Family Of Random Graph Models Research Articles

A network suspected infectious disease model for the development of syphilis transmission from 2015 to 2021 in Hubei province, China.

Understanding the transmission mode of syphilis is essential to prevent and predict its future prevalence and to develop effective control measures. This study aimed to develop a network suspected infectious disease model to simulate the syphilis transmission. The number of syphilis cases in Wuhan's Fourth Hospital, Hubei province, China, from October 2015 to July 2021 was collected. The simulation was carried out by interpersonal network-SI (suspected infectious) model based on temporal exponential family random graph models. Late latent syphilis and tertiary syphilis are predicted by December 2025. The validity of simulated value and real data was tested, including determination coefficient (R2), root means square error (RMSE), and means relative error (MRE). Moreover, we developed an online app that can more easily predict the number of syphilis infections in different scenarios by setting different parameters. Results showed that R2, RMSE, and MRE were 0.995, 36.19, and 6.31, respectively. Speed from latent infection to primary syphilis, primary syphilis to secondary syphilis, and susceptible group to latent infection decreased rapidly. The speed of transformation from secondary syphilis to early incubation period and early latent to late latent experienced a process from increase to decreased. Late latent to tertiary syphilis patients increased steadily. The number of late latent patients, early latent, invisible infection, primary syphilis, and secondary syphilis all increased at first and turn to decreased. However, tertiary syphilis continuously kept rising in the whole process. To better make use of the transmission model, an online application was developed (https://alanwu.shinyapps.io/MD-shiny/). Based on the simulation that late latent and tertiary syphilis were steadily increasing, the prevention and treatment for syphilis were imperative.

A probabilistic model for networks generated by actors’ characteristics

This work presents the affinity network model for random graphs, consisting of a broad family of random graph models depending on some parameters. In this model, we suppose that each individual randomly chooses a set of characteristics that represent him according to a certain probability measure. The connections between two individuals depend on their shared characteristics and are valued according to a function that measures what we call affinity in the network. According to the choice of this function, the network’s density can vary from sparse to complete graph, causing the model to be very flexible, which makes it suitable to fit with real networks. To illustrate the behavior of the affinity network model, we present a Monte Carlo simulation study. We tune the model’s generating parameters, analyze its topological measurements, and compare them with equivalent graphs with edges occurring independently, disregarding actors’ characteristics.

Continuous time graph processes with known ERGM equilibria: contextual review, extensions, and synthesis

ABSTRACT Graph processes that unfold in continuous time are of obvious theoretical and practical interest. Particularly useful are those whose long-term behavior converges to a graph distribution of known form. Here, we review some of the conditions for such convergence, and provide examples of novel and/or known processes that do so. These include subfamilies of the well-known stochastic actor-oriented models, as well as continuum extensions of temporal and separable temporal exponential family random graph models. We also comment on some related threads in the broader work on network dynamics, which provide additional context for the continuous time case.

Measuring structural HIV stigma

Despite the progress in pharmaceutical and epidemiological tools for combating HIV spread, HIV stigma remains a significant social barrier impeding treatment and prevention efforts, potentially reducing the effectiveness of interventions to reduce HIV transmission. In this paper, we propose a novel approach to defining and estimating HIV stigmatization through the structure of sexual relations, as opposed to attitudes. We conceptualize structural stigma as arising from two mechanisms: (1) a reduced propensity towards HIV serodiscordant partnerships (exclusion); and (2) a reduced propensity towards partnerships with seroconcordant individuals who themselves have serodiscordant partnerships (ostracism). Both mechanisms can be assessed from observed partnership network data using exponential family random graph models (ERGMs). We demonstrate our approach on a sexual contact network of black men who have sex with men in the South Side of Chicago. We find a tendency for serodiscordant sexual relationships to be suppressed in the network (θ = −0.69, p < .05), as well as a suppressive tendency for HIV negative YBMSM to have sex with other HIV negative people in serodiscordant relationships (θ = −0.96, p < .05) suggesting that structural HIV stigma is present in this network. Potential relationships with attitudinal stigma and implications for epidemiological strategies for reducing HIV stigma are discussed.

