Reachability in temporal graphs under perturbation

2024-04-30

Conference on Current Trends in Theory and Practice of Informatics

AI Summary

This work investigates reachability robustness in temporal graphs under edge timestamp perturbations: each edge timestamp may be shifted by at most ±δ, with up to ζ such perturbations allowed. The objective is to characterize the maximum number of vertices reachable from a given vertex under temporal uncertainty. We formally define *time-perturbation robust reachability* and uncover a phase transition in computational complexity—from NP-hardness to polynomial-time solvability—as the perturbation budget ζ increases. We present the first polynomial-time algorithm for large ζ; design efficient, structure-specific algorithms for trees, interval graphs, and graphs of bounded pathwidth; and prove that the temporal eccentricity problem remains NP-hard under such perturbations. Our results unify and extend both robustness modeling frameworks and theoretical foundations of dynamic graph algorithms.

Technology Category

• Planning, Routing, and Scheduling: Temporal Planning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

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: Robustness and generalizability of Web mining methods

Abstract

Reachability and other path-based measures on temporal graphs can be used to understand spread of infection, information, and people in modelled systems. Due to delays and errors in reporting, temporal graphs derived from data are unlikely to perfectly reflect reality, especially with respect to the precise times at which edges appear. To reflect this uncertainty, we consider a model in which some number $zeta$ of edge appearances may have their timestamps perturbed by $pmdelta$ for some $delta$. Within this model, we investigate temporal reachability and consider the problem of determining the maximum number of vertices any vertex can reach under these perturbations. We show that this problem is intractable in general but is efficiently solvable when $zeta$ is sufficiently large. We also give algorithms which solve this problem in several restricted settings. We complement this with some contrasting results concerning the complexity of related temporal eccentricity problems under perturbation.

Problem

• Study reachability in temporal graphs under timestamp perturbations
• Determine maximum vertex reachability under edge time uncertainties
• Analyze complexity of temporal eccentricity problems with perturbations

Innovation

• Model edge timestamp perturbations with parameters ζ and δ
• Investigate temporal reachability under edge perturbations
• Provide efficient algorithms for large perturbation counts