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Decision-Theoretic Robustness for Network Models

Marios Papamichalis, Regina Ruane, Simon Lunagomez, Swati Chandna

TL;DR

This work studies local decision-theoretic robustness by allowing the posterior to vary within a small Kullback-Leibler neighborhood and choosing actions that minimize worst-case posterior expected loss, and develops a nonparametric minimax theory for robust model selection between sparse Erdos-Renyi and block models.

Abstract

Bayesian network models (Erdos Renyi, stochastic block models, random dot product graphs, graphons) are widely used in neuroscience, epidemiology, and the social sciences, yet real networks are sparse, heterogeneous, and exhibit higher-order dependence. How stable are network-based decisions, model selection, and policy recommendations to small model misspecification? We study local decision-theoretic robustness by allowing the posterior to vary within a small Kullback-Leibler neighborhood and choosing actions that minimize worst-case posterior expected loss. Exploiting low-dimensional functionals available under exchangeability, we (i) adapt decision-theoretic robustness to exchangeable graphs via graphon limits and derive sharp small-radius expansions of robust posterior risk; under squared loss the leading inflation is controlled by the posterior variance of the loss, and for robustness indices that diverge at percolation/fragmentation thresholds we obtain a universal critical exponent describing the explosion of decision uncertainty near criticality. (ii) Develop a nonparametric minimax theory for robust model selection between sparse Erdos-Renyi and block models, showing-via robustness error exponents-that no Bayesian or frequentist method can uniformly improve upon the decision-theoretic limits over configuration models and sparse graphon classes for percolation-type functionals. (iii) Propose a practical algorithm based on entropic tilting of posterior or variational samples, and demonstrate it on functional brain connectivity and Karnataka village social networks.

Decision-Theoretic Robustness for Network Models

TL;DR

This work studies local decision-theoretic robustness by allowing the posterior to vary within a small Kullback-Leibler neighborhood and choosing actions that minimize worst-case posterior expected loss, and develops a nonparametric minimax theory for robust model selection between sparse Erdos-Renyi and block models.

Abstract

Bayesian network models (Erdos Renyi, stochastic block models, random dot product graphs, graphons) are widely used in neuroscience, epidemiology, and the social sciences, yet real networks are sparse, heterogeneous, and exhibit higher-order dependence. How stable are network-based decisions, model selection, and policy recommendations to small model misspecification? We study local decision-theoretic robustness by allowing the posterior to vary within a small Kullback-Leibler neighborhood and choosing actions that minimize worst-case posterior expected loss. Exploiting low-dimensional functionals available under exchangeability, we (i) adapt decision-theoretic robustness to exchangeable graphs via graphon limits and derive sharp small-radius expansions of robust posterior risk; under squared loss the leading inflation is controlled by the posterior variance of the loss, and for robustness indices that diverge at percolation/fragmentation thresholds we obtain a universal critical exponent describing the explosion of decision uncertainty near criticality. (ii) Develop a nonparametric minimax theory for robust model selection between sparse Erdos-Renyi and block models, showing-via robustness error exponents-that no Bayesian or frequentist method can uniformly improve upon the decision-theoretic limits over configuration models and sparse graphon classes for percolation-type functionals. (iii) Propose a practical algorithm based on entropic tilting of posterior or variational samples, and demonstrate it on functional brain connectivity and Karnataka village social networks.
Paper Structure (61 sections, 16 theorems, 371 equations, 12 figures, 4 tables, 2 algorithms)

This paper contains 61 sections, 16 theorems, 371 equations, 12 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Object--level versus decision--level robustness.
  3. Network functionals, critical behavior, and decision--level uncertainty.
  4. Background and setup
  5. Exchangeable network models and graph limits
  6. Graphon KL neighborhoods.
  7. Local perturbations inside the KL ball.
  8. Bayesian modelling for networks
  9. Parametric models.
  10. Nonparametric and graphon priors.
  11. Baseline posterior and pseudo--posterior inference.
  12. General decision--theoretic robustness theory for network functionals
  13. Decision--theoretic robustness for network functionals
  14. Decision problems driven by network functionals
  15. Generic critical exponent for fragmentation--type indices
  16. ...and 46 more sections

