Causal Edge Rees Algebras for Spatiotemporal Graphs

TL;DR

This paper introduces the Causal Edge Rees Algebra (CERA), an algebraic framework that encodes the evolution of connectivity in causal spatiotemporal graphs, linking dynamic topology with commutative algebra.

Contribution

It develops a novel algebraic construction, CERA, that captures the temporal evolution of connectivity in causal graphs, bridging graph dynamics and algebraic structures.

Findings

CERA encodes full connectivity history in a single algebraic object.

Successive quotients reveal emergence of new structural connections.

Temporal bridge modules identify edges responsible for component fusion.

Abstract

Understanding the evolution of connectivity in spatiotemporal systems requires mathematical frameworks capable of encoding not only instantaneous interactions but also their cumulative causal structure. In this work, we introduce the \emph{Causal Edge Rees Algebra} (CERA), a new algebraic construction associated with causal spatiotemporal graphs. Given a temporal filtration induced by causal constraints, we associate a sequence of edge ideals whose Rees algebra encodes the full history of connectivity evolution in a single graded object. This construction establishes a bridge between dynamic graph topology and commutative algebra. In particular, we show that successive quotients of the filtration capture the emergence of new structural connections, allowing the identification of critical edges responsible for the fusion of previously disconnected components. This leads to the definition of temporal bridge modules and to a bridge detection theorem, which relates the dimension of these modules to the reduction in the number of connected components over time. Unlike existing algebraic approaches in topological data analysis, which are primarily based on geometric filtrations, the proposed framework is driven by intrinsic causal constraints. As a result, the CERA captures not only topological features but also their temporal organization and mechanisms of coalescence. The theory provides a new algebraic perspective on causal network dynamics, connecting edge ideals, Rees algebras, and temporal graphs. Beyond its theoretical significance, the framework opens new directions for the analysis of spatiotemporal systems, including epidemic networks, transport systems, and information propagation processes.