Observational indistinguishability between classical diffusion and thermal quantum propagation on networks

Abstract The heat kernel of a graph Laplacian is widely used to describe diffusion, transport, correlations and geometry in complex networks. In this work, we show that the same heat-kernel operator emerges both as the propagator of classical diffusion on a network and as the thermal Green's function of a network of coupled quantum harmonic oscillators restricted to the one-particle sector. We demonstrate that the geometry induced by this operator can be reconstructed from multiple observational experiments without direct access to the microscopic physical substrate generating the dynamics. Consequently, the same experimentally observed heat-kernel geometry may arise from fundamentally different physical mechanisms. This implies that heat-kernel-based observations alone may be insufficient to discriminate between classical and quantum interpretations of hidden dynamics in black-box systems. The results are discussed in the context of complex networks and observational geometries reconstructed from indirect measurements, including potential implications for neuroscience and brain connectivity analysis.