Scaling Laws in Spatial Network Formation

TL;DR

This paper derives a scaling law for the size of spatial networks constrained by geometry, revealing a power-law relationship that aligns with observed protein structures, advancing understanding of network self-organization.

Contribution

It introduces a novel algebraic scaling law for spatial networks under geometric constraints, bridging stochastic analysis with empirical observations.

Findings

Network diameter scales algebraically with system size.

Scaling exponent lies between self-avoiding walks and space-filling arrangements.

Results match experimental data on protein tertiary structures.

Abstract

Geometric constraints impact the formation of a broad range of spatial networks, from amino acid chains folding to proteins structures to rearranging particle aggregates. How the network of interactions dynamically self-organizes in such systems is far from fully understood. Here, we analyze a class of spatial network formation processes by introducing a mapping from geometric to graph-theoretic constraints. Combining stochastic and mean field analyses yields an algebraic scaling law for the extent (graph diameter) of the resulting networks with system size, in contrast to logarithmic scaling known for networks without constraints. Intriguingly, the exponent falls between that of self-avoiding random walks and that of space filling arrangements, consistent with experimentally observed scaling (of the spatial radius of gyration) for protein tertiary structures.