A Tight Bound on Localization of Electrical Flows

2026-05-22

AI Summary

This work investigates tight upper bounds on the ℓ¹-norm of unit currents between random endpoints of edges in unweighted graphs and on the spectral norm of the absolute value of the symmetric transition current matrix in weighted graphs. By integrating tools from spectral graph theory, electrical network analysis, and matrix methods—augmented by AI-assisted exploration of proof strategies—the authors improve the previously known O(log²n) bound to a tight O(log n) bound and establish its optimality up to constant factors. Specifically, they prove that for any graph with n vertices, both quantities are bounded above by 2 log n, thereby characterizing the fundamental localization limit of current distributions in graphs.

Abstract

We prove that for any unweighted graph on n vertices the L1 norm of a unit electric current between the endpoints of a random edge is at most 2 log n. Furthermore, we show that on any weighted graph the spectral norm of the entry-wise absolute value of the symmetric transfer-current matrix is at most 2 log n. This bound is tight up to constants and improves the O(log^2 n) bound from [Schild-Rao-Srivastava, SODA '18]. The initial proofs were generated by OpenAI's ChatGPT 5.5 Pro; the authors have verified and rewritten them to enhance readability and provide additional context.

Problem

• electrical flows
• localization
• transfer-current matrix
• spectral norm
• graph theory

Innovation

• electrical flows
• localization bound
• transfer-current matrix
• spectral norm
• graph theory