The most promising class of statistical models for expressing structural properties of social networks observed at one moment in time, is the class of Exponential Random Graph Models (ERGMs), also known as p ∗ models. The strong point of these models is that they can represent a variety of structural tendencies, such as transitivity, that define complicated dependence patterns not easily modeled by more basic probability models. Recently, MCMC algorithms have been developed which produce approximate Maximum Likelihood estimators. Applying these models in their traditional specification to observed network data often has led to problems, however, which can be traced back to the fact that important parts of the parameter space correspond to nearly degenerate distributions, which may lead to convergence problems of estimation algorithms, and a poor fit to empirical data. This paper proposes new specifications of Exponential Random Graph Models. These specifications represent structural properties such as transitivity and heterogeneity of degrees by more complicated graph statistics than the traditional star and triangle counts. Three kinds of statistic are proposed: geometrically weighted degree distributions, alternating k-triangles, and alternating independent two-paths. Examples are presented both of modeling graphs and digraphs, in which the new specifications lead to much better results than the earlier existing specifications of the ERGM. It is concluded that the new specifications increase the range and applicability of the ERGM as a tool for the statistical analysis of social networks.

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the social network groups at the University of Groningen and the University of Melbourne, and for the helpful suggestions of an anonymous reviewer. This article reviews new specifications for exponential random graph models proposed by Snijders, Pattison, Robins & Handcock (2006) and demonstrates their improvement over homogeneous Markov random graph models in fitting empirical network data. Not only do the new specifications show improvements in goodness of fit for various data sets, they also help to avoid the problem of near-degeneracy that often afflicts the fitting of Markov random graph models in practice, particularly to network data exhibiting high levels of transitivity. The inclusion of a new higher order transitivity statistic allows estimation of parameters of exponential graph models for many (but not all) cases where it is impossible to estimate parameters of homogeneous Markov graph models. The new specifications were used to model a large number of classical smallscale network data sets and showed a dramatically better performance than Markov graph models. We also review three current programs for obtaining maximum likelihood estimates of model parameters and we compare these Monte Carlo maximum likelihood estimates with less accurate pseudo-likelihood estimates. Finally we discuss whether homogeneous Markov random graph models may be superseded by the new specifications, and how additional elaborations may further improve model performance. 2 In recent years, there has been growing interest in exponential random graph

Recent advances in statistical network analysis based on the family of exponential random graph (ERG) models have greatly improved our ability to conduct inference on dependence in large social networks

Curved exponential family models are a useful generalization of exponential random graph models (ERGMs). In particular, models involving the alternating k-star, alternating k-triangle, and alternating ktwopath statistics of Snijders et al. [Snijders, T.A.B., Pattison, P.E., Robins, G.L., Handcock, M.S., in press. New specifications for exponential random graph models. Sociological Methodology] may be viewed as curved exponential family models. This article unifies recent material in the literature regarding curved exponential family models for networks in general and models involving these alternating statistics in particular. It also discusses the intuition behind rewriting the three alternating statistics in terms of the degree distribution and the recently introduced shared partner distributions. This intuition suggests a redefinition of the alternating k-star statistic. Finally, this article demonstrates the use of the statnet package in R for fitting models of this sort, comparing new results on an oft-studied network dataset with results found in the literature.

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The current methods used to mine and analyze temporal social network data make two assumptions: all edges have the same strength, and all parameters are time-homogeneous. We show that those assumptions may not hold for social networks and propose an alternative model with two novel aspects: (1) the modeling of edges as multi-valued variables that can change in intensity, and (2) the use of a curved exponential family framework to capture time-inhomogeneous properties while retaining a parsimonious and interpretable model. We show that our model outperforms traditional models on two real-world social network data sets.

Abstract. Exposure and adoption of opinions in social networks are important questions in education, business, and government. We describe a novel application of pervasive computing based on using mobile phone sensors to measure and model the face-to-face interactions and subsequent opinion changes amongst undergraduates, during the 2008 US presidential election campaign. We find that self-reported political discussants have characteristic interaction patterns and can be predicted from sensor data. Mobile features can be used to estimate unique individual exposure to different opinions, and help discover surprising patterns of dynamic homophily related to external political events, such as election debates and election day. To our knowledge, this is the first time such dynamic homophily effects have been measured. Automatically estimated exposure explains individual opinions on election day. Finally, we report statistically significant differences in the daily activities of individuals that change political opinions versus those that do not, by modeling and discovering dominant activities using topic models. We find people who decrease their interest in politics are routinely exposed (face-to-face) to friends with little or no interest in politics. 1

A model for network panel data is discussed, based on the assumption that the observed data are discrete observations of a continuoustime Markov process on the space of all directed graphs on a given node set, in which changes in tie variables are independent conditional on the current graph. The model for tie changes is parametric and designed for applications to social network analysis, where the network dynamics can be interpreted as being generated by choices made by the social actors represented by the nodes of the graph. An algorithm for calculating the Maximum Likelihood estimator is presented, based on data augmentation and stochastic approximation. An application to an evolving friendship network is given and a small simulation study is presented which suggests that for small data sets the Maximum Likelihood estimator is more efficient than the earlier proposed Method of Moments estimator.

Network models are widely used to represent relational information among interacting units and the implications of these relations. In studies of social networks recent emphasis has been placed on random graph models where the nodes usually represent individual social actors and the edges represent a specified relationship between the actors. Most inference for models for social networks assumes that the presence or absence of all links in the network are completely observed, that the information is completely reliable and there are no measurement (e.g. recording) errors. This is clearly not true in practice, as much network data is collected though sample surveys. In addition even if a census of a population is attempted, individuals and links between individuals are missed (i.e., do not appear in the recorded data). In this paper we develop the conceptual and computational theory for inference based on sampled network information. We first review forms of network sampling designs used in practice and consider the various forms of out-of-design missing data. We consider inference from the likelihood framework, and develop a typology of network data that reflects their treatment within this frame. We then develop inference for social network models based on information from adaptive network mechanisms. We motivated and illustrate the ideas by analyzing the effect of link-tracing sampling designs on a collaboration network and by an analysis of social relations from the National Longitudinal Study of Adolescent Health subject to missing data. 1