  Urban street networks of unplanned or self-organized cities typically exhibit
astonishing scale-free patterns. This scale-freeness can be shown, within the
maximum entropy formalism (MaxEnt), as the manifestation of a fluctuating
system that preserves on average some amount of information. Monte Carlo
methods that can further this perspective are cruelly missing. Here we adapt to
self-organized urban street networks the Metropolis algorithm. The "coming to
equilibrium" distribution is established with MaxEnt by taking scale-freeness
as prior hypothesis along with symmetry-conservation arguments. The equilibrium
parameter is the scaling; its concomitant extensive quantity is, assuming our
lack of knowledge, an amount of information. To design an ergodic dynamics, we
disentangle the state-of-the-art street generating paradigms based on
nonoverlapping walks into layout-at-junction dynamics. Our adaptation
reminisces the single-spin-flip Metropolis algorithm for Ising models. We thus
expect Metropolis simulations to reveal that self-organized urban street
networks, besides sustaining scale-freeness over a wide range of scalings,
undergo a crossover as scaling varies -- literature argues for a small-world
crossover. Simulations for Central London are consistent against the
state-of-the-art outputs over a realistic range of scaling exponents. Our
illustrative Watts-Strogatz phase diagram with scaling as rewiring parameter
demonstrates a small-world crossover curving within the realistic window 2-3;
it also shows that the state-of-the-art outputs underlie relatively large
worlds. Our Metropolis adaptation to self-organized urban street networks
thusly appears as a scaling variant of the Watts-Strogatz model. Such insights
may ultimately allow the urban profession to anticipate self-organization or
unplanned evolution of urban street networks.
