Emergent Gravity from Local Geometric Order in Graph-Based Networks

This work investigates the emergence of gravitational behavior in discrete graph-based systems where geometry arises from connectivity rather than being predefined. Using graph Laplacians as discrete analogues of differential operators, a scalar field is defined from a localized source and used to construct an effective metric on the network. We show that gravitational-like scaling, characterized by a power-law dependence with distance, emerges robustly in regular lattice graphs. However, this behavior progressively degrades as local geometric disorder is introduced. While the spectral dimension remains approximately constant, a strong correlation is observed between gravitational strength and local geometric regularity. In particular, we define a disorder parameter based on degree fluctuations and find a high correlation between this measure and the weakening of gravitational behavior. These results suggest that gravity in such systems is not determined solely by global spectral properties, but depends critically on local geometric coherence. Finally, we discuss the possibility of a qualitative connection between geometric disorder and effective repulsive contributions analogous to a cosmological constant, highlighting a potential link between microscopic structure and large-scale emergent behavior.