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On equilibrium Metropolis simulations on selforganized urban street networks
Applied Network Science volume 6, Article number: 33 (2021)
Abstract
Urban street networks of unplanned or selforganized cities typically exhibit astonishing scalefree patterns. This scalefreeness 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 selforganized urban street networks the Metropolis algorithm. The “coming to equilibrium” distribution is established with MaxEnt by taking scalefreeness as prior hypothesis along with symmetryconservation 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 stateoftheart street generating paradigms based on nonoverlapping walks into layoutatjunction dynamics. Our adaptation reminisces the singlespinflip Metropolis algorithm for Ising models. We thus expect Metropolis simulations to reveal that selforganized urban street networks, besides sustaining scalefreeness over a wide range of scalings, undergo a crossover as scaling varies—literature argues for a smallworld crossover. Simulations for Central London are consistent against the stateoftheart outputs over a realistic range of scaling exponents. Our illustrative Watts–Strogatz phase diagram with scaling as rewiring parameter demonstrates a smallworld crossover curving within the realistic window 2–3; it also shows that the stateoftheart outputs underlie relatively large worlds. Our Metropolis adaptation to selforganized urban street networks thusly appears as a scaling variant of the Watts–Strogatz model. Such insights may ultimately allow the urban profession to anticipate selforganization or unplanned evolution of urban street networks.
Introduction
Unplanned or selforganized cities spontaneously undergo scaling coherences for which a comprehensive explanation is lacking (Rybski et al 2019). Scaling coherence, or scalefreeness, expresses apparent invariance under zoomingin or out transformations. The scaling coherence of the spatial organization of a city is reflected in its streets: the streets of a selforganized city typically follow a scalefree behaviour which has attracted much attention from observational and theoretical researchers (Rosvall et al 2005; Porta et al 2006; Crucitti et al 2006; Jiang et al 2008). We recently linked the scalefreeness of selforganized urban street networks to a preservation principle through a fluctuating mesoscopic model (Benoit and Jabari 2019a, b).
The invoked preservation principle is the Jaynes’s Maximum Entropy principle (Jaynes 1957, 2003; Lawrence 2019). This principle assesses the most plausible probability distribution of a fluctuating system according to moment constraints. We inversely applied it by envisioning streets as mesoscopic objects governed by social interactions (Benoit and Jabari 2019a, b). We reflect the scaling coherence by randomly distributing their numbers of configurations according to a scalefree distribution, specifically, a discrete Pareto distribution (Clauset et al 2009). The discrete Pareto distribution results from a constraint on the first logarithm moment (Dover 2004). Since their configurations are equally probable due to our lack of knowledge, this constraint interprets itself as an information measure preservation. The predominance of a number of vital connections among social connections asymptotically leads to a discrete Pareto distribution for the number of junctions per street. We have what is observed among selforganized urban street networks. However promising the approach appears, we need to investigate it completely with some specific tools.
To study such fluctuating models, analytical and simulational methods are usually employed as complementary methods to obtain more complete and accurate interpretations. Our analytical framework is the maximum entropy formalism, a general formalism of modern probability theory partially inherited from statistical physics (Jaynes 2003; Grandy 1987; Lawrence 2019). For simulating fluctuating systems, physicists mostly rely on random sampling algorithms based on Markov chain Monte Carlo methods, often abbreviated as Monte Carlo methods (Newman and Barkema 1999; Landau and Binder 2015). Each thusgenerated random sample enables us to obtain numerical results that we can confront to theoretical ones. The Monte Carlo method of first choice remains the algorithm pioneered by Nicolas Metropolis and his coworkers (Metropolis et al 1953; Newman and Barkema 1999).
Strictly speaking the Metropolis algorithm may apply to configurations of streets or their associated information networks. An information network (Rosvall et al 2005; Porta et al 2006) is a dual network representation of an urban street network that (i) associates each street to a node, and (ii) links each pair of nodes (streets) sharing a common junction (see Fig. 1 for illustration). It is this dual graph representation that reveals the underlying scalefreeness (Porta et al 2006; Crucitti et al 2006; Jiang et al 2008). For instance, the valence distribution of an information network associated to a selforganized urban street network typically follows a discrete Pareto distribution (Clauset et al 2009). This observed scalefreeness provides a clue to find the prior hypothesis (Jaynes 2003; Grandy 1987) necessary to construct a fluctuating mesoscopic model for the streets, that is, to model the probability distribution to which Monte Carlo simulations are “coming to equilibrium” (Newman and Barkema 1999; Landau and Binder 2015). For mimicking fluctuating transitions, we may use the property that one information network transforms into another when a junction alters its street layout.
Basically, a Monte Carlo simulation iterates a Markov process for generating a Markov chain of states, a sequence of states whose every state depends only on its predecessor (Newman and Barkema 1999; Landau and Binder 2015).^{Footnote 1} Here a state is any configuration of streets or its associated information network (see Fig. 1 for illustration). The Markov process is built so that the Markov chain reaches, when it is iterated enough times starting from any arbitrary state, the prescribed statistical equilibrium. To achieve this, the Markov process has to fulfil (i) the condition of detailed balance and (ii) the condition of ergodicity. The Metropolis algorithm is essentially an implementation choice for the former. The implementation of the condition of ergodicity relies on the details of the systems. The objective of this work is twofold. First, to present how a Metropolis algorithm adaptation can compel these two conditions for selforganized urban street networks. Second, to apprehend whether or not Metropolis simulations can provide pertinent “experimental” data to investigate their scaling coherence.
