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Stein's method of moment estimators for local dependency exponential random graph models

Adrian Fischer, Gesine Reinert, Wenkai Xu

TL;DR

This work tackles parameter estimation in exponential random graph models by introducing Stein estimators for local dependency ERGMs (LERGM). It constructs a Stein operator tailored to the LERGM and derives blockwise estimating equations that yield a computationally efficient estimator with theoretical guarantees. The authors establish nonasymptotic concentration bounds and asymptotic normality, with rates that depend on the largest block size and the number of blocks, and they demonstrate favorable performance relative to MLE, MPLE, and contrastive divergence in simulations. The framework provides explicit, verifiable conditions under which existence, uniqueness, concentration, and normal approximation hold, offering practical advantages for large or complex networks.

Abstract

Providing theoretical guarantees for parameter estimation in exponential random graph models is a largely open problem. While maximum likelihood estimation has theoretical guarantees in principle, verifying the assumptions for these guarantees to hold can be very difficult. Moreover, in complex networks, numerical maximum likelihood estimation is computer-intensive and may not converge in reasonable time. To ameliorate this issue, local dependency exponential random graph models have been introduced, which assume that the network consists of many independent exponential random graphs. In this setting, progress towards maximum likelihood estimation has been made. However the estimation is still computer-intensive. Instead, we propose to use so-called Stein estimators: we use the Stein characterizations to obtain new estimators for local dependency exponential random graph models.

Stein's method of moment estimators for local dependency exponential random graph models

TL;DR

This work tackles parameter estimation in exponential random graph models by introducing Stein estimators for local dependency ERGMs (LERGM). It constructs a Stein operator tailored to the LERGM and derives blockwise estimating equations that yield a computationally efficient estimator with theoretical guarantees. The authors establish nonasymptotic concentration bounds and asymptotic normality, with rates that depend on the largest block size and the number of blocks, and they demonstrate favorable performance relative to MLE, MPLE, and contrastive divergence in simulations. The framework provides explicit, verifiable conditions under which existence, uniqueness, concentration, and normal approximation hold, offering practical advantages for large or complex networks.

Abstract

Providing theoretical guarantees for parameter estimation in exponential random graph models is a largely open problem. While maximum likelihood estimation has theoretical guarantees in principle, verifying the assumptions for these guarantees to hold can be very difficult. Moreover, in complex networks, numerical maximum likelihood estimation is computer-intensive and may not converge in reasonable time. To ameliorate this issue, local dependency exponential random graph models have been introduced, which assume that the network consists of many independent exponential random graphs. In this setting, progress towards maximum likelihood estimation has been made. However the estimation is still computer-intensive. Instead, we propose to use so-called Stein estimators: we use the Stein characterizations to obtain new estimators for local dependency exponential random graph models.

Paper Structure

This paper contains 8 sections, 11 theorems, 106 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The local dependency exponential random graph model
  3. Stein's method of moments
  4. Convergence analysis
  5. Concentration bounds
  6. Asymptotic normality
  7. Auxiliary results for Section \ref{['section_standad_errors']}
  8. Auxiliary results for Section \ref{['section_asymptotic_normality']}

Key Result

Theorem 3.1

Let $\mathbf{X} \sim \mathrm{LERGM}(\beta)$ and $1 \le k \le l \le K$. Then, for all functions $f_{k,l}: \mathbb{X}_{k,l} \rightarrow \mathbb{R}$, $\mathbb{E}[\mathcal{A}_{\beta_{\bullet}}^{k,l} f_{k,l}(\mathbf{X}_{k,l})]=0$, and for all collections $f \in \mathcal{F}$, we have $\mathbb{E}[\mathcal{

Figures (1)

  • Figure 1: Estimated parameters for $\beta$, with $|A| = 20$, $K = 20$, and $30$ runs: red triangle SE, green triangle MLE, blue square CD, and black circle MPLE. The true parameters (True Param) are $\beta_W = (1.0, -1.0)$ and $\beta_B = (1.0, -1.0, -1.0)$ (magenta diamond).

Theorems & Definitions (36)

  • Definition 2.1: Exponential random graph model
  • Definition 2.2: Local dependency exponential random graph model
  • Theorem 3.1
  • proof
  • Definition 3.2: LERGM-Stein estimator
  • Example 3.3: Bernoulli random graph
  • Example 3.5: Bernoulli random graph
  • Proposition 3.6
  • proof
  • Example 3.8
  • ...and 26 more