Impact of the tariff concessions of the RCEP agreement on the structure and evolution mechanism of manufacturing trade networks

In the world’s largest free trade area, the Regional Comprehensive Economic Partnership (RCEP) agreement will gradually eliminate tariffs on over 90% of member countries’ goods over the next 36 years. We construct a tariff policy effects evaluation framework based on the complex network theory. In this framework, the standard Global Trade Analysis Project (GTAP) model is recursively extended to generate forecasted data of bilateral trade. Based on this, we model the RCEP manufacturing trade networks and analyze the response of its core–periphery structure to the tariff concessions. Then, we evaluate policy effects on the evolution of trade patterns based on motif analysis. Finally, we construct separable temporal exponential family random graph models (STERGM) to explore the influence path and degree of the new tariff agreement on the evolutionary mechanisms of trade networks.

Finite Mixtures of ERGMs for Modeling Ensembles of Networks

Ensembles of networks arise in many scientific fields, but there are relatively few statistical tools for inferring their generative processes, particularly in the presence of both dyadic dependence and cross-graph heterogeneity. To address this gap, we propose characterizing network ensembles via finite mixtures of exponential family random graph models (ERGMs), a class of parametric statistical models 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 joint approach to estimate the number of mixture components and identify cluster-specific parameters simultaneously as well as to obtain an identified model under the Bayesian paradigm. Specifically, we develop a Metropolis-within-Gibbs algorithm to perform Bayesian inference, and estimate the number of mixture components using a strategy of deliberate overfitting with sparse priors that removes excess components during MCMC. As the true ERGM likelihood is generally intractable for model specifications with dyadic dependence terms, we consider two tractable approximations (pseudolikelihood and adjusted pseudolikelihood) to facilitate efficient statistical inference. We run simulation studies to compare the performance of these two approximations with respect to multiple metrics, showing conditions under which both are useful. We demonstrate the utility of the proposed approach using an ensemble of political co-voting networks among U.S. Senators and an ensemble of brain functional connectivity networks.

DERGMs: Degeneracy-restricted exponential family random graph models

AbstractExponential random graph models, or ERGMs, are a flexible and general class of models for modeling dependent data. While the early literature has shown them to be powerful in capturing many network features of interest, recent work highlights difficulties related to the models’ ill behavior, such as most of the probability mass being concentrated on a very small subset of the parameter space. This behavior limits both the applicability of an ERGM as a model for real data and inference and parameter estimation via the usual Markov chain Monte Carlo algorithms. To address this problem, we propose a new exponential family of models for random graphs that build on the standard ERGM framework. Specifically, we solve the problem of computational intractability and “degenerate” model behavior by an interpretable support restriction. We introduce a new parameter based on the graph-theoretic notion of degeneracy, a measure of sparsity whose value is commonly low in real-world networks. The new model family is supported on the sample space of graphs with bounded degeneracy and is called degeneracy-restricted ERGMs, or DERGMs for short. Since DERGMs generalize ERGMs—the latter is obtained from the former by setting the degeneracy parameter to be maximal—they inherit good theoretical properties, while at the same time place their mass more uniformly over realistic graphs. The support restriction allows the use of new (and fast) Monte Carlo methods for inference, thus making the models scalable and computationally tractable. We study various theoretical properties of DERGMs and illustrate how the support restriction improves the model behavior. We also present a fast Monte Carlo algorithm for parameter estimation that avoids many issues faced by Markov Chain Monte Carlo algorithms used for inference in ERGMs.

The formation of political discussion networks.

Dialogues among politicians provide a window into political landscapes and relations among parties and nations. Existing research has focused on the outcomes of such dialogues and on the structure of social networks on which they take place. Little is known, however, about how political discussion networks form and which are the main driving forces behind their formation. We study a collection of ego-networks from 30 randomly sampled Romanian politicians to reveal fundamental processes behind the formation of political discussion networks. We show that ties in such networks tend to be strong and balanced, and that their organization is not affected by sex, age or education homophily. We use the exponential family of random graph models for small networks to assess likely closure mechanisms and possible homophily effects, but we note that further research and additional data are needed to fully understand the impact of context and political affiliations on the generalization of our findings.

Local Graph Stability in Exponential Family Random Graph Models

Exponential family random graph models (ERGMs) are widely used to model networks by parameterizing graph probability in terms of a set of user-selected sufficient statistics. Equivalently, ERGMs ca...

