Laplacian Network Optimization via Information Functions

TL;DR

This paper introduces a theoretical framework for Laplacian-based network optimization using information functions, enabling efficient algorithms for improving network robustness and performance metrics.

Contribution

It develops a unified approach to analyze and optimize spectral network measures through a new class of information functions called Kiefer's measures.

Findings

Unified treatment of gradients and subgradients in Laplacian optimization

Derived efficient rank-one update formulas for spectral criteria

Devised a new edge-exchange algorithm with reduced complexity

Abstract

Designing networks to optimize robustness and other performance metrics is a well-established problem with applications ranging from electrical engineering to communication networks. Many such performance measures rely on the Laplacian spectrum; notable examples include total effective resistance, the number of spanning trees, and algebraic connectivity. This paper advances the study of Laplacian-based network optimization by drawing on ideas from experimental design in statistics. We present a theoretical framework for analyzing performance measures by introducing the notion of information functions, which captures a set of their desirable properties. Then, we formulate a new parametric family of information functions, Kiefer's measures, which encompasses the three most common spectral objectives. We provide a regular reformulation of the Laplacian optimization problem, and we use this reformulation to compute directional derivatives of Kiefer's measures. The directional derivatives provide a unified treatment of quantities recurring in Laplacian optimization, such as gradients and subgradients, and we show that they are connected to Laplacian-based measures of node distance, which we call node dissimilarities. We apply the node dissimilarities to derive efficient rank-one update formulas for Kiefer's criteria, and to devise a new edge-exchange method for network optimization. These update formulas enable greedy and exchange algorithms with reduced asymptotic time complexity.