Abstract
We present a technique for approximating generic normalization constants subject to constraints. The method is then applied to derive the exact asymptotics for the conditional normalization constant of constrained exponential random graphs.
Highlights
Exponential random graph models are widely used to characterize the structure and behavior of real-world networks as they are able to predict the global structure of the networked system based on a set of tractable local features
The method is applied to derive the exact asymptotics for the conditional normalization constant of constrained exponential random graphs
Based on a large deviation principle for Erdos-Rényi graphs established in Chatterjee and Varadhan [6], Chatterjee and Diaconis [5] developed an asymptotic approximation for the normalization constant ψNζ as N → ∞ and connected the occurrence of a phase transition in the dense exponential model with the non-analyticity of the asymptotic limit of ψNζ
Summary
Exponential random graph models are widely used to characterize the structure and behavior of real-world networks as they are able to predict the global structure of the networked system based on a set of tractable local features. Based on a large deviation principle for Erdos-Rényi graphs established in Chatterjee and Varadhan [6], Chatterjee and Diaconis [5] developed an asymptotic approximation for the normalization constant ψNζ as N → ∞ and connected the occurrence of a phase transition in the dense exponential model with the non-analyticity of the asymptotic limit of ψNζ. Using the large deviation principle established in Chatterjee and Varadhan [6] and Chatterjee and Diaconis [5], we developed an asymptotic approximation for the conditional normalization constant ψNe,ζ,t as N → ∞ and t → 0, since it is in this limit that interesting singular behavior occurs [7] This approximation suffers from the same problem: the error bound on ψNe,ζ,t is of the order of some negative power of log∗ N and the method does not lead to an exact limit for ψNe,ζ,t in the sparse setting. Due to the imposed constraint, instead of working with a generic normalization constant of the form (1.7) as in Chatterjee and Dembo [4], we will work with a generic conditional normalization constant in Theorem 3.1 and apply this result to derive a concrete error bound for the conditional normalization constant ψNe,ζ,t of constrained exponential random graphs in Theorems 4.1 and 4.2
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