Disorder-Driven Degradation of Emergent Gravitational Scaling in Small-World Networks

This work investigates the emergence and stability of gravity-like scaling behavior in discrete graph-based systems. Using scalar fields defined via graph Laplacians on Watts–Strogatz networks, we observe a transition-like behavior from ordered regimes exhibiting power-law decay to disordered regimes where this behavior progressively degrades. A finite-size scaling analysis reveals that the apparent critical threshold depends strongly on system size, shifting toward zero as the system grows. This indicates that the observed transition is not governed by a universal critical point, but instead arises from finite-size effects and the loss of local geometric coherence. We further analyze the scaling collapse of the normalized observable and find partial agreement across different system sizes, with deviations attributable to finite-size limitations. These results suggest that emergent gravitational behavior is highly sensitive to local structural order and may not be a generic property of arbitrary discrete systems. This work contributes to the broader effort of understanding gravity as an emergent phenomenon arising from discrete underlying structures.