# Estimating and understanding exponential random graph models

@article{Chatterjee2013EstimatingAU, title={Estimating and understanding exponential random graph models}, author={Sourav Chatterjee and Persi Diaconis}, journal={Annals of Statistics}, year={2013}, volume={41}, pages={2428-2461} }

We introduce a method for the theoretical analysis of exponential random graph models. The method is based on a large-deviations approximation to the normalizing constant shown to be consistent using theory developed by Chatterjee and Varadhan [European J. Combin. 32 (2011) 1000-1017]. The theory explains a host of difficulties encountered by applied workers: many distinct models have essentially the same MLE, rendering the problems ``practically'' ill-posed. We give the first rigorous proofs… Expand

#### 276 Citations

Tractable and Consistent Random Graph Models

- Computer Science, Physics
- ArXiv
- 2012

This work provides the first general results on when these models' parameters estimated from the observation of a single network are consistent (i.e., become accurate as the number of nodes grows), and shows how choice-based (strategic) network formation models can be written as SERGMs and SUGMs, and apply the models and techniques to network data from rural Indian villages. Expand

DERGMs: Degeneracy-restricted exponential random graph models

- Mathematics, Computer Science
- ArXiv
- 2016

A new exponential family of models for random graphs that build on the standard ERGM framework is proposed, and a new parameter based on the graph-theoretic notion of degeneracy, a measure of sparsity whose value is low in real-worlds networks is introduced. Expand

Estimation of exponential random graph models for large social networks via graph limits

- Computer Science, Mathematics
- 2013 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM 2013)
- 2013

A new theoretical framework for estimating the parameters of ERGM by approximating the normalizing constant using the emerging tools in graph theory-graph limits is proposed and a new matching method to find a starting point for the iterative algorithm is proposed. Expand

Sub-critical Exponential random graphs: concentration of measure and some applications

- Mathematics, Physics
- 2019

The exponential random graph model (ERGM) is a central object in the study of clustering properties in social networks as well as canonical ensembles in statistical physics. Despite some breakthrough… Expand

CONSISTENCY UNDER SAMPLING OF EXPONENTIAL RANDOM GRAPH MODELS.

- Mathematics, Medicine
- Annals of statistics
- 2013

It is shown that this apparently trivial condition is in fact violated by many popular and scientifically appealing models, and that satisfying it drastically limits ERGM's expressive power. Expand

A survey on exponential random graph models: an application perspective

- Medicine, Computer Science
- PeerJ Comput. Sci.
- 2020

This review paper can be used as an introduction for scientists from various disciplines whose aim is to use ERGMs in some networked data in their field of expertise. Expand

Exponential-Family Models of Random Graphs: Inference in Finite-, Super-, and Infinite Population Scenarios

- Mathematics, Computer Science
- 2017

The core statistical notions of "sample" and "population" in the ERGM framework are clarified, and the process that generates the population graph from the observation process is separated, and likelihood-based inference in finite-, super-, and infinite-population scenarios are reviewed. Expand

Limit theorems for exponential random graphs

- Mathematics
- 2021

We consider the edge-triangle model, a two-parameter family of exponential random graphs in which dependence between edges is introduced through triangles. In the so-called replica symmetric regime,… Expand

The birth of geometry in exponential random graphs

- Physics, Mathematics
- 2021

Inspired by the prospect of having discretized spaces emerge from random graphs, we construct a collection of simple and explicit exponential random graph models that enjoy, in an appropriate… Expand

An introduction to large deviations for random graphs

- Mathematics
- 2016

This article gives an overview of the emerging literature on large deviations for random graphs. Written for the general mathematical audience, the article begins with a short introduction to the… Expand

#### References

SHOWING 1-10 OF 67 REFERENCES

On the geometry of discrete exponential families with application to exponential random graph models

- Mathematics
- 2008

There has been an explosion of interest in statistical models for analyzing network data, and considerable interest in the class of exponential random graph (ERG) models, especially in connection… Expand

Metrics for sparse graphs

- Mathematics
- 2007

Recently, Bollob\'as, Janson and Riordan introduced a very general family of random graph models, producing inhomogeneous random graphs with $\Theta(n)$ edges. Roughly speaking, there is one model… Expand

Random graphs with a given degree sequence

- Mathematics
- 2011

Large graphs are sometimes studied through their degree sequences (power law or regular graphs). We study graphs that are uniformly chosen with a given degree sequence. Under mild conditions, it is… Expand

New Specifications for Exponential Random Graph Models

- Mathematics
- 2006

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… Expand

On exchangeable random variables and the statistics of large graphs and hypergraphs

- Mathematics
- 2008

De Finetti’s classical result of [18] identifying the law of an
exchangeable family of random variables as a mixture of i.i.d. laws was
extended to structure theorems for more complex notions of… Expand

Markov Chain Monte Carlo Estimation of Exponential Random Graph Models

- Computer Science
- J. Soc. Struct.
- 2002

This paper is about estimating the parameters of the exponential random graph model using frequentist Markov chain Monte Carlo (MCMC) methods, based on the Robbins-Monro algorithm for approximating a solution to the likelihood equation. Expand

Assessing Degeneracy in Statistical Models of Social Networks

- 2003

This paper presents recent advances in the statistical modeling of random graphs that have an impact on the empirical study of social networks. Statistical exponential family models (Wasserman and… Expand

On replica symmetry of large deviations in random graphs

- Mathematics, Computer Science
- Random Struct. Algorithms
- 2015

The replica symmetry phase consists of all p,r such that rd,hpr lies on the convex minorant of xi¾?hpx1/d where hp is the rate function of a binomial with parameter p. Expand

The large deviation principle for the Erdős-Rényi random graph

- Computer Science, Mathematics
- Eur. J. Comb.
- 2011

The formulation and proof of the main result uses the recent development of the theory of graph limits by Lovasz and coauthors and Szemeredi's regularity lemma from graph theory to establish a large deviation principle under an appropriate topology. Expand