Main path equals geodesic conjecture

Establish whether the main path identified by Search Path Count (SPC) main path analysis on interval directed acyclic graphs embedded in D-dimensional Euclidean space is a geodesic—specifically, the straight line between the source node s and the sink node t.

Background

Main path analysis aims to identify influential citation trajectories in directed acyclic graphs (DAGs). The paper proposes grounding this heuristic in geometry by relating main paths to geodesics of an embedding Euclidean space, arguing that geodesics represent paths of least effort and thus plausible routes for information flow.

To explore this idea, the authors study toy models (hypercubic lattices and random geometric DAGs) and compare SPC and entropy-based (SPE) main paths against geodesics. While they provide evidence of near-geodesic behavior, they explicitly present the geodesic equivalence of main paths as a conjecture to be established.