The rest of the paper presents our innovative modeling approach as follows. The second section carries out the two requested conditions. Firstly, once the probability distribution to come to equilibrium is established, the condition of detailed balance reduces to writing down the emblematic Metropolis acceptance ratio. Secondly, a short analysis enables us to disentangle the stateoftheart paradigms for generating information networks into a constrained ergodic dynamics, which nonetheless recalls the classical singlespinflip ergodic dynamics. This dynamics can potentially become unconstrained. Eventually, our Metropolis adaptation implements itself and compares easily against the classical singlespinflip adaptation for Ising models. Next, the third section compares, over a wide range of scaling exponents, Metropolis generation series against stateoftheart outputs for Central London (United Kingdom). The range of consistency renders scaling investigations around their accepted scaling values feasible. As illustration, we plot the Watts–Strogatz phase diagram with scaling as rewiring parameter. We demonstrate thusly a smallworld crossover curving at realistic scaling values. Accordingly the stateoftheart outputs underlie relatively large worlds. In the concluding section, after a summary of the findings, we point how the presented methodology may contribute, as part of a fluctuating system approach, to change our perspective on urban street networks and, by extension, on cities.
Implementation of the Metropolis algorithm
This section shows how we can apply the classical Metropolis algorithm on unplanned or selforganized urban street networks to generate scalefree streets. We first adapt the most emblematic part, then we design two appropriate dynamics. Each dynamics aims to create from any current configuration of streets a new one. The emblematic part tells us whether or not to accept the new configurations of streets in order for their sequences to tend to a prescribed statistical equilibrium.
The emblematic Metropolis acceptance ratio
Typically Monte Carlo methods are applied to thermal systems. So applying them to a nonthermal system requires the extra preliminary work to frame the statistics of its steady fluctuations. The framework provided by the maximum entropy formalism allows us to derive an equilibrium distribution which is relevant to our scalefree system. This first achievement of our paper is necessary to implement any Monte Carlo method. The resulting Metropolis acceptance ratio takes a typical form.
Scalefreeness as available information
In the classic literature, the prescribed equilibrium distribution is de facto the Boltzmann distribution (Newman and Barkema 1999; Landau and Binder 2015). The same modern tools that derive the Boltzmann distribution from a conservation argument allows us to establish the prescribed equilibrium distribution of a scalefree system through a symmetry argument. We obtain a discrete Pareto distribution of an undefined quantity. This result should be folklore in some area, but we could not locate it in the literature.
At thermal equilibrium, the probability \(p_{\mu }\) for a thermal system to occupy any state (see Footnote 1) \(\mu\) is assumed to yield the Boltzmann distribution
with \(E_{\mu }\) the energy of state \(\mu\) and \(\beta\) the inverse temperature (Newman and Barkema 1999; Landau and Binder 2015; Grandy 1987). We have \(\beta =1/kT\) with k the Boltzmann constant and T the temperature. Nowadays the probability distribution (1) can easily be derived by applying the principle of maximum entropy (MaxEnt) formulated by Jaynes (1957) as a general principle of probability theory (Jaynes 2003; Grandy 1987; Lawrence 2019). Within the maximum entropy formalism, Boltzmann probability (1) becomes the most plausible probability distribution that preserves the total energy of the system on average. This preservation is formally a constraint imposed on the mean of the energy. In practice the constraint is treated as a standard variational problem (Jaynes 2003; Grandy 1987) using the method of Lagrangian multipliers (see for example Applebaum 2008, App. 2). The Lagrangian writes (Jaynes 2003)
where \(p_{\mu }\) is our unknown probability distribution and the first Lagrange multiplier \(\nu\) forces its normalization, while the second Lagrange multiplier \(\beta\) imposes the mean energy to have the constant energy value \({\left\langle {E}\right\rangle }\). The stationary solution of Lagrangian (2) is the desired probability distribution \(p_{\mu }\) (Jaynes 2003); we have
for arbitrarily small variations \(\delta {p_{\mu }}\) of \(p_{\mu }\). Resolving (3) immediately gives
as partition function (Jaynes 2003; Grandy 1987). Probability distribution (4) is Boltzmann probability (1) expressed in its canonical form (Jaynes 2003; Grandy 1987). If the maximum entropy formalism tells us how to treat total energy preservation, noticeably it does not tell us why we choose this constraint over another. Formally the preservation of the total energy is part of the initial hypothesis or available information (Jaynes 2003; Grandy 1987) that we have on systems in thermal equilibrium.
For selforganized urban street networks, our only available information is scalefreeness. However scalefreeness is not a preserved quantity but rather a property (Stanley 1971). But, at the same time, scalefreeness of a selforganized information network may result from a selfsimilarity inherited from its selforganized city (Kalapala et al 2006; Batty 2008). Selfsimilarity is a symmetry (Mandelbrot 1982), a transformation that lets an object or a system stay invariant. Symmetries play a fundamental role in modern physics (Gross 1996; RomeroMaltrana 2015; GarcíaPérez et al 2018). A general consensus in physics is that an invariance to a transformation underlies a preserved entity, and vice versa (Gross 1996; RomeroMaltrana 2015). Let us see how this idea applies here. For our purpose, we must first rewrite Lagrangian (2) in the more generic form
where X is an extensive quantity whose each value \(X_{\mu }\) describes state \(\mu\). An extensive quantity scales linearly under scaling transformations. The new second Lagrange multiplier \(\lambda\) imposes our unknown constraint which expresses in terms of an unknown function \({f}\) acting on X. It literally coerces the mean value of \({{f}({X})}\) to have the constant value \({\left\langle {{f}({X})}\right\rangle }\). For the sake of demonstration, we will assume exact selfsimilarity. Accordingly, under the scaling transformation
a selfsimilar (or homogeneous) function \({{{\Phi }}}(x)\) will transform as
with \(\alpha\) a scaling exponent (Stanley 1971, sec. 11.1). Here, the selfsimilarity invariance holds in the unknown probability distribution \({p_{\mu }={p}(X_\mu )}\). Under transformation (6), \(p_{\mu }\) remains unchanged as expected; we have
If we demand that Lagrangian (5) stays invariant under transformation (6), then
for any scaling factor \({\mathfrak{s}}\) and any possible probability distribution \(p_{\mu }\). Hence, the unknown function \({f}\) satisfies the functional equation
When X takes only positive values x, the most general solution of (10) which is continuous is
with K a constant (Aczél 1966, Th. 2.1.2(2)). Substituting solution (11) into the generic Lagrangian (5) gives the selfsimilar Lagrangian
once the useless constant K is absorbed. One easily verifies that (12) remains indeed unchanged under the scaling transformation (6). The corresponding most plausible probability distribution \(p_{\mu }\) yields the stationary equation
whose solution readily writes
in the canonical form. One quickly checks that probability distribution (14) is invariant under the scaling transformation (6), as expected. This probability distribution is known as the discrete Pareto probability distribution (Clauset et al 2009). Let us summarize our result as follows. What the maximum entropy formalism (Jaynes 2003; Grandy 1987; Lawrence 2019) combined with the symmetryconservation correspondence idea (Gross 1996; RomeroMaltrana 2015) tells us about statistically selfsimilar steady fluctuations is threefold:

