Fig. 5

Single-junction-switch Metropolis algorithm for configurations of streets: the frames (\({\mathrm{g}}_{\star }\)) show how this algorithm may evolve from the configuration of streets on street map (\({\mathrm{m}}^{\prime }\)) to the one on street map (\({\mathrm{m}}\)) from the notional example in Fig. 1. The subscript of each frame label indicate the generation time. The figure reads from top to bottom. It actually separates generated configurations of streets from intermediate computational steps in two complementary ways: it shifts (resp. epochs) the formers to the left column (resp. with integers) and the latters to the right column (resp. with half-integers). The right frames display more information. In particular each switchable junction is marked with an extended regular polygon whose the corners (sides) enumerate its maximum matchings (see Fig. 4): when the polygon is in grey, the junction is resting; when the polygon is in red and notably bigger, the junction is switching. As for the state-of-the-art construction paradigm in Fig. 3, we attribute to each street a particular colour. In clear contrast, however, all streets are here fully constructed. The streets in dashed lines are not under construction but instead under challenge as follows. The dashing actually indicates coexistence of old streets with new ones. Each old street keeps its colour, its circle mark, and its label. Each new street emerges with a new color, a wavy-circle mark, and a tilded label. The new streets result from the new street layout at the switching junction. This change leads so to a new configuration of streets \(\nu\) that competes with the old one \(\mu\). This is the actual ongoing challenge. The Metropolis algorithm resolves such challenges by either accepting or rejecting change \(\mu \rightarrow \nu\) with an acceptance ratio \(A(\mu \rightarrow \nu )\) in an optimal way. After a sufficient number of generations, the configurations of streets reach a prescribed statistical equilibrium—provided the equilibrium is sustainable. Our prescribed statistical equilibrium follows from the assumption that self-organized urban street networks are statistically self-similar. It is a Boltzmann-like distribution with a total amount of surprisal (information) instead of energy and the scaling as equilibrium parameter (see formula (16)). Our odds of accepting or rejecting new configurations of streets favour the less surprising ones (see formulae (23) and (30)). The algorithm goes like this (see “Informal implementation” section). Note first that only the four junctions \(j_{1}\), \(j_{2}\), \(j_{3}\), and \(j_{8}\) are actually switchable: junction \(j_{3}\) has \({{\mathscr {Y}}_{3}=6}\) maximum matchings (see Fig. 4c); junctions \(j_{1}\), \(j_{2}\), and \(j_{8}\) have \({{\mathscr {Y}}_{1}={\mathscr {Y}}_{2}={\mathscr {Y}}_{8}=2}\) maximum matchings (see Fig. 4a, b); the remaining junctions \(j_{\circ }\) have obviously \({{\mathscr {Y}}_{\circ }=1}\) maximum matching. So each move will first choose randomly one junction among junctions \(j_{1}\), \(j_{2}\), \(j_{3}\), and \(j_{8}\) with probabilities \(\tfrac{1}{8}\), \(\tfrac{1}{8}\), \(\tfrac{5}{8}\), and \(\tfrac{1}{8}\), respectively. Each move will second pick uniformly at random a new maximum matching. There will be \({{\mathscr {Y}}_{3}-1=5}\) choices for \(j_{3}\), and \({{\mathscr {Y}}_{1}-1}={{\mathscr {Y}}_{2}-1}={{\mathscr {Y}}_{8}-1}=1\) choice for \(j_{1}\), \(j_{2}\), and \(j_{8}\). Each move will third calculate its change in amount of surprisal \(\Delta {S}\) in view to compute its acceptance ratio A. Here the surprisal changes at mid-steps (\({\mathrm{g}}_{{\frac{1}{2}}}\)), (\({\mathrm{g}}_{1{\frac{1}{2}}}\)), and (\({\mathrm{g}}_{2{\frac{1}{2}}}\)) are respectively \(\Delta {\widetilde{S}} = \ln \frac{5}{9}\), \(\Delta {\widetilde{S}} = 0\), and \(\Delta {\widetilde{S}} = \ln \frac{6}{5}\) (see “Working assumptions” section). Ultimately each move will either accept or reject the change with probability A. The layout changes at (\({\mathrm{g}}_{{\frac{1}{2}}}\)) and (\({\mathrm{g}}_{1{\frac{1}{2}}}\)) are certain since they are less or equally surprising, the one at (\({\mathrm{g}}_{2{\frac{1}{2}}}\)) is accepted with probability \(\exp (-{\widetilde{\lambda }}\ln \frac{6}{5})\) where \({\widetilde{\lambda }}\) is our effective equilibrium parameter (see formula (30)). Our illustration actually rejects the last move. Supplementary Animation A2 (Additional file 2) shows a longer sequence