[Paper Review] Resilient Continuous-Time Consensus in Fractional Robust Networks

Heath J. LeBlanc, Haotian Zhang+2 more|arXiv (Cornell University)|Mar 11, 2013

Distributed Control Multi-Agent Systems22 references4 citations

TL;DR

This paper proposes a continuous-time resilient consensus protocol, ARC-P, for multi-agent networks under fractional adversary models where up to a fraction $f$ of each normal agent's neighbors may be adversarial. Using a novel graph-theoretic metric called fractional robustness, the authors prove that consensus is achieved if the network is $2f$-fraction robust (sufficient condition) and necessary if $f$-fraction robust, under time-varying topologies with dwell time constraints.

ABSTRACT

In this paper, we study the continuous-time consensus problem in the presence of adversaries. The networked multi-agent system is modeled as a switched system, where the normal agents have integrator dynamics and the switching signal determines the topology of the network. We consider several models of omniscient adversaries under the assumption that at most a fraction of any normal agent's neighbors may be adversaries. Under this fractional assumption on the interaction between normal and adversary agents, we show that a novel graph theoretic metric, called fractional robustness, is useful for analyzing the network topologies under which the normal agents achieve consensus.

Motivation & Objective

• Address the lack of topological conditions for resilient consensus in continuous-time systems under fractional adversary models.
• Overcome limitations of traditional graph metrics like connectivity, which fail to capture redundancy in information flow under extreme value removal.
• Develop a continuous-time version of the MSR algorithm (ARC-P) that operates under local information and fractional adversary assumptions.
• Establish necessary and sufficient conditions for resilient consensus using a fractional robustness metric tailored to time-varying network topologies.
• Analyze convergence under three adversary models: Byzantine, malicious, and crash, under the assumption that at most a fraction $f$ of neighbors are compromised.

Proposed method

• Model the network as a switched system with time-varying digraphs $\mathcal{D}(t)$ governed by a piecewise constant switching signal $\sigma(t)$.
• Adapt the Adversarial Robust Consensus Protocol (ARC-P) to continuous time, where each normal agent updates its state based on a weighted average of its neighbors’ states after removing extreme values.
• Introduce a fractional robustness metric, defined as $p$-fraction robustness, where $p > 2f$, to characterize network resilience under the $f$-fraction adversary model.
• Use a contradiction argument based on the evolution of the maximum and minimum normal agent states over discrete time intervals $\Delta$, tracking the reduction in the set of agents with extreme values.
• Apply a dwell time assumption ($t_{k+1} - t_k \geq \tau$) to ensure sufficient time for convergence between topology switches.
• Define $\Psi(t) = M_{\mathcal{N}}(t) - m_{\mathcal{N}}(t)$ as the spread between the maximum and minimum values among normal agents, and show that $\Psi(t)$ strictly decreases over time intervals under the robustness condition.

Experimental results

Research questions

• RQ1 What topological condition ensures resilient consensus in continuous-time multi-agent systems when adversaries constitute a fraction $f$ of each normal agent’s neighbors?
• RQ2 How does the fractional robustness metric relate to the convergence of the ARC-P protocol under time-varying network topologies?
• RQ3 Can necessary and sufficient conditions for resilient consensus be established under the $f$-fraction adversary model for continuous-time systems?
• RQ4 How do different adversary models—Byzantine, malicious, and crash—affect the resilience of the consensus protocol under the fractional threat assumption?
• RQ5 What role does the dwell time between topology switches play in ensuring convergence under time-varying networks?

Key findings

• The ARC-P protocol achieves resilient consensus in continuous time if the network is $2f$-fraction robust, which is a sufficient condition for convergence.
• A necessary condition for resilient consensus is that the network must be $f$-fraction robust, establishing tightness of the bound under the given adversary model.
• The protocol ensures that the spread $\Psi(t) = M_{\mathcal{N}}(t) - m_{\mathcal{N}}(t)$ between the maximum and minimum values among normal agents strictly decreases over time intervals $\Delta$, leading to consensus.
• For time-varying networks, convergence is guaranteed if the network is $p$-fraction robust with $2f < p \leq 1$ for all $t \geq t'$, provided the switching intervals are bounded below by a dwell time $\tau$.
• The protocol maintains resilience even when adversaries are omniscient and can adaptively manipulate values, as long as the fraction of compromised neighbors per normal agent is bounded by $f < 1/2$.
• The analysis shows that the number of agents with extreme values ($\mathcal{X}_M^j$ and $\mathcal{X}_m^j$) decreases over time, ensuring that $\Psi(t)$ eventually drops below any $\epsilon > 0$, implying consensus.