On equilibrium Metropolis simulations on self-organized urban street networks

Jérôme G.M. Benoit, Saif Eddin G. Jabari

Research output: Contribution to journalArticlepeer-review


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 non-overlapping 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.

Original languageEnglish (US)
Article number33
JournalApplied Network Science
Issue number1
StatePublished - Dec 2021


  • Conserved quantities
  • Graph matchings
  • Ising model
  • MaxEnt
  • Metropolis algorithm
  • Scale-freeness
  • Self-organization
  • Self-similarity
  • Small-world crossover
  • Surprisal
  • Symmetries
  • Urban street networks
  • Watts–Strogatz model

ASJC Scopus subject areas

  • General
  • Computer Networks and Communications
  • Computational Mathematics


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