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An information-theoretic proof of the Shannon-Hagelbarger theorem

Venkat Anantharam

TL;DR

This work provides an information-theoretic proof of the Shannon-Hagelbarger theorem by leveraging the Gaussian free field associated with a resistive network: edge resistances $R_e$ induce independent Gaussians with variance $R_e$, and conditioning on circuit sums yields a Gaussian free field whose pairwise differences $U_{a\to b}$ have variance equal to the effective resistance $R^{\rm eff}_{ab}$. By introducing independent edge-variance perturbations $(\bar{R}_e)$ and forming $\hat{X}_e=X_e+\bar{X}_e$, the paper shows through entropic arguments that $\hat{R}^{\rm eff}_{ab} \ge R^{\rm eff}_{ab}+\bar{R}^{\rm eff}_{ab}$, which implies concavity under linear scaling of edge resistances. The approach extends to determinants of cross effective resistances, using Minkowski's determinantal inequality to obtain concavity of the $n$-th root of $\det R^{\rm eff}$ for $n$ ordered pairs. Overall, the work reveals a natural information-theoretic structure in a classical circuit-theoretic result and suggests broad generalizations to matrix-functionals of resistive networks.

Abstract

The Shannon-Hagelbarger theorem states that the effective resistance across any pair of nodes in a resistive network is a concave function of the edge resistances. We give an information-theoretic proof of this result, building on the theory of the Gaussian free field. This also allows us to prove an extension of the result to determinants of matrices of cross effective resistances.

An information-theoretic proof of the Shannon-Hagelbarger theorem

TL;DR

This work provides an information-theoretic proof of the Shannon-Hagelbarger theorem by leveraging the Gaussian free field associated with a resistive network: edge resistances induce independent Gaussians with variance , and conditioning on circuit sums yields a Gaussian free field whose pairwise differences have variance equal to the effective resistance . By introducing independent edge-variance perturbations and forming , the paper shows through entropic arguments that , which implies concavity under linear scaling of edge resistances. The approach extends to determinants of cross effective resistances, using Minkowski's determinantal inequality to obtain concavity of the -th root of for ordered pairs. Overall, the work reveals a natural information-theoretic structure in a classical circuit-theoretic result and suggests broad generalizations to matrix-functionals of resistive networks.

Abstract

The Shannon-Hagelbarger theorem states that the effective resistance across any pair of nodes in a resistive network is a concave function of the edge resistances. We give an information-theoretic proof of this result, building on the theory of the Gaussian free field. This also allows us to prove an extension of the result to determinants of matrices of cross effective resistances.
Paper Structure (5 sections, 6 equations, 1 figure)

This paper contains 5 sections, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Existing proofs
  3. A probabilistic point of view
  4. An information-theoretic proof
  5. Concluding remarks

Figures (1)

  • Figure 1: A resistive network with vertex set $V = \{1, 2, 3, 4\}$ and six edges, the resistance of each of which is indicated. The effective resistances can be computed to be $R^{\rm eff}_{12} = \frac{11}{17}$, $R^{\rm eff}_{13} = \frac{11}{17}$, $R^{\rm eff}_{14} = \frac{29}{17}$, $R^{\rm eff}_{23} = \frac{10}{17}$, $R^{\rm eff}_{24} = \frac{22}{17}$, and $R^{\rm eff}_{34} = \frac{24}{17}$.