Part I: Scaling Laws & Infrastructure
How do cities change as they grow? This part derives the universal scaling exponents that govern urban infrastructure and social output, grounding them in hierarchical network optimization, interaction integrals, and non-equilibrium thermodynamics.
Part Overview
Derives scaling exponents from hierarchical network optimization and interaction integrals. \(\beta = 3/4\) for infrastructure (sublinear), \(\beta = 5/4\) for social metrics (superlinear), plus Glansdorff-Prigogine entropy production as the thermodynamic backbone of urban dissipation.
Key Topics
- • Hierarchical network optimization
- • Kleiber's law derivation
- • Interaction integral for superlinearity
- • Glansdorff-Prigogine theory
- • Python scaling fits
- • Fortran log-log regression
3 chapters | From networks to thermodynamics | Quantifying how cities scale
Chapters
Chapter 1: Infrastructure Scaling (\(\beta = 3/4\))
Derives the sublinear exponent from hierarchical fractal networks that minimize transport cost. Kleiber's law for biological metabolic rate generalizes to urban infrastructure: roads, cables, and pipes scale as \(Y \propto N^{3/4}\).
Chapter 2: Social Superlinearity (\(\beta = 5/4\))
The interaction integral over pairwise encounters in a mixing population yields a superlinear exponent. Innovation, wages, and crime all scale faster than population: \(Y \propto N^{5/4}\).
Chapter 3: Entropy Production
Cities as dissipative structures far from equilibrium. The Glansdorff-Prigogine stability criterion governs which urban configurations persist, linking thermodynamic entropy production to observable scaling laws.