Weighted Exponential Random Graph Models: Scope and Large Network Limits
We study models of weighted exponential random graphs in the large network limit. These models have recently been proposed to model weighted network data arising from a host of applications including socio-econometric data such as migration flows and neuroscience. Analogous to fundamental results derived for standard (unweighted) exponential random graph models in the work of Chatterjee and Diaconis, we derive limiting results for the structure of these models as the number of nodes goes to infinity. Our results are applicable for a wide variety of base measures including measures with unbounded support. We also derive sufficient conditions for continuity of functionals in the specification of the model including conditions on nodal covariates. Finally we include a number of open problems to spur further understanding of this model especially in the context of applications.
KeywordsExponential random graph models Weighted graph Large deviations Random networks Dense graph limits Graphons Markov Chain Monte Carlo
Mathematics Subject ClassificationPrimary: 60C05 05C80
SB and SuC have been supported by NSF DMS-1613072, DMS-1606839 and ARO Grant W911NF-17-1-0010. SB, SC and BD have been partially supported by NSF SES Grant 1357622. SC has been partially supported by NSF SES-1461493, and SES-1514750. BD has been partially supported by NSF grants SES-1558661, SES-1619644, SES-1637089, CISE-1320219, SMA-1360104. We would like to thank Mathew Denny, James Wilson and Sayan Banerjee for illuminating discussions on the relevance of the results in this paper for applications. We thank four referees for reading the paper closely and providing many valuable suggestions.
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