Document Type : Research Paper

Authors

• Madjid Eshaghi ¹
• Ali Jabbari ²

¹ Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran

² Applied Mathematics and Informatics, Kyrgyz-Turkish Manas University, Bishkek, Kyrgyzstan

Abstract

Modelling complex systems where connections are transient requires a departure from static graph theory. This paper proposes a comprehensive mathematical framework for {Temporal Graphs}, extending classical concepts to capture dynamic interactions and causality. We identify fundamental limitations in static representations, specifically the "Path Existence Fallacy" and "Flow Indistinguishability." To resolve these, we introduce the novel concept of {$(f, g)$-homomorphism}, a structural mapping $f$ coupled with a temporal mapping $g$, which allows for the modelling of time dilation, contraction, and reversal within network flows. Furthermore, we define hierarchical temporal graphs (HTGs) for multi-scale analysis. Finally, we present a rigorous list of open problems in pure mathematics—ranging from temporal fixed point theory to persistent homology—and future applications in quantum computing and blockchain dynamics.

Keywords

• Temporal Graphs
• (f,g)-Homomorphism
• Dynamic Networks
• Time-Varying Topology
• Causal Paths
• Fixed Point Theory