Asymptotics of Parking Search in Hyperfractal Networks

TL;DR

This paper analyzes the asymptotic behavior of parking search distances in hyperfractal networks, revealing a power-law decay influenced by the network's large-scale geometry.

Contribution

It establishes a power-law decay of expected parking distance with a universal exponent linked to the hyperfractal dimension, robust under stochastic heterogeneity.

Findings

Expected parking distance decays as a power law with network intensity.

The decay exponent equals the inverse of the hyperfractal dimension.

Variance and search complexity also follow similar scaling laws.

Abstract

We study the asymptotic behaviour of the distance to the first available parking slot in a recursive Manhattan street network endowed with a hyperfractal intensity structure, where slot-release events occur according to Poisson processes along the streets. We establish, by analysing the associated self-similar harmonic sums via Mellin-transform asymptotics, a power-law decay of the expected distance as the total intensity grows, with exponent equal to the inverse of the hyperfractal dimension. In particular, the scaling exponent depends only on the large-scale geometry of the network. We further prove that this exponent is robust under random multiplicative modulations of the street intensities: mild stochastic heterogeneity affects only the multiplicative constant. Similar scaling behaviour holds for the variance, the number of turns before parking, and for a jump-over variant of the search strategy.