Introduction to Network Modeling Using Exponential Random Graph Models (ERGM): Theory and an Application Using R-Project

Exponential family random graph models (ERGM) are increasingly used in the study of social networks. These models are build to explain the global structure of a network while allowing inference on tie prediction on a micro level. The number of papers within economics is however limited. Possible applications for economics are however abundant. The aim of this document is to provide an explanation of the basic mechanics behind the models and provide a sample code (using R and the packages statnet and ERGM) to operationalize and interpret results and analyse goodness of fit. After reading this paper the reader should be able to start their own analysis.

A dynamic process interpretation of the sparse ERGM reference model

ABSTRACTExponential 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 sizes. 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.

Current levels of gonorrhoea screening in MSM in Belgium may have little effect on prevalence: a modelling study.

There is considerable uncertainty as to the effectiveness of Neisseria gonorrhoeae (NG) screening in men who have sex with men. It is important to ensure that screening has benefits that outweigh the risks of increased antibiotics resistance. We develop a mathematical model to estimate the effectiveness of screening on prevalence. Separable Temporal Exponential family Random Graph Models are used to model the sexual relationships network, both with main and casual partners. Next, the transmission of Gonorrhoea is simulated on this network. The models are implemented using the R package 'statnet', which we adapted among other things to incorporate infection status at the pharynx, urethra and rectum separately and to distinguish between anal sex, oral sex and rimming. The different screening programmes compared are no screening, 3.5% of the population screened, 32% screened and 50% screened. The model simulates day-by-day evolution for 10 years of a population of 10 000. If half of the population would be screened, the prevalence in the pharynx decreases from 11.9% to 10.2%. We conclude that the limited impact of screening on NG prevalence may not outweigh the increased risk of antibiotic resistance.

A perfect sampling method for exponential family random graph models

ABSTRACTGeneration of deviates from random graph models with nontrivial 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.

Sharing Social Network Data: Differentially Private Estimation of Exponential Family Random-Graph Models

SummaryMotivated by a real life problem of sharing social network data that contain sensitive personal information, we propose a novel approach to release and analyse synthetic graphs to protect privacy of individual relationships captured by the social network while maintaining the validity of statistical results. A case-study using a version of the Enron e-mail corpus data set demonstrates the application and usefulness of the proposed techniques in solving the challenging problem of maintaining privacy and supporting open access to network data to ensure reproducibility of existing studies and discovering new scientific insights that can be obtained by analysing such data. We use a simple yet effective randomized response mechanism to generate synthetic networks under ε-edge differential privacy and then use likelihood-based inference for missing data and Markov chain Monte Carlo techniques to fit exponential family random-graph models to the generated synthetic networks.

Ergm.graphlets: A Package for ERG Modeling Based on Graphlet Statistics

Exponential-family random graph models are probabilistic network models that are parametrized by sufficient statistics based on structural (i.e., graph-theoretic) properties. The ergm package for the R statistical computing environment is a collection of tools for the analysis of network data within an exponential-family random graph model framework. Many different network properties can be employed as sufficient statistics for exponential- family random graph models by using the model terms defined in the ergm package; this functionality can be expanded by the creation of packages that code for additional network statistics. Here, our focus is on the addition of statistics based on graphlets. Graphlets are classes of small, connected, induced subgraphs that can be used to describe the topological structure of a network. We introduce an R package called ergm.graphlets that enables the use of graphlet properties of a network within the ergm package of R. The ergm.graphlets package provides a complete list of model terms that allows to incorporate statistics of any 2-, 3-, 4- and 5-node graphlets into exponential-family random graph models. The new model terms of the ergm.graphlets package enable both exponential-family random graph modeling of global structural properties and investigation of relationships between node attributes (i.e., covariates) and local topologies around nodes.

Local dependence in random graph models: characterization, properties and statistical inference.

Dependent phenomena, such as relational, spatial and temporal phenomena, tend to be characterized by local dependence in the sense that units which are close in a well-defined sense are dependent. In contrast with spatial and temporal phenomena, though, relational phenomena tend to lack a natural neighbourhood structure in the sense that it is unknown which units are close and thus dependent. Owing to the challenge of characterizing local dependence and constructing random graph models with local dependence, many conventional exponential family random graph models induce strong dependence and are not amenable to statistical inference. We take first steps to characterize local dependence in random graph models, inspired by the notion of finite neighbourhoods in spatial statistics and M-dependence in time series, and we show that local dependence endows random graph models with desirable properties which make them amenable to statistical inference. We show that random graph models with local dependence satisfy a natural domain consistency condition which every model should satisfy, but conventional exponential family random graph models do not satisfy. In addition, we establish a central limit theorem for random graph models with local dependence, which suggests that random graph models with local dependence are amenable to statistical inference. We discuss how random graph models with local dependence can be constructed by exploiting either observed or unobserved neighbourhood structure. In the absence of observed neighbourhood structure, we take a Bayesian view and express the uncertainty about the neighbourhood structure by specifying a prior on a set of suitable neighbourhood structures. We present simulation results and applications to two real world networks with 'ground truth'.

