High Dimensional Robust Consensus over Networks with Limited Capacity

TL;DR

This paper studies how to achieve robust consensus in high-dimensional networks with limited communication capacity by optimally assigning edges to state dimensions and allocating additional capacity, ensuring small steady state variance.

Contribution

It introduces a method for optimal edge-to-dimension assignment in capacity-constrained networks and formulates capacity allocation as a submodular maximization problem.

Findings

Small steady state variance requires large network capacity.

Optimal edge-dimension assignment improves consensus robustness.

Capacity augmentation can be effectively approximated using greedy algorithms.

Abstract

We investigate robust linear consensus over networks under capacity-constrained communication. The capacity of each edge is encoded as an upper bound on the number of state variables that can be communicated instantaneously. When the edge capacities are small compared to the dimensionality of the state vectors, it is not possible to instantaneously communicate full state information over every edge. We investigate how robust consensus (small steady state variance of the states) can be achieved within a linear time-invariant setting by optimally assigning edges to state-dimensions. We show that if a finite steady state variance of the states can be achieved, then both the minimum cut capacity and the total capacity of the network should be sufficiently large. Optimal and approximate solutions are provided for some special classes of graphs. We also consider the related problem of optimally allocating additional capacity on a feasible initial solution. We show that this problem corresponds to the maximization of a submodular function subject to a matroid constraint, which can be approximated via a greedy algorithm.