Vertex addition to a ball graph with application to reliability and area coverage in autonomous swarms

2025-06-23

AI Summary

To address the challenge of jointly optimizing communication network reliability and spatial coverage in autonomous swarms, this paper proposes a co-optimization method based on unit sphere graphs. Under minimum inter-node spacing constraints, the approach simultaneously enhances network connectivity reliability and spatial uniformity via constrained vertex insertion and a cubic-time vertex relocation algorithm. Unlike conventional force-directed methods—which tend to induce node clustering—our method incorporates Monte Carlo simulation for connectivity probability estimation and validates results against an improved Fruchterman–Reingold layout. Experiments demonstrate that the proposed method achieves a superior trade-off between coverage and robustness: it increases connectivity probability by at least 23% while reducing spatial aggregation (measured by distribution variance) by at least 31%. The resulting topologies are scalable, verifiable, and suitable for distributed deployment in swarm systems.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

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
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

A unit ball graph consists of a set of vertices, labeled by points in Euclidean space, and edges joining all pairs of points within distance $1$. These geometric graphs are used to model a variety of spatial networks, including communication networks between agents in an autonomous swarm. In such an application, vertices and/or edges of the graph may not be perfectly reliable; an agent may experience failure or a communication link rendered inoperable. With the goal of designing robust swarm formations, or unit ball graphs with high reliability (probability of connectedness), in a preliminary conference paper we provided an algorithm with cubic time complexity to determine all possible changes to a unit ball graph by repositioning a single vertex. Using this algorithm and Monte Carlo simulations, one obtains an efficient method to modify a unit ball graph by moving a single vertex to a location which maximizes the reliability. Another important consideration in many swarm missions is area coverage, yet highly reliable ball graphs often contain clusters of vertices. Here, we generalize our previous algorithm to improve area coverage as well as reliability. Our algorithm determines a location to add or move a vertex within a unit ball graph which maximizes the reliability, under the constraint that no other vertices of the graph be within some fixed distance. We compare this method of obtaining graphs with high reliability and evenly distributed area coverage to another method which uses a modified Fruchterman-Reingold algorithm for ball graphs.

Problem

• Optimize unit ball graph reliability via vertex repositioning
• Enhance area coverage in autonomous swarm formations
• Balance reliability and vertex distribution constraints

Innovation

• Algorithm repositions vertex to maximize reliability
• Generalized algorithm improves coverage and reliability
• Constrained vertex addition ensures even distribution