Discrete-Time Consensus Networks: Scalability, Grounding and Countermeasures

TL;DR

This paper studies how grounding a node in discrete-time consensus networks impacts their scalability and performance, showing that it can deteriorate connectivity and proposing countermeasures to mitigate these effects.

Contribution

It analyzes the effects of grounding on expander graphs and introduces strategies for selecting nodes to ground to restore consensus.

Findings

Grounding reduces algebraic connectivity and eigenratio.

Grounding can cause loss of consensusability.

Countermeasures can partially recover network performance.

Abstract

We investigate the disruption of discrete-time consensus problems via grounding. Loosely speaking, grounding a network occurs if the state of one agent no longer responds to inputs from other agents and/or changes its dynamics. Then, the agent becomes a leader or a so-called stubborn agent. The disruption of the agent can be caused by internal faults, safety protocols or due to an external malicious attack. In this paper we investigate how grounding affects expander graph families that usually exhibit good scaling properties with increasing network size. It is shown that the algebraic connectivity and eigenratio of the network decrease due to the grounding causing the performance and scalability of the network to deteriorate, even to the point of losing consensusability. We then present possible countermeasures to such disruptions and discuss their practicality and limitations. In particular, for a specific countermeasure of deliberately grounding additional nodes, we investigate extensively how to select additional nodes to ground and how many nodes we need to ground to recover the consensus performance. Our findings are supported by a wide range of numerical simulations.