(i)
They follow a discrete Pareto probability distribution with the selfsimilar scaling exponent as scaling exponent.

(ii)
They preserve on average the logarithm of an extensive quantity.

(iii)
Their equilibrium parameter is the selfsimilar scaling exponent itself.
However, it can tell us nothing about the nature of the extensive quantity.
Thusly, the prerequisite to Monte Carlo methods for selforganized urban street networks can be expressed as follows. At scaling equilibrium, the probability \(p_{\mu }\) for a selforganized urban street network to develop its streets in any state \(\mu\) is assumed to yield the discrete Pareto distribution
with \(X_{\mu }\) the value at state \(\mu\) of an extensive quantity X and \(\lambda\) the scaling exponent. Still, it remains to make a genuine hypothesis on the extensive quantity X.
A surprisaldriven system
In our context a state \(\mu\) is a possible information network, namely a possible configuration of streets, that an urban street network can develop (see Fig. 1 for illustration). Previous investigations show that an information network of a selforganized urban street network typically underlies scalefreeness (Porta et al 2006; Crucitti et al 2006; Jiang et al 2008). Therefore, as shown in previous section, the distribution of their nodes (streets) preserves on average the logarithm of an extensive quantity, so that this distribution is most plausibly a discrete Pareto distribution of this extensive quantity. This extensive quantity cannot be specified due to our lack of knowledge on information networks of selforganized urban street networks.
However, the simplest assumption we can make is that a selforganized urban street network is a selfsimilar mesoscopic system whose mesoscopic objects have equiprobable configurations. Namely, we apply to our mesoscopic objects the principle of indifference (Jaynes 2003; Lawrence 2019). We may call such a system a selfsimilar Boltzmannmesoscopic system. Our extensive quantity becomes then the number of equiprobable configurations of the mesoscopic objects. Let us denote by \(\Pr (\Omega )\) the probability for a mesoscopic object to have \(\Omega\) possible equiprobable configurations, and by \(o(\Omega )\) a mesoscopic object having effectively \(\Omega\) possible equiprobable configurations. With these notations, we may say that each mesoscopic object \(o(\Omega )\) has \(\Omega\) as extensive quantity. Thence, for each mesoscopic object \(o(\Omega )\), our extensive quantity logarithm \(\ln {\Omega }\) interprets itself either as the Boltzmann entropy of \(o(\Omega )\) or as the surprisal associated to each configuration of \(o(\Omega )\). Surprisal (or surprise, or information content) \({{{\,\mathrm{Su}\,}}=\ln \circ \Pr }\) measures uncertainty, astonishment, and knowledge attached to an event (Tribus 1961; MacKay 2003; Applebaum 2008; Stone 2015; Lawrence 2019). While the average of surprisal over all the possible events gives their (Shannon) entropy, the surprisal attached to a possible event pertains its cognitive magnitude. When an event expected to be rare occurs, we are surprised and we feel that we learn a lot: the larger the uncertainty before the event, the greater the astonishment at the event, the wider the knowledge after the event (MacKay 2003; Applebaum 2008; Lawrence 2019). And vice versa. So that, compared to the entropy interpretation, the surprisal interpretation appears in essence finer and more cognitive. For these reasons, we may favour the surprisal interpretation. The preserved moment \({\sum _{\Omega } \Pr (\Omega )\ln {\Omega }}\) interprets then itself as an amount of surprisal that equilibria preserve on average. We interpret thusly steady fluctuations as a manifestation of uncertainties, astonishments, and knowledges whose the magnitudes remain on average the same. Presuming that this manifestation actually reflects a social process, each equilibrium becomes then a match between steady fluctuating configurations of streets and how citydwellers comprehend their own urban street network (Dover 2004; Benoit and Jabari 2019a, b). We may expect that their comprehension reflects their agility and proficiency to navigate their own urban street network in normal or disrupted traffic.
With this assumption, the probability \(p_{\mu }\) for a selforganized urban street network to develop an information network (or a configuration of streets) \(\mu\) yields
with
the total amount of surprisal for information network \(\mu\); the product (the sum) is over the streets \(s_{\mu }\) and junctions \(j_{\mu }\) of information network \(\mu\). Along the interpretation developed in the previous paragraph, the total amount of surprisal \(S_{\mu }\) (17) quantifies the comprehension of the citydwellers for information network \(\mu\). Thus, accordingly, it is their comprehension that drives probability distribution (16), that is, the statistical equilibrium of their own urban street network.
The Metropolis acceptance ratio
Now that we have set up the fluctuating statistical model of our system, we are ready to implement the emblematic part of the Metropolis algorithm. The Metropolis algorithm holds its specificity among Monte Carlo methods in the implementation details of the condition of detailed balance (Newman and Barkema 1999; Landau and Binder 2015). This condition assures both that (i) each Markov chain (or sequence) reaches an equilibrium and that (ii) the equilibrium states follow the prescribed probability distribution. It applies, technically, to the probability \(P(\mu \rightarrow \nu )\) of generating a state \(\nu\) from a given state \(\mu\) which is called the transition probability; along the constraint
the transition probabilities \({P(\mu \rightarrow \nu )}\) must satisfy the detailed balance equation
Each transition probability \(P(\mu \rightarrow \nu )\) may be split into two parts as
The selection probability \(g(\mu \rightarrow \nu )\) is a probability imposed to our algorithm for generating a new state \(\nu\) given a state \(\mu\), while the acceptance ratio \(A(\mu \rightarrow \nu )\) gives the odds of accepting or rejecting the move to state \(\nu\) from state \(\mu\). For the Metropolis algorithm, the selection probabilities \(g(\mu \rightarrow \nu )\) for all permitted transitions are equal. Scheme (20) along this choice reduces the detailed balance equation (19) into a ratio equation for the acceptance ratios \({A(\mu \rightarrow \nu )}\); we have
The last equality tells us that the odds of accepting or rejecting a move between two states are in favour to the more likely of them. This is common sense. Nonetheless, this still leaves open numerous possibilities. For the Metropolis algorithm, the more likely moves are assumed certain, while the less likely moves get their odds adjusted with respect to (21); we read
For our statistical model (16), the Metropolis acceptance ratio \({A(\mu \rightarrow \nu )}\) takes the more familiar form
