[Paper Review] An Introduction to Temporal Graphs: An Algorithmic Perspective

Othon Michail|arXiv (Cornell University)|Mar 1, 2015

Opportunistic and Delay-Tolerant Networks3 citations

TL;DR

This paper introduces temporal graphs as dynamic networks where edge availability varies over discrete time steps, proposing an algorithmic framework to analyze temporal connectivity and diameter. It establishes that in random temporal cliques with lifetime r=n, the temporal diameter is O(log n) with high probability, using a bidirectional front expansion algorithm to construct journeys efficiently.

ABSTRACT

A \emph{temporal graph} is, informally speaking, a graph that changes with time. When time is discrete and only the relationships between the participating entities may change and not the entities themselves, a temporal graph may be viewed as a sequence $G_1,G_2\ldots,G_l$ of static graphs over the same (static) set of nodes $V$. Though static graphs have been extensively studied, for their temporal generalization we are still far from having a concrete set of structural and algorithmic principles. Recent research shows that many graph properties and problems become radically different and usually substantially more difficult when an extra time dimension in added to them. Moreover, there is already a rich and rapidly growing set of modern systems and applications that can be naturally modeled and studied via temporal graphs. This, further motivates the need for the development of a temporal extension of graph theory. We survey here recent results on temporal graphs and temporal graph problems that have appeared in the Computer Science community.

Motivation & Objective

• To develop a foundational algorithmic framework for temporal graphs, a dynamic extension of static graph theory.
• To address the challenge of temporal connectivity and journey existence in networks where edges appear only at specific times.
• To analyze the temporal diameter of random temporal graphs and establish tight bounds on reachability.
• To model real-world systems such as communication, social, and transportation networks that evolve over time.
• To provide algorithmic tools for efficient information dissemination in dynamic, adversarial network topologies.

Proposed method

• Models temporal graphs as sequences of static graphs G₁, G₂, ..., Gₗ over the same node set V, with edges appearing only at specific time steps.
• Applies the Erdős-Rényi random graph model G(n,p) to analyze connectivity thresholds in temporal subgraphs.
• Employs a bidirectional journey construction algorithm: expanding a forward front from source s and a backward front from target t over time intervals of length c₂.
• Uses probabilistic analysis to show that both fronts grow to Θ(√n) nodes with high probability in O(log n) steps.
• Establishes temporal connectivity by proving the existence of a connecting edge between the final fronts of s and t with high probability.
• Applies concentration bounds to show that the probability of failure for any node pair is at most 3/n, leading to a high-probability bound on the temporal diameter.

Experimental results

Research questions

• RQ1 What is the temporal diameter of a random temporal clique where each edge appears independently at random times?
• RQ2 How can efficient journeys be constructed in dynamic networks with arbitrary, time-varying topologies?
• RQ3 What is the minimum lifetime r required for a temporal clique to ensure temporal diameter O(log n) with high probability?
• RQ4 How does the temporal diameter scale with the number of nodes n and the edge availability lifetime r?
• RQ5 What is the price of randomness in ensuring temporal reachability across all node pairs in a random temporal graph?

Key findings

• The temporal diameter of a random temporal clique with edge lifetime r=n is O(log n) with high probability.
• For r=Ω(n), almost every temporal graph has temporal diameter ω(log n), indicating a sharp threshold around this value.
• The bidirectional front expansion algorithm succeeds in constructing an s-t journey of duration O(log n) with probability at least 1−3/n³ for any fixed pair s,t.
• The final front of s and the final front of t each consist of Θ(√n) nodes with high probability, enabling high-likelihood connection.
• The existence of a connecting edge between the final fronts of s and t is guaranteed with high probability, ensuring journey completion.
• The overall temporal diameter is O(log n) with probability at least 1−3/n, confirming fast information dissemination in such networks.