Key Result

Theorem 3.3

Suppose Assumptions ass:critical-R and ass:local-BvM hold. Let $\Delta_0:=\Delta(\theta_0)>0$ be the distance of the true network to the fragmentation threshold, and allow $\Delta_0=\Delta_0^{(n)}\downarrow 0$ with $\Delta_0^{(n)}\gg r_n$ as $n\to\infty$. Assume, moreover, that: Then:

Figures (12)

  • Figure 1: Synthetic small--radius robustness sensitivity curves. (a) ER vs. two--block SBM test with $n=400$, $c=3$, $\lambda=0.4$: normalized excess robust misclassification probability $(R_{\mathrm{rob},n}(C)-R_{0,n})/ (\sqrt{2 R_{0,n}(1-R_{0,n})}\sqrt{C})$ versus $\sqrt{C}$. (b) Configuration model with Poisson degrees of mean $1-\Delta$ ($n=5000$, $\Delta=0.2$): normalized excess robust susceptibility $(\rho_{\mathrm{rob},n}(C)-\rho_{0,n})/(\rho_{0,n}\sqrt{C})$ versus $\sqrt{C}$. Dashed horizontal lines mark reference levels corresponding to the leading $\sqrt{C}$ coefficients predicted by Theorem S.2 (panel a) and Theorem 3.3 (panel b).
  • Figure 2: Configuration--model susceptibility: log--log scaling in the distance to criticality. Left: $-\log\Delta \mapsto \log(n\,\rho_{0,n})$ with fitted least--squares line (dashed); right: $-\log\Delta \mapsto \log\bigl(n(\rho_{\mathrm{rob},n}(C)-\rho_{0,n})/\sqrt{C}\bigr)$ at $C\approx 10^{-3}$. The regression slopes $\widehat{\kappa}_{\mathrm{base}} \approx 4.49$ and $\widehat{\kappa}_{\mathrm{rob}} \approx 4.65$ are close to the theoretical exponent 4 from Theorem \ref{['thm:critical-exponent']}.
  • Figure 3: Misspecification stress test (DCSBM truth, SBM working model). Robustification is performed by exponential tilting within a KL ball of radius $C$ around the working posterior draws. Left: Estimation performance (MSE) vs. KL radius $C$ for $\lambda_1$. Right: Threshold-decision regret vs. KL radius $C$; robustification lowers regret by hedging against false negatives under SBM misspecification.
  • Figure 4: A) Left Up: Local $\sqrt{C}$ normalization. Deviations at large $n$ and strong $\lambda$ indicate a nonlocal radius grid. B) Right Up: Exponent estimate $-\log(R)/n$ vs. $n$ for baseline and $\mathcal{C}_n=\exp(-2\alpha n)$. C) Down:Effective exponent across radius scalings (exponent collapse under polynomial/constant radii).Synthetic ER vs. SBM validation of locality and exponent behavior under posterior KL-robustification. The key qualitative prediction is that exponentially shrinking radii can preserve an exponential rate, while polynomial/constant radii destroy the exponent.
  • Figure 5: Brain connectivity experiment, SBM($K_1$) vs SBM($K_2$). Left: normalized robust risk increase $(\rho_{\text{rob}}(C)-\rho_0)/\sqrt{C}$ for the model--selection decision in a representative scan, plotted against $\sqrt{C}$. Right: observed small--world index $S$ versus tempered posterior probability $p_{\mathrm{SBM}(K_2),\tau}$ across scans.
  • ...and 7 more figures

Theorems & Definitions (30)

  • Theorem 3.3: Robust critical exponent for fragmentation--type indices
  • Definition 3.4: $\phi$--divergence and $\phi$--ball
  • Remark 3.5: Local equivalence of KL and $\phi$--balls
  • Remark 3.6: Extension of decision--theoretic robustness exponents
  • Lemma 4.1: KL divergence and per--vertex information
  • Lemma 4.2: Chernoff information and error exponent
  • Remark 4.3: Small--signal information exponents
  • Lemma 4.4: Information and decision--theoretic robustness noise indices under graphon representation
  • Theorem 4.5: Decision--theoretic robust Bayes factor testing for graphons
  • Theorem 4.6: Nonparametric graphon decision--theoretic robustness minimax testing
  • ...and 20 more