That is to say, if the newly selected information network \(\nu\) has a total amount of surprisal \(S_{\nu }\) strictly greater than the current one \(S_{\mu }\), we accept to replace the current information network \(\mu\) by the newly selected one \(\nu\) with the probability given above; otherwise, we accept with certainty.
Two simple ergodic singlejunction dynamics
The stateoftheart generating paradigms are not dynamics. This is primarily because they build each information network from scratch. To be a dynamics, they should instead create a new information network from the current one. An analysis of their streetoriented paradigm gives us clues to design relevant ergodic dynamics. This second achievement of our paper permits us to concretely adapt the Metropolis algorithm to selforganized urban street networks.
A street is an exclusive joined sequence of streetsegments
For information networks, nodes are streets, basically an exclusive sequence of successive streetsegments that are joined at junctions according to some paradigms. By exclusive we mean that a streetsegment can only belong to a single street. This is the perspective used in the stateoftheart literature (Jiang and Claramunt 2004; Rosvall et al 2005; Porta et al 2006; Jiang et al 2008; Masucci et al 2014).
An immediate paradigm is the “named street” paradigm (Jiang and Claramunt 2004; Jiang et al 2008) which simply reproduces cadasters^{Footnote 2}. Since for some cities a cadaster may not exist, or simply reflect local habits and customs, some studies have considered generic substitutes instead. The choice of the paradigm may then ponder social and geographical phenomena. A relevant parameter has appeared to be the deflection angle between two adjacent streetsegments (Jiang et al 2008; Molinero et al 2017). Figure 2 illustrates the notion of deflection angle in our context through two typical junctions. If beyond some threshold angle any joining has to be excluded, many possibilities remain open.
Three paradigms based on deflection angles have been mainly used to generate information networks. Basically these paradigms are nonoverlapping walks governed by a join principle. The everybestfit join principle (Porta et al 2006; Jiang et al 2008) acts at every junction by joining its streetsegment pairs in increasing order of their deflection angles, until applicable. The selfbestfit join principle (Viana et al 2013; Jiang et al 2008) and self[random]fit join principle (Jiang et al 2008) act sequentially on growing streets, until applicable, by randomly seeding them with a notyetselected streetsegment before recursively appending, until applicable, one of the notyetappended streetsegments. The self join principles differ only in the choice of the notyetselected streetsegment to append. Figure 3 illustrates how the inner recursion can construct an entire street; supplementary Animation A1 (Additional file 1) shows how the full machinery can achieve a complete configuration of streets. The selfbestfit join principle selects the one forming the smallest deflection angle; the self[random]fit join principle selects at random. By construction, these three joint principles fall into two categories. The everybestfit join principle is local and almost deterministic^{Footnote 3}; the two self join principles are global and random. The latters clearly differ nevertheless in the degree of their randomness. Unsurprisingly, due to their walkoriented construction, the two self join principles have appeared, against wellfounded cadasters and transportation traffic in terms of correlation, more realistic (Jiang et al 2008). They thusly show that the deflection angle is a suitable parameter for generating information networks. However, the same walkoriented construction renders them not easily tractable. In short, even though it provides a suitable parameter, the stateoftheart approach can not be used to build an easily tractable dynamics.
A junction is a matching of streetsegments
For information networks, edges are junctions, essentially an exclusive set of singletons and pairs of streetsegments that are isolated or paired according to the ongoing streets. By exclusive we mean that a streetsegment can only belong either to one singleton or to one pair. Such a set is, in graph theory, a matching (Pemmaraju and Skiena 2003). To the best of our knowledge, this is the first work that mentions this perspective.
The graph theory perspective can apply on junctions as follows. First, inspired by the dual network representation of urban street networks, we may represent every streetsegment attached to a junction by a node. Let us put each node at the intersection of its associated streetsegment with a circle centred at the junction. Second, we may link pair of nodes whose associated streetsegments have a deflection angle smaller than the deflection angle threshold. Figure 4 illustrates in its two first rows these two steps for three realistic junctions. The resulting graph clearly depends on the deflection angle threshold: when it is set to the flat angle \(\pi\), the graph is a complete graph; when it is set to the zero angle 0, the graph is an empty graph; otherwise the graph is an incomplete graph. We will call such a graph a junction graph. In general, a junction graph has no direct application for our purpose in the sense that any bunch of edges that share a common node (or adjacent edges) corresponds to a set of overlapping streets. In practice, we want a graph without any adjacent edge so that the graph corresponds to a set of nonoverlapping streets. Such a graph is, for a given junction, a matching subgraph (or matching for short) (Pemmaraju and Skiena 2003) of its a junction graph. In short, we are interested by the set of matchings of the junction graphs. The number of matchings of a graph is called the Hosoya index (Hosoya 1971). We will denote the Hosoya index of the junction graph of a junction j by \({\mathscr {Z}}_{j}\). Also notice that a matching can be saturated in the sense that it cannot be expanded to another matching by adding any edge of the underlying graph. Such a matching is called a maximal matching (Pemmaraju and Skiena 2003). Figure 4 gives in its fourth row the set of maximum matchings we can derive for each of its junctions. We will denote the number of maximal matchings of the junction graph of a junction j by \({\mathscr {Y}}_{j}\); we have
