Estimating and understanding exponential random graph models
Sourav Chatterjee,Persi Diaconis +1 more
Reads0
Chats0
TLDR
In this paper, the authors introduce a method for the theoretical analysis of exponential random graph models based on a large deviation approximation to the normalizing constant shown to be consistent using theory developed by Chatterjee and Varadhan [European J. Combin. 32 (2011) 1000-1017].Abstract:
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 of “degeneracy” observed in these models. Here, almost all graphs have essentially no edges or are essentially complete. We supplement recent work of Bhamidi, Bresler and Sly [2008 IEEE 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS) (2008) 803–812 IEEE] showing that for many models, the extra sufficient statistics are useless: most realizations look like the results of a simple Erdős–Renyi model. We also find classes of models where the limiting graphs differ from Erdős–Renyi graphs. A limitation of our approach, inherited from the limitation of graph limit theory, is that it works only for dense graphs.read more
Citations
More filters
Journal ArticleDOI
Weighted Exponential Random Graph Models: Scope and Large Network Limits
TL;DR: Limiting results for the structure of these models as the number of nodes goes to infinity are derived, applicable for a wide variety of base measures including measures with unbounded support.
Posted Content
Motif Estimation via Subgraph Sampling: The Fourth Moment Phenomenon
TL;DR: This paper develops a framework for statistical inference for counting network motifs, such as edges, triangles, and wedges, in the widely used subgraph sampling model, and derives the exact thresholds for consistency and asymptotic normality of the HT estimator in various natural graph ensembles.
Journal ArticleDOI
Logarithmic Sobolev inequalities for finite spin systems and applications
Holger Sambale,Arthur Sinulis +1 more
TL;DR: In this article, sufficient conditions for a probability measure on a finite product space (a spin system) to satisfy a (modified) logarithmic Sobolev inequality were derived for various examples, such as the (vertex-weighted) exponential random graph model, the random coloring and the hard core model with fugacity.
Journal ArticleDOI
A Framework for Reconstructing Archaeological Networks Using Exponential Random Graph Models
TL;DR: A general framework in which exponential random graph models are combined with archaeological substantiations of mechanisms that may be responsible for network formation is presented, which may be applied to infer the structure of ancient networks in a large variety of archaeological settings.
Posted Content
Bipodal structure in oversaturated random graphs
TL;DR: It is proved that, for all but finitely many values of the edge density, the typical large graph is bipodal with parameters varying analytically with the densities.
References
More filters
Book
Social Network Analysis: Methods and Applications
TL;DR: This paper presents mathematical representation of social networks in the social and behavioral sciences through the lens of Dyadic and Triadic Interaction Models, which describes the relationships between actor and group measures and the structure of networks.
Journal ArticleDOI
A Stochastic Approximation Method
Herbert Robbins,Sutton Monro +1 more
TL;DR: In this article, a method for making successive experiments at levels x1, x2, ··· in such a way that xn will tend to θ in probability is presented.
Journal ArticleDOI
Statistical Analysis of Non-Lattice Data
TL;DR: In this article, a fixed system of n sites, labelled by the first n positive integers, and an associated vector x of observations, Xi,..., Xn, which, in turn, is assumed to be a realization of a vector X of (dependent) random variables, Xi,.., Xn, X.. In practice the sites may represent points or regions in space and the random variables may be either continuous or discrete.
Journal ArticleDOI
An Exponential Family of Probability Distributions for Directed Graphs
Paul W. Holland,Samuel Leinhardt +1 more
TL;DR: An exponential family of distributions that can be used for analyzing directed graph data is described, and several special cases are discussed along with some possible substantive interpretations.
Related Papers (5)
An Exponential Family of Probability Distributions for Directed Graphs
Paul W. Holland,Samuel Leinhardt +1 more