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Computational Complexity pp 2953–2967Cite as

Social Networks, Exponential Random Graph (p *) Models for

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Article Outline

Glossary

Definition of the Subject

Introduction

Notation and Terminology

Dependence Hypotheses

Bernoulli Random Graph (Erdös–Rényi) Models

Dyadic Independence Models

Markov Random Graphs

Simulation and Model Degeneracy

Social Circuit Dependence: Partial Conditional Dependence Hypotheses

Social Circuit Specifications

Estimation

Goodness of Fit and Comparisons with Markov Models

Further Extensions and Future Directions

Bibliography

Exponential random graph models , also known as p ∗ models , constitute a family of statistical models for socialnetworks. The importance of this modeling framework lies in its capacity to represent social structural effects commonly observed in many human socialnetworks, including general degree‐based effects as well as reciprocity and transitivity, and at the node-level, homophily and attribute‐basedactivity and popularity effects.The models can be derived from explicit hypotheses about dependencies among network ties. They are parametrized in termsof the prevalence of small subgraphs (configurations) in the network and can be interpreted as describing the combinations of local social processes fromwhich a given network emerges. The models are estimable from data and readily simulated.Versions of the models have been proposed for univariateand multivariate networks, valued networks, bipartite graphs and for longitudinal network data. Nodal attribute data can be incorporated in socialselection models, and through an analogous framework for social influence models.

The modeling approach was first proposed in the statistical literature in the mid-1980s, building on previous work in the spatial statistics andstatistical mechanics literature. In the 1990s, the models were picked up and extended by the social networks research community. In this century, withthe development of effective estimation and simulation procedures, there has been a growing understanding of certain inadequacies in the originalform of the models. Recently developed specifications for these models have shown a substantial improvement in fitting real social network data, tothe point where for many network data sets a large number of graph features can be successfully reproduced by the fitted models.

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Abbreviations

Alternating independent-2-paths:

A parameter (and statistic) in new specification models; a particular combination of k‑independent-2-path counts into the one statistic.

Alternating k-stars:

A Markov parameter (and statistic) in the new specification models; a particular combination of Markov k‑star counts into the one statistic; equivalent to geometrically weighted degree counts; useful for modeling the degree distribution.

Alternating k‑triangles:

A parameter (and statistic) in the new specification models; a particular combination of k‑triangle counts into the one statistic; equivalent to weighted shared partners.

Cyclic triad:

A Markov graph configuration: in a directed network, ties ij, jk and ki are observed among actors \( { i, j } \), and k.

Degeneracy (or near‐degeneracy):

When a model implies that very few distinct graphs are probable, often only empty or complete graphs; degenerate models cannot be good models for social network data.

Dependence assumption:

Theoretical assumption about dependencies among possible network ties; determines the type of parameters in the model.

Dyad independence:

Assumes that dyads are independent of one another; the model includes edge and reciprocity parameters, and possibly also node or dyad attributes.

Dyad-wise shared partners:

A parameter (and statistic) in the higher order models; equivalent to alternating independent 2-paths.

Edge-wise shared partner distribution:

Distribution of the number of dyads who are themselves related and who have a fixed number of shared partners.

Edge-wise shared partners:

A parameter (and statistic) in the higher order models; equivalent to alternating k‑triangles.

Geometrically weighted degree counts:

A statistic (and parameter) in the new specification models: a sum of degree counts with geometrically decreasing weights; equivalent to alternating k-stars.

Homogeneity assumption:

Assumption about which parameters to equate, to make a model identifiable.

k‑independent-2-paths:

Configurations in the higher order models; equivalent to k‑triangles but without the base.

k‑in-star:

A Markov graph configuration: in a directed graph, k arcs are directed to the one actor.

k‑out-star:

A Markov graph configuration: in a directed graph, k arcs are expressed by the one actor.

k‑triangle:

A configuration in higher order models; in a non‐directed graph, the combination of k triangles, each sharing the one edge (the base of the k‑triangle).

k‑star:

A Markov graph configuration: in a non‐directed graph, k edges are expressed by the one actor.

Markov dependence assumption:

Introduced by Frank and Strauss [9], proposes that, conditional on the rest of the graph, two possible ties are independent of each other unless they share an actor.

Mixed-star:

A Markov graph configuration: a two path in a directed graph.

Monte Carlo Markov chain maximum likelihood estimation (MCMCMLE):

Method of estimation based on computer simulation; more principled than pseudolikelihood.

Network configuration:

A small subgraph that may be observed in the data and that is represented by parameters in the model: e. g. reciprocated ties, triangles.

Parameters:

Relate to specific network configurations that may be observed in the graph; a large positive parameter is interpreted as the presence of more of the configurations than might be expected from chance (given the other effects in the model); a large negative parameter signifies the relative absence of the configuration.

Partial dependence assumption:

Assumption for dependencies among possible ties created by the presence of other ties; permits models with higher order configurations than Markov configurations.

Pseudo‐likelihood estimation:

An approximate method of estimation using logistic regression; does not produce reliable standard errors.

p 1 Model:

An early dyad independence model, including popularity and expansiveness effects.

p 2 Model:

Elaboration of p 1 model, where popularity and expansiveness effects are random, and independent variables may be used to predict ties.

Simple random graphs, Bernoulli graphs, Erdös–Rényi graphs:

Assume that edges are independent of one another and are observed with a given probability.

Social circuit dependence:

Two possible ties are conditionally dependent when, if observed, they would create a 4-cycle.

Transitive triad:

A Markov graph configuration: in a directed network, ties ij, jk and ik are observed among actors \( { i, j } \), and k.

Triangle:

A Markov graph configuration: in a non‐directed network, a clique of three actors, ties ij, jk and ik are observed among actors \( { i, j } \), and k.

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Authors and Affiliations

  1. School of Behavioural Science, University of Melbourne, Melbourne, Australia

    Garry Robins

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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Robins, G. (2012). Social Networks, Exponential Random Graph (p *) Models for. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_182

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