Let us now describe the previous generating paradigms in terms of matchings. The “named street” paradigm selects the matching as implicitly recorded in cadasters. The everybestfit join principle chooses for each junction the maximal matching which is optimal in terms of deflection angle distribution. The two self join principles operate at every junction on the set of matchings by successive visits. This becomes more apparent when we interpret their concrete implementations as nonoverlapping walks that haphazardly visit every junction several times. Each visit either steps forward or terminates the walk, that is, each visit selects a subset of matchings. This selection process reveals itself in supplementary Animation A1 (Additional file 1). For the selfbestfit join principle, the move is optimal in terms of deflection angle; for the self[random]fit join principle, the move is random. Over the visits the subset of matchings decreases until it contains only one matching. This remaining matching is a maximal matching since every walk terminates only when no more streetsegment is attachable. Figure 4 draws in its third row the end results of these repeated visits along their maximum matching for each of its junctions. Actually, Fig. 4 sketches, through three realistic junctions, why and how to any junction corresponds a set of maximum matching. The so isolated maximal matchings give the generated information network. In other words, the self join paradigms interpret themselves now as an intricate haphazard fashion to pick for every junction a maximal matching. Thusly, the matching viewpoint allows us to slightly disentangle the two most pertinent join paradigms.
The singlejunctionswitch and flip dynamics
The previous slightly untangled description actually leads to a disembodied form of the self join paradigms with all their underlying principles removed. This is exactly what an ergodic dynamics is about. To the best of our knowledge, no ergodic dynamics has been reported so far for generating configurations of streets.
To begin with, let us deliberately ignore for a while the nonoverlapping walk machineries. The self join paradigms reduce then to choose for every junction a maximal matching regardless of the matchings of the other junctions. So the elementary disembodied dynamics that occurs at junctions is to set up in an independent way a maximal matching. The new set up will generally change the maximal matching into another maximal one. For clarity, this dynamics does not alter the urban street network but rather transforms the information network into another, since the new maximal matching sets a new layout for some of the streets that cross the junction. By now we are able to tell that this dynamics is ergodic. An ergodic dynamics is a dynamics which from any state can reach any other state after a finite number of iterations. It is indeed obvious that we can get from any information network to any other by changing one by one each of the maximal matchings by which the two information networks differ. We coined this dynamics, following the literature (Newman and Barkema 1999) and as an obvious analogy to railroad switches, the singlejunctionswitch dynamics. The restriction to consider only maximal matchings is inherited from the join principles. This restriction is arbitrary in the sense that it is not actually imposed by physical constraints. In fact, the reasoning held above for the singlejunctionswitch dynamics evidently holds for any arbitrary choice of subset of matchings. For completeness, we coined the dynamics that involves all matchings the singlejunctionflip dynamics. Let us recap along these lines. Assuming an entire urban street network, the singlejunctionswitch dynamics is an ergodic dynamics which switches the maximal matching of a single junction into another maximal one, while the singlejunctionflip dynamics is an ergodic dynamics which flips the matching of a single junction into another.
Using the singlejunctionswitch or flip dynamics ensures that our Metropolis algorithm fulfils the condition of ergodicity. It remains however to specify how we select from a given information network a new one which differs by only one dynamics step. For the sake of simplicity, we will only consider the singlejunctionswitch dynamics in the following. The choice of the Metropolis algorithm imposes (Newman and Barkema 1999; Landau and Binder 2015) that the selection probabilities \(g(\mu \rightarrow \nu )\) for each possible new information network \(\nu\) after one dynamics step are all chosen equal—the selection probabilities for all other information networks are set to zero. For an entire urban street network, one dynamics step enables to reach each of the \({\mathscr {Y}}_{j}1\) new maximal matchings of every junction j. Hence the number of possible new information networks that we can reach after one dynamics step from a given information network is the total number of maximal matchings
minus the number of junction N. Therefore we count \({\mathscr {N}}N\) nonzero selection probabilities \(g(\mu \rightarrow \nu )\), and each of them takes the value
In practice we can realize this selection in two easy steps. First we pick at random a junction j with probability proportional to \({{\mathscr {Y}}_{j}1}\). Then we choose at random a new maximal matching among the \({{\mathscr {Y}}_{j}1}\) possible new maximal matchings of junction j.
Another variant of the singlespinflip Metropolis algorithm ?
When all junctions have two maximal matchings, the singlejunctionswitch dynamics is formally equivalent to the singlespinflip dynamics on the original Ising model. Our two above achievements actually combine to give another variation on the singlespinflip Metropolis algorithm theme. This algorithm is a computational interpretation of the Ising model. A brief comparison provides basic physical insights and a simple clue for a crossover as scaling varies.
Informal implementation
As summary of our above results, let us informally implement our adaptation of the Metropolis algorithm to urban street networks as follows.
First, we choose randomly a junction j with probability proportional to its number of maximal matchings \({\mathscr {Y}}_{j}\) minus 1, \({{\mathscr {Y}}_{j}1}\); its streetsegments will be laid out according to some maximal matching \({\mathsf {M}}_{j}\). Second, we pick at random a new maximal matching \(\widetilde{{\mathsf {M}}}_{j}\) not identical to \({\mathsf {M}}_{j}\) among the remaining \({{\mathscr {Y}}_{j}1}\) available possibilities. Third, we calculate the change in the total amount of surprisal \(\Delta {S}\) that would result if we were to lay out this change to this junction. Ultimately, with acceptance probability
$$\begin{aligned} A= {\left\{ \begin{array}{ll} {\mathrm{e}}^{\lambda {\Delta {S}}} &{} \text {if } {\Delta {S}}>0\\ 1 &{} \text {otherwise}, \end{array}\right. } \end{aligned}$$either accept or reject the change.
Properly speaking, this informal implementation encodes the singlejunctionswitch Metropolis algorithm for urban street networks. We let the reader to elaborate the corresponding singlejunctionflip Metropolis algorithm. Meanwhile, the reader may refer to Fig. 5 and supplementary Animation A2 (Additional file 2) for illustration.
A brief comparison with Ising models
The singlejunctionswitch (resp. singlejunctionflip) Metropolis algorithm for our urban street network model mimics the classical singlespinflip Metropolis algorithm for Ising models (Newman and Barkema 1999; Landau and Binder 2015; Berlinsky and Harris 2019; MacKay 2003). Nonetheless our model differs from them in three basic aspects:

