Estimating and understanding exponential random graph models
Sourav Chatterjee,Persi Diaconis +1 more
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
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Bayesian exponential random graph models with nodal random effects
TL;DR: This work extends the well-known and widely used exponential random graph model (ERGM) by including nodal random effects to compensate for heterogeneity in the nodes of a network by developing an approximate but feasible calculation of the Bayes factor which allows one to pursue model selection.
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How random are complex networks.
Chiara Orsini,Marija Mitrović Dankulov,Almerima Jamakovic,Priya Mahadevan,Pol Colomer-de-Simón,Amin Vahdat,Kevin E. Bassler,Zoltán Toroczkai,Marián Boguñá,Guido Caldarelli,Santo Fortunato,Dmitri Krioukov +11 more
TL;DR: This work considers six real networks and finds that many important local and global structural properties of these networks are closely reproduced by $dk$-random graphs whose degree distributions, degree correlations, and clustering are as in the corresponding real network.
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Asymptotic Structure of Graphs with the Minimum Number of Triangles
TL;DR: This work describes the asymptotic structure of extremal graphs by characterizing the set of flag algebra homomorphisms that minimize the triangle density by considering the problem of minimizing the number of triangles in a graph of given order and size.
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Multipodal Structure and Phase Transitions in Large Constrained Graphs
TL;DR: In this paper, the authors studied the asymptotics of large, simple, labeled graphs constrained by the densities of two subgraphs and showed that such graphs are multipodal.
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Stochastic Weighted Graphs: Flexible Model Specification and Simulation
TL;DR: The generalized exponential random graph model (GERGM) as mentioned in this paper is a recently proposed method used to simulate and model the edges of a weighted graph, which can be used to avoid likelihood degeneracy and efficiently capture network structure.
References
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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.
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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.
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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.
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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.
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