Bayesian Exponential Random Graph Models with Nodal Random Effects

We extend the well-known and widely used Exponential Random Graph Model (ERGM) by including nodal random effects to compensate for heterogeneity in the nodes of a network. The Bayesian framework for ERGMs proposed by Caimo and Friel (2011) yields the basis of our modelling algorithm. A central question in network models is the question of model selection and following the Bayesian paradigm we focus on estimating Bayes factors. To do so we develop an approximate but feasible calculation of the Bayes factor which allows one to pursue model selection. Two data examples and a small simulation study illustrate our mixed model approach and the corresponding model selection.Comment: 23 pages, 9 figures, 3 table

Adjusting for Network Size and Composition Effects in Exponential-Family Random Graph Models

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).Comment: 37 pages, 2 figures, 5 tables; notation revised and clarified, some sections (particularly 4.3 and 5) made more rigorous, some derivations moved into the appendix, typos fixed, some wording change

A spatial model for social networks

We study spatial embeddings of random graphs in which nodes are randomly distributed in geographical space. We let the edge probability between any two nodes to be dependent on the spatial distance between them and demonstrate that this model captures many generic properties of social networks, including the ``small-world'' properties, skewed degree distribution, and most distinctively the existence of community structures.Comment: To be published in Physica A (2005

Quantifying structure in networks

We investigate exponential families of random graph distributions as a framework for systematic quantification of structure in networks. In this paper we restrict ourselves to undirected unlabeled graphs. For these graphs, the counts of subgraphs with no more than k links are a sufficient statistics for the exponential families of graphs with interactions between at most k links. In this framework we investigate the dependencies between several observables commonly used to quantify structure in networks, such as the degree distribution, cluster and assortativity coefficients.Comment: 17 pages, 3 figure

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.Comment: 28 pages (including a 4-page appendix); a substantial rewrite, with many corrections, changes in terminology, and a different analysis for the exampl

Exponential Random Graph Modeling for Complex Brain Networks

Exponential random graph models (ERGMs), also known as p* models, have been utilized extensively in the social science literature to study complex networks and how their global structure depends on underlying structural components. However, the literature on their use in biological networks (especially brain networks) has remained sparse. Descriptive models based on a specific feature of the graph (clustering coefficient, degree distribution, etc.) have dominated connectivity research in neuroscience. Corresponding generative models have been developed to reproduce one of these features. However, the complexity inherent in whole-brain network data necessitates the development and use of tools that allow the systematic exploration of several features simultaneously and how they interact to form the global network architecture. ERGMs provide a statistically principled approach to the assessment of how a set of interacting local brain network features gives rise to the global structure. We illustrate the utility of ERGMs for modeling, analyzing, and simulating complex whole-brain networks with network data from normal subjects. We also provide a foundation for the selection of important local features through the implementation and assessment of three selection approaches: a traditional p-value based backward selection approach, an information criterion approach (AIC), and a graphical goodness of fit (GOF) approach. The graphical GOF approach serves as the best method given the scientific interest in being able to capture and reproduce the structure of fitted brain networks

Differentially Private Exponential Random Graphs

We propose methods to release and analyze synthetic graphs in order to protect privacy of individual relationships captured by the social network. Proposed techniques aim at fitting and estimating a wide class of exponential random graph models (ERGMs) in a differentially private manner, and thus offer rigorous privacy guarantees. More specifically, we use the randomized response mechanism to release networks under $\epsilon$-edge differential privacy. To maintain utility for statistical inference, treating the original graph as missing, we propose a way to use likelihood based inference and Markov chain Monte Carlo (MCMC) techniques to fit ERGMs to the produced synthetic networks. We demonstrate the usefulness of the proposed techniques on a real data example.Comment: minor edit

A Statistical Social Network Model for Consumption Data in Food Webs

We adapt existing statistical modeling techniques for social networks to study consumption data observed in trophic food webs. These data describe the feeding volume (non-negative) among organisms grouped into nodes, called trophic species, that form the food web. Model complexity arises due to the extensive amount of zeros in the data, as each node in the web is predator/prey to only a small number of other trophic species. Many of the zeros are regarded as structural (non-random) in the context of feeding behavior. The presence of basal prey and top predator nodes (those who never consume and those who are never consumed, with probability 1) creates additional complexity to the statistical modeling. We develop a special statistical social network model to account for such network features. The model is applied to two empirical food webs; focus is on the web for which the population size of seals is of concern to various commercial fisheries.Comment: On 2013-09-05, a revised version entitled "A Statistical Social Network Model for Consumption Data in Trophic Food Webs" was accepted for publication in the upcoming Special Issue "Statistical Methods for Ecology" in the journal Statistical Methodolog

Characterization and Management of Food Loss and Waste in North America

Policies and programs on food loss and waste (FLW) are gaining momentum across North America as awareness of the issue continues to grow. The Commission for Environmental Cooperation (CEC) established the North American Initiative on Food Waste Reduction and Recovery as part of its Green Economy and Climate Change project areas. This white paper characterizes FLW in Canada, Mexico and the United States and identifies opportunities for the industrial, commercial and institutional (ICI) sector, governments, and nongovernmental organizations (NGOs) to take action across the three countries. The scope of this research included post-harvest to pre-consumer stages of the food supply chain (i.e., post-harvest food production; processing; distribution; retail; and food service). Pre-harvest food production and the consumer stages of the food supply chain are beyond the scope of this study. This project complements the CEC's North American Initiative on Organic Waste Diversion and Processing, which examines composting, anaerobic digestion, and other industrial processes (e.g. rendering, biofuel) for FLW and other organic waste. The content of this white paper was compiled from primary and secondary sources of information in Canada, Mexico, the United States and countries outside of North America. Primary sources included interviews and email exchanges with 167 stakeholders representing various locations, organization types and sizes, and stages of the food supply chain. Secondary sources included reports, white papers, academic papers, news articles, media recordings and government databases, as well as a review of on-the-ground programs and projects implemented by the ICI sector, governments and NGOs. North American and international experts on the subject matter also vetted key findings during a three-day stakeholder session held in Canada, in February 2017