(i)
Our model is driven by scaling and surprisal (information) whereas Ising models are driven by temperature and energy. The parallel scalinginformation versus temperatureenergy \((\lambda ,{S})\leftrightarrow (\beta ,{E})\) pours into the discipline the all maturity of thermodynamics and statistical physics. This parallel is actually superseded by the maximum entropy formalism “in a disembodied form with all the physics removed” (Jaynes 2003). This formalism provides, for instance, numerical tools to compute for any information network measure (Newman 2018; Porta et al 2006) its linear response to arbitrary small scaling changes, namely its specificheatcapacitylike coefficient or susceptibility (Newman and Barkema 1999; Grandy 1987; Jaynes 2003).

(ii)
Junctions are nodes of a finite arbitrary planar graph while spins are classically attached to sites of an “infinite” regular lattice. Finiteness means that collective phenomena will get smoother. Arbitrariness renders our model closer to the Ising spinglass models for which the values of the spinspin interactions are no more constant but random (Newman and Barkema 1999; Landau and Binder 2015; MacKay 2003). Collective phenomena in Ising spinglass models are more subtle and more intricate (Newman and Barkema 1999; Landau and Binder 2015).

(iii)
The distribution of maximal matchings (resp. matchings) among junctions is heterogeneous while the distribution of spins among sites is classically homogeneous. That is, junctions have different numbers of maximal matchings (resp. numbers of matchings (Hosoya indices)) while spins have classically the same number of states or the same dimension. Because the number of streetsegments attached to a junction is mostly three or four, the distribution of matching is expected to be statistically homogeneous with a belllike distribution. This contributes to make our model even closer to the Ising spinglass models.
On the other hand, it is noteworthy that the Ising models have became a toy model to crack phase transition and crossover phenomena (Berlinsky and Harris 2019; MacKay 2003). This raises the obvious question whether our model may actually undergo a crossover as scaling varies:

(iv)
Our model experiences, as scaling increases, an ultrasmall to smallworld crossover around the scaling value of 3. The smallworld effect is in effect a statement on geodesic (or shortest) distances between node pairs (Newman 2018). Their mean \(\ell\) behaves in smallworld networks as the logarithm of the number of nodes N, \({\ell \sim \ln {N}}\) (Newman 2018). The smallworld effect becomes extreme in scalefree networks as the scaling \(\lambda\) get smaller than 3 (Cohen and Havlin 2003): the mean geodesic distance \(\ell\) behaves as \({\ell \sim \ln \ln {N}}\) when \({2<\lambda <3}\), as \({\ell \sim \ln {N}/\ln \ln {N}}\) at \({\lambda =3}\), and as \({\ell \sim \ln {N}}\) for \({3<\lambda }\). Thusly, since our generated information networks are scalefree, our model effectively undergoes a smallworld crossover as scaling varies. Clearly its manifestation relies on the behaviour of the number of nodes N. A more substantial analytical work is however beyond the scope of the present paper. Meanwhile, notice that a geodesic distance counts in our context how many changes of street are required for a particular journey. That is, the mean geodesic distance reflects how rapidly on average citydwellers can travel. Accordingly, the crossover diagram of the mean geodesic distance interprets itself as an efficiency diagram. This means, for instance, that our approach provides a method to analyze the relative efficiency of an actual configuration of streets.
Equilibrium Metropolis simulations
What we have achieved in the previous section is adapting to unplanned or selforganized urban street networks the Metropolis algorithm. Now, in this section, we exercise this adaptation in a case study. As case study, we select the urban street network of Central London (United Kingdom), which is a classical example of selforganized urban street network (Jacobs 1993).
Working assumptions
For the sake of illustration, we have made two suppositions. First, we have assumed that streets predominate junctions. Second, we have described the mesoscopic streets as asymptotic agent systems driven by social interactions (Dover 2004; Benoit and Jabari 2019a, b). According to this agent model, a number of vital connections \(\upsilon\) dominates among the possible numbers of connection between agents. So that, the number of configurations \(\Omega _{s_{\mu }}\) of street \(s_{\mu }\) in configuration of streets \(\mu\) becomes proportional to a power of its number of junctions \(n_{s_{\mu }}\) (Benoit and Jabari 2019a, b); we have
So the total amount of surprisal \(S_{\mu }\) (17) in configuration of streets \(\mu\) becomes
up to an irrelevant constant; the sum is over the streets \(s_{\mu }\) of configuration of streets \(\mu\). Thusly our working assumptions bring out an effective scaling exponent \({\widetilde{\lambda }}\) along an effective total amount of surprisal \(\widetilde{S}_{\mu }\); we read
The corresponding effective Metropolis acceptance ratio is literally the tilde version of formula (23); we get
Figure 5 along with supplementary Animation A2 (Additional file 2) show how our Metropolis adaptation can actually generate a sequence of configurations of streets.
Equilibria
Singlejunctionswitch Metropolis generation series versus selffit outputs
Central London offers, as shown in Fig. 6, singlejunctionswitch Metropolis generation series that come to equilibria. The equilibria were attained from selffit outputs through a basic algebraic annealing schedule (Newman and Barkema 1999; Press et al 2007; Galassi et al 2009). To paraphrase: increase (resp. decrease) the control effective scaling exponent \({\widetilde{\lambda }}_{c}\) to \({\widetilde{\lambda }}_{c}(1+\epsilon )\) (resp. \({\widetilde{\lambda }}_{c}/(1+\epsilon )\)) after every m accepted/rejected singlejunctionswitch moves up (resp. down) to the desired equilibrium effective scaling exponent \({\widetilde{\lambda }}\); the initial control effective scaling exponent \({\widetilde{\lambda }}_{0}\) and the parameters \(\epsilon\) and m are determined by experiment. This annealing schedule allowed us to reach equilibria for a range of effective scaling exponent \({\widetilde{\lambda }}\) values large enough to capture the features of our system as follows.
The singlejunctionswitch Metropolis generation series exhibited in Fig. 6 show at least four noticeable properties:

(i)
The sustained Metropolis equilibria (a–f) are clearly comparable to the selffit outputs in terms of order of magnitude of their means and fluctuations. This property holds, as shown Fig. 7, within a window grossly comprised between 1 and 5. We must always bear in mind that scaling exponents of realworld networks are typically comprised between 2 and 3 (Newman 2018).

(ii)
The ground state, namely the sustained Metropolis equilibrium (h) attained for \({{\widetilde{\lambda }}=\infty }\), lays below the selffit outputs by about eleven times their standarddeviation. The ground state was obtained through a ‘simulated annealing’ (Press et al 2007; Galassi et al 2009).

(iii)
The sustained Metropolis equilibrium (a) shows that there also exist equilibria that detach significantly from the selffit outputs from above.

(iv)
The sustained Metropolis equilibrium (d) shows that the singlejunctionswitch Metropolis algorithm can mimic quite well sequences of selffit outputs.
These four properties lead us to claim that the singlejunctionswitch Metropolis algorithm generates series that are consistent with the selffit outputs.
The effective total and average amounts of surprisal, \(\widetilde{S}_{\mu }\) and \(\langle \widetilde{S}_{\mu }\rangle\) respectively, exhibit in Fig. 7 at least two promising properties:

(i)
Their means and their standarddeviations vary smoothly as a function of the effective scaling exponent \({\widetilde{\lambda }}\) at least for values smaller than 4.5. For greater effective scaling exponent \({\widetilde{\lambda }}\) values, our simulations get subject to noise: the effective total amounts of surprisal \(\widetilde{S}_{\mu }\) and its standarddeviation \({{\,\mathrm{SD}\,}}(\widetilde{S}_{\mu })\) continue to vary smoothly while they tend asymptotically to a constant; however, their average counterparts \(\langle \widetilde{S}_{\mu }\rangle\) and \({{\,\mathrm{SD}\,}}(\langle \widetilde{S}_{\mu }\rangle )\) experience noisy variations.^{Footnote 4}

