On Robustness of Consensus over Pseudo-Undirected Path Graphs

2025-09-24

AI Summary

Traditional consensus algorithms confine the consensus value to the convex hull of initial states under undirected/directed graphs and suffer from limited robustness. To address this, we propose a pseudo-undirected graph topology framework: it preserves bidirectional connectivity among nodes while permitting asymmetric and even negative edge weights. Leveraging the asymmetric Laplacian matrix and stability theory, we derive bounded admissibility conditions for negative weights on path graphs, guaranteeing consensus convergence. Theoretically, this framework enables stable consensus values *outside* the initial state convex hull—significantly expanding the solution space. Experimental validation demonstrates its effectiveness in mobile target synchronized interception scenarios. Our core contribution lies in breaking the geometric constraint imposed by conventional graph structures on consensus values, thereby achieving both topological flexibility and enhanced system robustness.

Technology Category

• Search and Optimization: Non-convex Optimization
• Intelligent Robots: State Estimation
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Consensus over networked agents is typically studied using undirected or directed communication graphs. Undirected graphs enforce symmetry in information exchange, leading to convergence to the average of initial states, while directed graphs permit asymmetry but make consensus dependent on root nodes and their influence. Both paradigms impose inherent restrictions on achievable consensus values and network robustness. This paper introduces a theoretical framework for achieving consensus over a class of network topologies, termed pseudo-undirected graphs, which retains bidirectional connectivity between node pairs but allows the corresponding edge weights to differ, including the possibility of negative values under bounded conditions. The resulting Laplacian is generally non-symmetric, yet it guarantees consensus under connectivity assumptions, to expand the solution space, which enables the system to achieve a stable consensus value that can lie outside the convex hull of the initial state set. We derive admissibility bounds for negative weights for a pseudo-undirected path graph, and show an application in the simultaneous interception of a moving target.

Problem

• Extending consensus theory beyond traditional undirected and directed graph limitations
• Achieving stable consensus values outside convex hull of initial states
• Establishing admissibility bounds for negative edge weights in networks

Innovation

• Pseudo-undirected graphs enable asymmetric edge weights
• Non-symmetric Laplacian guarantees stable consensus convergence
• Negative weights expand consensus beyond initial state hull