Temporal Exploration of Random Spanning Tree Models

2025-08-05

AI Summary

This paper studies vertex exploration on stochastic temporal graphs—specifically, sequences of independent and identically distributed random spanning trees—aiming to visit all vertices in minimum time. We introduce the Random Spanning Tree (RST) model, the first systematic framework for analyzing temporal graph exploration under stochastic assumptions, revealing fundamental differences from adversarial settings. Using probabilistic analysis, random graph theory, and high-probability (w.h.p.) asymptotic methods, we prove that any n-vertex RST model is fully explorable in O(n^{3/2}) time, and this upper bound is tight. Moreover, if each tree is a subgraph of an m-edge graph, the exploration time improves to O(m). Our primary contribution is establishing the first theoretical foundation for exploration on random temporal graphs, accompanied by precise asymptotic complexity characterizations—namely, tight bounds on worst-case exploration time under stochastic edge dynamics.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

The Temporal Graph Exploration problem (TEXP) takes as input a temporal graph, i.e., a sequence of graphs $(G_i)_{iin mathbb{N}}$ on the same vertex set, and asks for a walk of shortest length visiting all vertices, where the $i$-th step uses an edge from $G_i$. If each such $G_i$ is connected, then an exploration of length $n^2$ exists, and this is known to be the best possible up to a constant. More fine-grained lower and upper bounds have been obtained for restricted temporal graph classes, however, for several fundamental classes, a large gap persists between known bounds, and it remains unclear which properties of a temporal graph make it inherently difficult to explore. Motivated by this limited understanding and the central role of the Temporal Graph Exploration problem in temporal graph theory, we study the problem in a randomised setting. We introduce the Random Spanning Tree (RST) model, which consists of a set of $n$-vertex trees together with an arbitrary probability distribution $μ$ over this set. A random temporal graph generated by the RST model is a sequence of independent samples drawn from $μ$. We initiate a systematic study of the Temporal Graph Exploration problem in such random temporal graphs and establish tight general bounds on exploration time. Our first main result proves that any RST model can, with high probability (w.h.p.), be explored in $O(n^{3/2})$ time, and we show that this bound is tight up to a constant factor. This demonstrates a fundamental difference between the adversarial and random settings. Our second main result shows that if all trees of an RST are subgraphs of a fixed graph with $m$ edges then, w.h.p. , it can be explored in $O(m)$ time.

Problem

• Explores shortest walk in temporal graphs with connected components
• Investigates exploration time in Random Spanning Tree models
• Compares adversarial vs random temporal graph exploration bounds

Innovation

• Random Spanning Tree model with probability distribution
• Exploration in O(n^(3/2)) time with high probability
• Exploration in O(m) time for fixed subgraphs