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Register a new userExponential-Family Random Graph Models for Valued Networks
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Added by
Pavel Krivitsky
Publication date
2011
fields
Mathematical Statistics
and research's language is
English
Authors
Pavel N. Krivitsky
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Exponential-family random graph models (ERGMs) provide a principled and flexible way to model and simulate features common in social networks, such as propensities for homophily, mutuality, and friend-of-a-friend triad closure, through choice of model terms (sufficient statistics). However, those ERGMs modeling the more complex features have, to date, been limited to binary data: presence or absence of ties. Thus, analysis of valued networks, such as those where counts, measurements, or ranks are observed, has necessitated dichotomizing them, losing information and introducing biases. In this work, we generalize ERGMs to valued networks. Focusing on modeling counts, we formulate an ERGM for networks whose ties are counts and discuss issues that arise when moving beyond the binary case. We introduce model terms that generalize and model common social network features for such data and apply these methods to a network dataset whose values are counts of interactions.
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Rank-order relational data, in which each actor ranks the others according to some criterion, often arise from sociometric measurements of judgment (e.g., self-reported interpersonal interaction) or preference (e.g., relative liking). We propose a class of exponential-family models for rank-order relational data and derive a new class of sufficient statistics for such data, which assume no more than within-subject ordinal properties. Application of MCMC MLE to this family allows us to estimate effects for a variety of plausible mechanisms governing rank structure in cross-sectional context, and to model the evolution of such structures over time. We apply this framework to model the evolution of relative liking judgments in an acquaintance process, and to model recall of relative volume of interpersonal interaction among members of a technology education program.
Exponential-family random graph models (ERGMs) provide a principled way to model and simulate features common in human social networks, such as propensities for homophily and friend-of-a-friend triad closure. We show that, without adjustment, ERGMs preserve density as network size increases. Density invariance is often not appropriate for social networks. We suggest a simple modification based on an offset which instead preserves the mean degree and accommodates changes in network composition asymptotically. We demonstrate that this approach allows ERGMs to be applied to the important situation of egocentrically sampled data. We analyze data from the National Health and Social Life Survey (NHSLS).
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 models 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.
Traditional Bayesian random partition models assume that the size of each cluster grows linearly with the number of data points. While this is appealing for some applications, this assumption is not appropriate for other tasks such as entity resolution, modeling of sparse networks, and DNA sequencing tasks. Such applications require models that yield clusters whose sizes grow sublinearly with the total number of data points -- the microclustering property. Motivated by these issues, we propose a general class of random partition models that satisfy the microclustering property with well-characterized theoretical properties. Our proposed models overcome major limitations in the existing literature on microclustering models, namely a lack of interpretability, identifiability, and full characterization of model asymptotic properties. Crucially, we drop the classical assumption of having an exchangeable sequence of data points, and instead assume an exchangeable sequence of clusters. In addition, our framework provides flexibility in terms of the prior distribution of cluster sizes, computational tractability, and applicability to a large number of microclustering tasks. We establish theoretical properties of the resulting class of priors, where we characterize the asymptotic behavior of the number of clusters and of the proportion of clusters of a given size. Our framework allows a simple and efficient Markov chain Monte Carlo algorithm to perform statistical inference. We illustrate our proposed methodology on the microclustering task of entity resolution, where we provide a simulation study and real experiments on survey panel data.
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.
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