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
Concentration Inequalities for Bounded Functionals via Log-Sobolev-Type Inequalities
TL;DR: In this paper, the authors prove multilevel concentration inequalities for bounded functionals for both dependent and independent random variables, including deviation inequalities for empirical processes and homogeneous chaos in bounded random variables.
Proceedings Article
A deterministic partition function approximation for exponential random graph models
TL;DR: A new quadratic time deterministic approximation to ERGM partition functions is introduced, enabling subgraph statistics to derive a lower bound for partition functions given that the model is not dominated by a few graphs.
Journal ArticleDOI
Finite-size effects in exponential random graphs
Alexander Gorsky,O. Valba +1 more
TL;DR: Numerically the strong finite-size effects in exponential random graphs are shown and it is found that there exists the critical value of number of nodes when the ground state undergoes clear-cut crossover.
Journal ArticleDOI
Equilibrium homophily in networks
TL;DR: In this paper, the equilibrium effects of homophily in network formation games having positive externalities were studied. And they showed that very weak preferences for network effects (such as a preference for being part of a clique) may result in very high levels of homomorphism at equilibrium.
Journal ArticleDOI
Elusive extremal graphs
TL;DR: This paper presents a counterexample to Lovasz's conjecture that every finite feasible set of subgraph density constraints can be extended further by a finite set of density constraints so that the resulting set is satisfied by an asymptotically unique graph.
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