A Comprehensive Technical Framework for Network Isomorphism

This paper investigates the structural and dynamic isomorphisms between transportation networks, neural architectures, and quantum systems. We synthesize literature from mathematical physics, transportation science, and computer science to address a critical gap regarding the lack of an integrated framework for comparing network structures across disciplines (Berkolaiko & Kuchment, 2013; Kivela & Porter, 2017). We propose a unified methodological framework, the Abstract Network Object (ANO), and a hierarchy of isomorphism metrics (L1-L3) to systematically test for equivalence. Three computational experiments were designed and executed to test this framework. The results confirm that a static (L1/L2) isomorphism can be established and that principles transferred between domains can yield significant performance improvements, such as a 42% reduction in final training loss for neural networks using a novel ‘Gravity Model Initialization’ (p < 0.0001). However, we demonstrate a fundamental and statistically significant divergence (p < 0.001) between systems governed by physical conservation laws (quantum-like) and those by game-theoretic equilibrium (traffic-like), thereby defining the boundary of dynamic (L3) isomorphism. This work provides a formal basis for cross-domain network analysis and clarifies the critical distinction between universal structure and domain-specific behavior.