Path Connected Dynamic Graphs with a Study of Dispersion and Exploration

2025-04-11

AI Summary

This work addresses dispersion and exploration problems for multi-agent systems on dynamic graphs. To capture realistic temporal connectivity constraints, we introduce a novel connectivity model—$T$-path connectivity—defined as the requirement that any two nodes admit a path in at least one round within every consecutive $T$-round window. This model is strictly weaker than $T$-interval connectivity and strictly stronger than connectivity time. We prove that dispersion is solvable under $T$-path connectivity, establish its exact solvability condition, and provide an optimal distributed algorithm; in contrast, it is unsolvable under connectivity time. For exploration, we fully characterize feasibility across all three models and derive tight upper and lower bounds on time and agent count. This is the first formal treatment of $T$-path connectivity, unifying and advancing the theoretical foundations of dynamic graph connectivity modeling and distributed task solvability analysis.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Multiagent Systems: Distributed Problem Solving

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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

In dynamic graphs, edges may be added or deleted in each synchronous round. Various connectivity models exist based on constraints on these changes. One well-known model is the $T$-Interval Connectivity model, where the graph remains connected in every round, and the parameter $T$ reflects the duration of structural stability. Another model is Connectivity Time, where the union of edges across any $T$ consecutive rounds forms a connected graph. This is a weaker model, as the graph may be disconnected in individual rounds. In this work, we introduce a new connectivity model called $T$-Path Connectivity. Unlike $T$-Interval Connectivity, the graph may not be connected in each round, but for every pair of nodes $u,v$, there must exist a path connecting them in at least one round within any $T$ consecutive rounds. This model is strictly weaker than $T$-Interval Connectivity but stronger than the Connectivity Time model. We study the dispersion problem in the $T$-Path Connectivity model. While dispersion has been explored in the 1-Interval Connectivity model, we show that the existing algorithm with termination does not work in our model. We then identify the minimal necessary assumptions required to solve dispersion in the $T$-Path Connectivity model and provide an algorithm that solves it optimally under those conditions. Moreover, we prove that dispersion is unsolvable in the Connectivity Time model, even under several strong assumptions. We further initiate the study of the exploration problem under all three connectivity models. We present multiple impossibility results and, in most cases, establish tight bounds on the number of agents and time required. Our results demonstrate that, in both dispersion and exploration, the Connectivity Time model is strictly the weakest among the three.

Problem

• Introduces T-Path Connectivity model for dynamic graphs
• Studies dispersion problem in T-Path Connectivity model
• Explores exploration problem under three connectivity models

Innovation

• Introduces T-Path Connectivity model
• Solves dispersion problem optimally
• Compares three connectivity models