Algebraic Representation and Min-Plus Matrix Formulation of Living Temporal Connectivity

This record contains the preprint version of the manuscript entitled “Algebraic Representation and Min-Plus Matrix Formulation of Living Temporal Connectivity”. This paper develops an algebraic viewpoint on living temporal graphs using the min-plus semiring ((\mathbb{R} \cup {+\infty}, \min, +)). We define the latency projection matrix (L(\mathcal{G})), whose entries record minimum direct latencies while discarding temporal ordering and waiting information. We prove that the min-plus powers of (L(\mathcal{G})) describe projected static walk costs, and that the min-plus closure (L^*) provides a universal static lower bound on true temporal distances. We further show that (L(\mathcal{G})) alone does not determine exact temporal distances, thereby motivating a richer set-valued block adjacency matrix (\mathbb{B}(\mathcal{G})) that preserves complete temporal edge information. The manuscript establishes a strictness result showing that the gap between projected and temporal distances can arise from two independent sources: temporal infeasibility of projected walks and positive waiting required for feasible walks. Connections with the time-expanded graph developed in Paper III are made explicit. Numerical experiments on random temporal graphs confirm the theoretical bounds and quantify the gap between projected and temporal distances. Finally, the paper proposes several conjectural connections between tropical spectral theory and temporal resilience, centrality, periodicity, and algebraic compression. This paper is the fourth contribution in a twelve-part series on living temporal graphs.