A Dynamic Data Structure for Representing Timed Transitive Closures on Disk

TL;DR

This paper introduces a disk-based data structure for efficiently maintaining and querying temporal reachability in large temporal graphs, supporting dynamic updates and various reachability queries with optimized disk access.

Contribution

It presents a novel disk-based data structure for dynamic temporal reachability that efficiently handles non-chronological contact addition and supports multiple query types.

Findings

Outperforms existing approaches on synthetic datasets

Supports dynamic updates with efficient disk access

Handles various reachability queries effectively

Abstract

Temporal graphs represent interactions between entities over time. These interactions may be direct, a contact between two vertices at some time instant, or indirect, through sequences of contacts called journeys. Deciding whether an entity can reach another through a journey is useful for various applications in complex networks. In this paper, we present a disk-based data structure that maintains temporal reachability information under the addition of new contacts in a non-chronological order. It represents the \emph{timed transitive closure} (TTC) by a set of \emph{expanded} R-tuples of the form $(u, v, t^-, t^+)$, which encodes the existence of journeys from vertex $u$ to vertex $v$ with departure at time $t^-$ and arrival at time $t^+$. Let $n$ be the number of vertices and $\tau$ be the number of timestamps in the lifetime of the temporal graph. Our data structure explicitly maintains this information in linear arrays using $O(n^2\tau)$ space so that sequential accesses on disk are prioritized. Furthermore, it adds a new unsorted contact $(u, v, t)$ accessing $O\left(\frac{n^2\tau}{B}\right)$ sequential pages in the worst-case, where $B$ is the of pages on disk; it answers whether there is of a journey from a vertex $u$ to a vertex $v$ within a time interval $[t_1, t_2]$ accessing a single page; it answers whether all vertices can reach each other in $[t_1, t_2]$; and it reconstructs a valid journey that validates the reachability from a vertex $u$ to a vertex $v$ within $[t_1, t_1]$ accessing $O\left(\frac{n\tau}{B}\right)$ pages. Our experiments show that our novel data structure are better that the best known approach for the majority of cases using synthetic and real world datasets.