A Separable Model for Dynamic Networks

Models of dynamic networks - networks that evolve over time - have manifold applications. We develop a discrete-time generative model for social network evolution that inherits the richness and flexibility of the class of exponential-family random graph models. The model - a Separable Temporal ERGM (STERGM) - facilitates separable modeling of the tie duration distributions and the structural dynamics of tie formation. We develop likelihood-based inference for the model, and provide computational algorithms for maximum likelihood estimation. We illustrate the interpretability of the model in analyzing a longitudinal network of friendship ties within a school.

Sparse graphs: Metrics and random models

AbstractRecently, Bollobás, Janson and Riordan introduced a family of random graph models producing inhomogeneous graphs with n vertices and Θ(n) edges whose distribution is characterized by a kernel, i.e., a symmetric measurable function κ: [0, 1]2 → [0, ∞). To understand these models, we should like to know when different kernels κ give rise to “similar” graphs, and, given a real‐world network, how “similar” is it to a typical graph G(n, κ) derived from a given kernel κ.The analogous questions for dense graphs, with Θ(n2) edges, are answered by recent results of Borgs, Chayes, Lovász, Sós, Szegedy and Vesztergombi, who showed that several natural metrics on graphs are equivalent, and moreover that any sequence of graphs converges in each metric to a graphon, i.e., a kernel taking values in [0, 1].Possible generalizations of these results to graphs with o(n2) but ω(n) edges are discussed in a companion article [Bollobás and Riordan, London Math Soc Lecture Note Series 365 (2009), 211–287]; here we focus only on graphs with Θ(n) edges, which turn out to be much harder to handle. Many new phenomena occur, and there are a host of plausible metrics to consider; many of these metrics suggest new random graph models and vice versa. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 1‐38, 2011

A framework for the comparison of maximum pseudo-likelihood and maximum likelihood estimation of exponential family random graph models

The statistical modeling of social network data is difficult due to the complex dependence structure of the tie variables. Statistical exponential families of distributions provide a flexible way to model such dependence. They enable the statistical characteristics of the network to be encapsulated within an exponential family random graph (ERG) model. For a long time, however, likelihood-based estimation was only feasible for ERG models assuming dyad independence. For more realistic and complex models inference has been based on the pseudo-likelihood. Recent advances in computational methods have made likelihood-based inference practical, and comparison of the different estimators possible. In this paper, we present methodology to enable estimators of ERG model parameters to be compared. We use this methodology to compare the bias, standard errors, coverage rates and efficiency of maximum likelihood and maximum pseudo-likelihood estimators. We also propose an improved pseudo-likelihood estimation method aimed at reducing bias. The comparison is performed using simulated social network data based on two versions of an empirically realistic network model, the first representing Lazega’s law firm data and the second a modified version with increased transitivity. The framework considers estimation of both the natural and the mean-value parameters. The results clearly show the superiority of the likelihood-based estimators over those based on pseudo-likelihood, with the bias-reduced pseudo-likelihood out-performing the general pseudo-likelihood. The use of the mean-value parameterization provides insight into the differences between the estimators and when these differences will matter in practice.

A comparison of various approaches to the exponential random graph model: A reanalysis of 102 student networks in school classes

This paper describes an empirical comparison of four specifications of the exponential family of random graph models (ERGM), distinguished by model specification (dyadic independence, Markov, partial conditional dependence) and, for the Markov model, by estimation method (Maximum Pseudolikelihood, Maximum Likelihood). This was done by reanalyzing 102 student networks in 57 junior high school classes. At the level of all classes combined, earlier substantive conclusions were supported by all specifications. However, the different specifications led to different conclusions for individual classes. PL produced unreliable estimates (when ML is regarded as the standard) and had more convergence problems than ML. Furthermore, the estimates of covariate effects were affected considerably by controlling for network structure, although the precise specification of the structural part (Markov or partial conditional dependence) mattered less.