Growing Linear Consensus Networks Endowed by Spectral Systemic Performance Measures

TL;DR

This paper introduces a spectral systemic performance measure for designing noisy linear consensus networks, providing new algorithms and theoretical limits for optimizing network performance efficiently.

Contribution

It defines a new class of spectral performance measures and develops polynomial-time algorithms for network growth optimization based on these measures.

Findings

Several existing performance measures are shown to belong to the new class.

Two polynomial-time algorithms are proposed for network synthesis.

Theoretical limits on optimal performance are derived.

Abstract

We propose an axiomatic approach for design and performance analysis of noisy linear consensus networks by introducing a notion of systemic performance measure. This class of measures are spectral functions of Laplacian eigenvalues of the network that are monotone, convex, and orthogonally invariant with respect to the Laplacian matrix of the network. It is shown that several existing gold-standard and widely used performance measures in the literature belong to this new class of measures. We build upon this new notion and investigate a general form of combinatorial problem of growing a linear consensus network via minimizing a given systemic performance measure. Two efficient polynomial-time approximation algorithms are devised to tackle this network synthesis problem: a linearization-based method and a simple greedy algorithm based on rank-one updates. Several theoretical fundamental limits on the best achievable performance for the combinatorial problem is derived that assist us to evaluate optimality gaps of our proposed algorithms. A detailed complexity analysis confirms the effectiveness and viability of our algorithms to handle large-scale consensus networks.