Characterizing Trust and Resilience in Distributed Consensus for Cyberphysical Systems

TL;DR

This work addresses resilient distributed consensus in multi-agent systems by exploiting stochastic trust signals between agents, modeled as $\alpha_{ij}\in[0,1]$. The authors develop a unified probabilistic framework that uses trust observations to form a trust-aware weight matrix $W(t)$, enabling convergence even when malicious agents exceed classical tolerances and providing probabilistic bounds on deviation from nominal consensus. Key contributions include almost-sure convergence with arbitrary numbers of adversaries under conditions on the trust expectations $c=E(\alpha_{ij}-\tfrac{1}{2})<0$ and $d=E(\alpha_{ij}-\tfrac{1}{2})>0$, finite-time correct classification of malicious nodes, and exponential decay rates for convergence errors tied to the second-largest eigenvalue modulus $\rho_2$ of the limit weight matrix $\overline{W}_{\mathcal{L}}$. The work also shows how additional information on trust variance can tighten probabilistic bounds via refined concentration inequalities, illustrating the practical impact of leveraging physical-layer trust in CPS for robust coordination.

Abstract

This work considers the problem of resilient consensus where stochastic values of trust between agents are available. Specifically, we derive a unified mathematical framework to characterize convergence, deviation of the consensus from the true consensus value, and expected convergence rate, when there exists additional information of trust between agents. We show that under certain conditions on the stochastic trust values and consensus protocol: 1) almost sure convergence to a common limit value is possible even when malicious agents constitute more than half of the network connectivity, 2) the deviation of the converged limit, from the case where there is no attack, i.e., the true consensus value, can be bounded with probability that approaches 1 exponentially, and 3) correct classification of malicious and legitimate agents can be attained in finite time almost surely. Further, the expected convergence rate decays exponentially as a function of the quality of the trust observations between agents.

Figures (8)

• Figure 1: A schematic of a multi-robot system with a consensus protocol modified to use agent transmitted values $x_i$ and $x_j$ as well as observations of inter-agent trust $\alpha_{ij}$.
• Figure 2: A schematic showing the deviation from the nominal average in the case of a malicious agent attack and no resilience in the consensus protocol.
• Figure 3: A depiction of the $\beta_{ij}(t)$ values for legitimate and malicious nodes becoming more separated and accurately classifiable with higher probability as a longer history of $\alpha_{ij}(t)$ becomes available (i.e. as $t$ gets larger).
• Figure 4: The undirected subgraph $\mathbb{G}$ of legitimate and malicious agents. Two agents are neighbors if they are connected by an edge. Legitimate agents are depicted by blue nodes and malicious agents are depicted by a red node. Edges between legitimate agents are depicted by black solid lines. Edges between legitimate and malicious agents are depicted by red dashed lines.
• Figure 5: This figure depicts the agent value with the maximal deviation from true consensus value for a single system realization for a system with 15 malicious agents, $\ell=0.4$ and maximal deviation malicious inputs.

Theorems & Definitions (25)

• Definition 2.1: Malicious agent
• Definition 2.2: $\alpha_{ij}$
• Remark : Arbitrary starting time $T_0$ of the consensus.
• Definition 2.3: Convergence of the consensus protocol
• Definition 3.1: Primitive matrix
• Lemma 1
• Lemma 2
• Corollary 1: Bounds on Expectation of $\alpha_{ij}(t)$
• Proposition 1
• Proposition 2: Convergence of legitimate agents' values