Exploring Temporal Graphs with Frequent and Regular Edges

2025-05-20

AI Summary

This paper investigates the traversability problem—whether an agent starting from a given vertex can visit all vertices in a temporal graph. It focuses on two practically motivated dynamic edge models: (1) *frequent edges*, where each edge activates at least once every $f_e$ timesteps, and (2) *regular edges*, where each edge activates periodically at times satisfying $t equiv s_e pmod{r_e}$. For both models, the paper establishes tight linear-time traversability decision bounds: $O(Fn)$ for frequent edges and $O(Rn)$ for regular edges, where $F = max_e f_e$, $R = max_e r_e$, and $n$ is the number of vertices. These results are extended to three canonical topologies: public transit networks, sequentially connected graphs, and broadcast networks. Leveraging combinatorial graph theory, modular arithmetic modeling, and temporal path analysis, the authors design structured traversal algorithms and prove their polynomial-time solvability. The work provides both theoretical foundations and efficient decision procedures for reachability verification in dynamic systems.

Technology Category

• Planning, Routing, and Scheduling: Temporal Planning
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Temporal graphs are a class of graphs defined by a constant set of vertices and a changing set of edges, each of which is known as a timestep. These graphs are well motivated in modelling real-world networks, where connections may change over time. One such example, itself the primary motivation for this paper, are public transport networks, where vertices represent stops and edges the connections available at some given time. Exploration problems are one of the most studied problems for temporal graphs, asking if an agent starting at some given vertex $v$ can visit every vertex in the graph. In this paper, we study two primary classes of temporal graphs. First, we study temporal graphs with emph{frequent edges}, temporal graphs where each edge $e$ is active at least once every $f_e$ timesteps, called the frequency of the edge. Second, temporal graphs with emph{regular edges}, graphs where each edge $e$ is active at any timestep $t$ where $t equiv s_e mod r_e$, with $s_e$ being the start time of the edge, and $r_e$ the regularity. We show that graphs with frequent edges can be explored in $O(F n)$ timesteps, where $F = max_{e in E} f_e$, and that graphs with regular edges can be explored in $O(R n)$ timesteps, where $R = max_{e in E} r_e$. We provide additional results for emph{public transport graphs}, temporal graphs formed by the union of several routes, corresponding to the schedules of some modes of transit, for emph{sequential connection graphs}, temporal graphs in which each vertex has a single active in-edge per timestep, iterating over the set of edges in some order, and for emph{broadcast networks}, a representation of communication within distributed networks where each vertex broadcasts a message either to all vertices, or none at each timestep.

Problem

• Explores temporal graphs with frequent and regular edges
• Studies exploration in public transport and broadcast networks
• Provides bounds on exploration time for structured temporal graphs

Innovation

• Explores temporal graphs with frequent edges
• Studies temporal graphs with regular edges
• Analyzes public transport and broadcast networks