(ii)
They all experience a noticeable change of behaviour within the window comprised between 1 and 4. The means of \(\widetilde{S}_{\mu }\) and \(\langle \widetilde{S}_{\mu }\rangle\) experience both a change of rate that leads them to their respective asymptotic plateau. The standarddeviations \({{\,\mathrm{SD}\,}}(\widetilde{S}_{\mu })\) reach a maximum at right of 1 before decreasing towards an asymptotic plateau; the standarddeviations \({{\,\mathrm{SD}\,}}(\langle \widetilde{S}_{\mu }\rangle )\) has the left profile of a Mexicanhat—left shape of a biquadratic curve—with a minimum around 2.
The latter property strongly suggests that the relevant physics of our system occurs within the window comprised between 1 and 4, while the former property means that Monte Carlo studies within this window are feasible.
The singlejunctionswitch versus flip dynamics
Singlejunctionflip Metropolis generation series came also, by applying the same annealing schedule scheme, to equilibria. Nevertheless, the singlejunctionflip Metropolis generation series contrast with the singlejunctionswitch Metropolis generation series into two major ways:

(i)
The total amounts of surprisal at singlejunctionflip equilibria within the realworld window 2–3 (Newman 2018) appear to be greater than the total amounts of surprisal of the selffit outputs by about 300 times their standarddeviations—while the corresponding ones at singlejunctionswitch equilibria are greater by at most 8 times their standarddeviations. This makes the singlejunctionflip Metropolis algorithm clearly inconsistent with the selffit join principle, hence unrealistic.

(ii)
For large effective scaling exponent \({\widetilde{\lambda }}\) values, the singlejunctionflip simulations appear much less subject to noise.
In brief: restricting matchings to maximal matchings renders our system realistic but numerically unstable for relatively large scaling exponents; vice versa, loosing matchings renders our system unrealistic but numerically stable for a relatively wider range of scaling exponents.
To explain this, we must keep in mind that our particular working assumptions neglect junctions. In fact, in one hand, the singlejunctionswitch dynamics provides to our working assumptions a “hardcoded” constraint on junction layouts so that our system becomes more realistic. On the other hand, the singlejunctionflip dynamics allows the Metropolis algorithm to reject maximal matchings in favour of nonmaximal matchings so that our algorithm becomes numerically more stable. To resolve this dilemma, we may replace the hard constraint on junction layouts with a soft constraint. This may take, in the total amount of surprisal \(S_{\mu }\) (28), the form of additional surprisal terms involving junctions or mixing streets and junctions. The derivation of such terms is however outside the scope of the present paper.
Unorthodox Watts–Strogatz phase diagram
So far we have demonstrated that Central London sustains scalefree configurations of streets over a wide range of scalings. This means that the selection of a realistic scalefree configuration of street involves other criteria than just scalefreeness. An appealing explanation might hold with the smallworld crossover, which may happen as we exposed in our “A brief comparison with Ising models” section. This hypothesis illustrates well the new class of explorations that the method presented in the present paper brings in the field. We keep our hypothesis for future work. Meanwhile, to emphasize our contribution, we demonstrate the smallworld crossover by adopting the phase diagram used for Watts–Strogatz smallworld models (Watts and Strogatz 1998; Newman 2003, 2018).
Tworegime phase diagram
The Watts–Strogatz phase diagram for Central London plotted in Fig. 8 shows two crossovers which occurs simultaneously at the effective scaling value of 3. This phase diagram plots for the information networks of Central London the averages of their mean geodesic distance (or mean vertexvertex distance) \(\ell\) and of their mean local transitivity (or clustering coefficient) C as functions of the rewiring parameter (Watts and Strogatz 1998; Newman 2018). The rewiring parameter is here the effective scaling exponent \({\widetilde{\lambda }}\). These two functions experience a qualitative change of behaviour in the vicinity of effective scaling \({{\widetilde{\lambda }}=3}\), that is, they exhibit a crossover at effective scaling \({{\widetilde{\lambda }}=3}\) (Gluzman and Yukalov 1998). Our claim that the crossovers precisely happen at effective scaling \({{\widetilde{\lambda }}=3}\) relies on the arguments given by Cohen and Havlin (2003). The two simultaneous crossovers indicate two distinct phases or regimes:

(i)
A uniform regime takes place as effective scaling increases from 3. As effective scaling gets higher and higher starting from 3, the mean geodesic distance between node pairs \(\ell\) (resp. the mean local transitivity C) tends on average asymptotically towards a slightlydecreasing (resp. a slightlyincreasing) plateau. The asymptotic behaviours become obvious around the effective scaling value of 4. This means that in this regime the involving phenomena are saturating.

(ii)
An emergent/reduction regime occurs as effective scaling decreases from 3. As effective scaling gets lower and lower starting from 3, the mean geodesic distance between node pairs \(\ell\) (resp. the mean local transitivity C) increases (resp. decreases) on average to reach a linear behaviour around the effective scaling value of 3/2. The decreasing on average of the mean geodesic distance between node pairs \(\ell\) as scaling increases means that scaling is inducing a smaller world (Newman 2018). The increasing on average of the mean local transitivity C as scaling increases means that scaling is inducing a denser world or a world with less “structural holes” (Newman 2018). In our context, a smaller world means journeys with lesser changes of streets ( see end of point (iv) in “A brief comparison with Ising models” section); a denser world means more local alternative routes.
We may regard the two linear behaviours for small and large scalings as degenerate or extreme. In this sense the relevant part of the phase diagram yields between the effective scaling values of 3/2 and 4. This is consistent with our previous expectation in “Equilibria” section that the relevant physics of our system may occur within the window 1–4.
A brief comparison with the classical Watts–Strogatz phase diagram
The Watts–Strogatz phase diagram for Central London confirms that our urban street network model underlies the smallworld effect. Our expectation was sketched in point (iv) of our “A brief comparison with Ising models” section. Nonetheless the obtained Watts–Strogatz phase diagram differs from the classical Watts–Strogatz phase diagram (Newman 2003, Fig. 6.2; Watts and Strogatz 1998, Fig. 2) in three essential characteristics:

(i)
A smaller world means a denser world, not a less dense one. This is because on average the mean local transitivity C increases instead of decreasing.

(ii)
The two crossovers coincide. In other words, no emergent/reduction regime overlaps with an uniform regime and vice versa.

