Partial Resilient Leader-Follower Consensus in Time-Varying Graphs

TL;DR

This work tackles resilient leader-follower consensus over sequences of time-varying digraphs with $F$-local adversaries by relaxing global robustness requirements. It introduces the BP-MSR algorithm, which couples Bootstrap Percolation with MSR updates and uses the threshold $r=2F+1$ to gate participation, achieving partial resilient leader-follower consensus for a nonempty convergent set $\\mathcal{F}_{\\mathcal{C}}$. The authors prove that followers who are active infinitely often and belong to strongly $(2F+1)$-robust subgraphs converge to the leader's reference state, while others remain within the convex hull; full resilience is recovered under additional conditions such as all followers being active or time-window robustness. Simulations on non-robust graph sequences demonstrate partial convergence where full resilience would fail under traditional algorithms, highlighting the practical value of local participation and percolation-based gating in dynamic networks.

Abstract

This work studies resilient leader-follower consensus with a bounded number of adversaries. Existing approaches typically require robustness conditions of the entire network to guarantee resilient consensus. However, the behavior of such systems when these conditions are not fully met remains unexplored. To address this gap, we introduce the notion of partial leader-follower consensus, in which a subset of non-adversarial followers successfully tracks the leader's reference state despite insufficient robustness. We propose a novel distributed algorithm - the Bootstrap Percolation and Mean Subsequence Reduced (BP-MSR) algorithm - and establish sufficient conditions for individual followers to achieve consensus via the BP-MSR algorithm in arbitrary time-varying graphs. We validate our findings through simulations, demonstrating that our method guarantees partial leader-follower consensus, even when standard resilient consensus algorithms fail.

Figures (3)

• Figure 1: Three digraphs where blue, green, and red nodes represent normal leaders, normal followers, and adversaries, respectively, each under a $1$-local attack. Yellow-shaded subgraphs are strongly $3$-robust with respect to $\mathcal{L}_{\mathcal{M}}$, while the gray-shaded subgraph is strongly $3$-robust with respect to $\mathcal{L} \cup \mathcal{A}$.
• Figure 2: Consensus performance of (a) W-MSR, (b) SW-MSR ($T=2$), and (c) BP-MSR algorithms, all with $F=1$, under the graph sequence $\mathbb{G}=(\mathcal{G}[t])_{t\in \mathbb{Z}_{\geq 0}}$, where $\mathcal{G}[2\tau]=\mathcal{G}_1$ and $\mathcal{G}[2\tau+1]=\mathcal{G}_2$, for $\tau\in\mathbb{Z}_{\geq 0}$ (see \ref{['fig:sequence']} (a)-(b) for $\mathcal{G}_1$ and $\mathcal{G}_2$). Partial leader-follower consensus is achieved by followers $\{6,7,8\}$ only via the BP-MSR algorithm. For clarity, only the adversary state received by follower $5$ is plotted.
• Figure 3: Consensus performance of the BP-MSR algorithm under the periodic graph sequence $\mathbb{G}$, where $\mathcal{G}_1$, $\mathcal{G}_2$, and $\mathcal{G}_3$ from \ref{['fig:sequence']} repeat periodically. The adversary $0$ (a) always shares $q_0^u[t,k]=0$ for all time with out-neighbors $u\in \mathcal{N}_0^o[t]$, yielding the smallest $\mathcal{F}_{\mathcal{C}}$ possible, and (b) always shares $q_0^u[t,k]=1$ for all time, yielding the largest $\mathcal{F}_{\mathcal{C}}$ possible. For clarity, we plot only the adversary’s state received by follower 5.

Theorems & Definitions (21)

• Definition 1: Adversarial agent
• Definition 2: $\mathbf F$-local
• Definition 3
• Definition 4: $\mathbf r$-reachable LeBlanc13
• Definition 5: strongly $\mathbf r$-robust zhang2012
• Lemma 1: ICRA2025
• Remark 1
• Definition 6
• Remark 2
• Remark 3