Fourth Draft
In this work, we present a Geometric Causal Model, a framework for the causal analysis of biological pathways through their systematic conversion into a combinatorial DAG and subsequent embedding in a three-dimensional geometric space. Within this framework, nodes are represented by coordinates defined by causal dependency, causal effect, and hierarchical causal depth, enabling causal relationships to be analyzed in terms of directions, angles, and magnitudes of causal flow.
The proposed methodology starts from a signaling pathway and proceeds with a graph-theoretical decomposition of the resulting Directed Acyclic Graph (hereinafter DAG) into vertical (hierarchical) and retrolateral (cross-edge) subgraphs. These structures are then transformed into a three-dimensional topological representation and into a topoempirical geometric model by incorporating statistical information derived from empirical data through controlled geometric transformations.
Causal validation is performed through direct comparison with the gold-standard study of Sachs et al. (2005), focusing on two competing structural scenarios: one in which a previously reported absent edge is enforced and another in which it remains excluded. The impact of this choice is assessed by analyzing changes in local conditional independence patterns and in the structural coherence of the inferred causal graph.
The geometric embedding further enables a topological closure of causal directions through a toroidal representation (torification), which provides a criterion for evaluating global causal coherence. This closure is not a trivial property, but emerges from the joint configuration of the graph, particularly the organization of the retrolateral subgraph. The causal flow remains globally directed and confined within the modeled functional domain, without introducing external dependencies or causal loops.
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