We compare our estimators on a variety of real-world graphs and sampling methods and provide suggestions for their use. Simulation shows that our method outperforms the state-of-the-art approach for relative subgraph frequencies by up to an order of magnitude for the same sample size. Finally, we apply our methodology to a rare sample of Facebook users across the social graph to estimate and interpret the clique size distribution and gender composition of cliques.

cs.SI↗