Theory 19/Urban Systems & Cities /Urban scaling/3 min
City Scaling Is Network Depth
Every city seems to scale the same way. The exponent may be a property of how many layers the city has.
The superlinear scaling of urban output is set by the depth of the infrastructure and interaction hierarchy rather than by population size, so cities with deeper hierarchies should have larger exponents and the exponent should be predictable from network topology.
01The evidence
- Scaling analysis of urban infrastructure has related the exponent to the geometry and hierarchy of the network, with the infrastructure area scaling sublinearly in a way that constrains the output exponent.PIK, urban scaling
- The hierarchy of urban road networks has been analysed from a scaling-law perspective, which supplies the topological quantity an exponent-from-depth account requires.Cities, road network hierarchy
02The tension
Urban scaling is reported as a universal law with an exponent near 1.15 for output and 0.85 for infrastructure. Universality is the puzzle: cities with very different histories and geometries produce similar exponents, and the deviations are treated as measurement error rather than as structure.
03The theory
A city's output depends on interactions, and interactions are organised by a hierarchy: local clusters, districts, and a centre. The number of interactions available grows with the depth of that hierarchy, because each level multiplies the possible pairings.
If the exponent reflects hierarchy depth, then cities with more levels should scale more steeply, and the universal value is the average depth of the sample rather than a law of cities. That makes the exponent a measurable structural parameter.
The prediction is that the exponent should correlate with a measured hierarchy index, such as the number of distinct levels in the road or transit network, and that cities engineered to add a level should shift their exponent.
This matters for policy because it suggests that the returns to scale are not automatic. A city that grows without deepening its hierarchy will scale like a large town, not like a metropolis.
04What this is not
The law treats the exponent as a constant of urbanism. This treats it as a function of hierarchy depth, which explains the deviations and predicts that the exponent can be changed by design.
05Predictions
- P1The output exponent should correlate with a measured hierarchy depth index.
- P2Adding a hierarchy level should increase the exponent.
- P3Cities with similar populations but different depths should differ in per-capita output.
06What would refute it
Exponents invariant to hierarchy depth across well-measured cities would restore the universal law.
07What it would change
It would make urban scaling a design variable, and change how infrastructure investment is justified.
08Evidence base
- Urban scaling and infrastructure geometry — PIKhttps://publications.pik-potsdam.de/rest/items/item_28210_4/component/file_28255/content?download=true
- Hierarchy of urban road networks from a scaling-law perspective — Citieshttps://www.sciencedirect.com/science/article/abs/pii/S0264275125005165