(iii)
The smallworld effect predominates. The overall variation of the average of the mean local transitivity C is of order 0.04, so we may regard the local transitivity evolution as insignificant. Meanwhile the average of the mean geodesic distance between node pairs \(\ell\) gains overall 2.5 nodes and 0.9 nodes within the relevant window from 3/2 to 4, that is, the smallworld effect is actually the substantial phenomenon.
Therefore, contrary to Watts–Strogatz smallworld networks (Watts and Strogatz 1998; Newman 2003, 2018), the information networks of Central London experience no balance between local transitivity and the smallworld effect. Actually, among both, only the smallworld effect is relevant as scaling varies.
The selffit configurations of streets are inefficient
Generation series (d) in Fig. 6 shows that, for Central London, selffit outputs are hardly distinguishable from Metropolis inequilibrium generations at effective scaling \({\widetilde{\lambda }}_{0}=1.495\). This value can be regarded as a measurement of the effective scaling at which Central London sustains selffit configurations of streets. This measurement is represented on the Watts–Strogatz phase diagram for Central London in Fig. 8 by the vertical dotted line. The phase diagram immediately tells us on selffit information networks for Central London three noteworthy facts:

(i)
Their worlds are on average of a magnitude one node larger. As natural reference, we take here the highscaling asymptotic configurations of streets.

(ii)
They occur around the end of the lowscaling linear behaviour. That is, they occur around the lowscaling boundary of the relevant window 3/2–4.

(iii)
There is room for information networks with significantly smaller worlds. A quick check shows that information network worlds at effective scaling 2.50 (centre of realistic window 2–3) and 2.75 (centre of the relevant window 3/2–4) are on average, respectively, 0.6 and 0.7 nodes smaller than the selffit worlds. These offset drops are, respectively, of the order of \(60\%\) and \(70\%\). Namely, they are substantial.
To summarize, the Watts–Strogatz phase diagram describes selffit information networks for Central London as being on average relatively large worlds.
However, Central London dwellers may rather want to know whether their selffit configurations of streets are efficient. Efficiency means here for citydwellers that they can complete their journeys as fast as possible. This can be partially achieved by decreasing as much as possible the number of street changes required per journey. This means to develop information network whose worlds are as small as possible. This corresponds on the Watts–Strogatz phase diagram to information networks having on average relatively small worlds. In effect, this involves the information networks that actually experience the crossover. As we have seen, quite the opposite actually happens to the selffit information networks of Central London: they take place where the smallscaling linear behaviour ceases and they underlie on average relatively large worlds. In brief, the selffit configurations of streets for Central London are inefficient.
Conclusions and future works
Unplanned or selforganized urban street networks undergo a scalefree coherence that we interpret in terms of a fluctuating system. This paper sketches how the Metropolis algorithm, which embodies well the idea of fluctuating systems (Newman and Barkema 1999; Landau and Binder 2015), can apply to selforganized urban street networks once our interpretation is embraced. The Metropolis algorithm is a classical entrypoint for more elaborate Monte Carlo methods. These methods are the natural numerical companions for theoretical studies on fluctuating systems, and vice versa. Our theoretical framework is the maximum entropy formalism (MaxEnt) (Jaynes 2003; Grandy 1987; Lawrence 2019).
Our prior hypothesis (Jaynes 2003; Grandy 1987) is scalefreeness (Stanley 1971). Assuming this property as the result of an underlying selfsimilarity symmetry (Mandelbrot 1982; Batty 2008) paves the way to a symmetryconservation correspondence as used in physics (Gross 1996; RomeroMaltrana 2015). This physical idea effortlessly adapts itself to MaxEnt. This allows us, as required for implementing any Monte Carlo method, to set up our prescribed statistical equilibrium. The selfsimilarity symmetry demands the conservation on average of the logarithm of an extensive quantity which, by virtue of MaxEnt, most plausibly underlies a discrete Pareto distribution (Clauset et al 2009). The scaling exponent is our equilibrium parameter. Meanwhile, the best we can tell on any information network is that it is a mesoscopic system whose objects, nodes (streets) and edges (junctions), have equiprobable configurations. So, the best we can assume about our extensive quantity is that it is a number of equiprobable configurations. The conserved quantity becomes then an average of Boltzmann entropies. However we may rather interpret this information measure as an amount of surprisal (Tribus 1961; MacKay 2003; Applebaum 2008; Stone 2015; Lawrence 2019) that actually quantifies the comprehension of the citydwellers for their own urban street network (Benoit and Jabari 2019a, b). Once our prescribed statistical equilibrium is fully set up, we can readily implement our Metropolis acceptance ratio.
As concerns the ergodic dynamics, its counterpart, the nonoverlapping walk approaches found in the literature (Porta et al 2006; Jiang et al 2008; Viana et al 2013) appear inappropriate but nonetheless inspirational. We imagine information networks not in terms of haphazard nonoverlapping walks along streetsegments, but in term of random street layout at junctions. Our approach readily leads to dynamics that mimic the classical singlespinflip dynamics in Ising models (Newman and Barkema 1999; Landau and Binder 2015; Berlinsky and Harris 2019). At every junction, each pair of streetsegments that can hold a street is a link of a graph where streetsegments map to nodes, so that each matching (Pemmaraju and Skiena 2003) of this graph represents a possible layout. As the singlespinflip dynamics changes the state of a spin into another possible state, our dynamics changes the matching (layout) of a junction into another possible matching (layout). We named singlejunctionflip the dynamics that involves any matchings, and singlejunctionswitch the dynamics that involves only maximal matchings (Pemmaraju and Skiena 2003). If our approach implicitly implies that selforganized urban street networks might sustain scaling coherence over a wide range of scalings, finding dynamics reminiscent of Ising models suggests first and foremost that they might undergo a crossover as scaling varies. Since large scalefree networks exhibit ultrasmall and smallworld behaviours for scaling values respectively smaller and greater that 3 (Cohen and Havlin 2003), selforganized urban street networks might actually experience as scaling increases a smallworld crossover around the scaling value of 3.
We choose as case study the recognized selforganized urban street network of Central London (United Kingdom) (Jacobs 1993). Simulations based on predominant streets and an asymptotic agent model driven by social interactions (Dover 2004; Benoit and Jabari 2019a, b) show that the singlejunctionswitch Metropolis algorithm generates equilibria that are consistent with the aforementioned nonoverlapping walk approaches. The simulations remain consistent over a range of scaling exponents large enough to contain the realistic window from 2 to 3 (Newman 2018) and to capture changes of behaviour in their total and average amounts of surprisal. Thusly, the singlejunctionswitch Metropolis algorithm allows simulational investigations. The singlejunctionflip dynamics also leads to equilibria, but with unrealistic amounts of surprisal. We explain this, given that our model neglects junctions while the singlejunctionswitch dynamics coerces junctions to have maximal layouts, by a lack of constraints on junctions. Along this explanation, the singlejunctionflip dynamics may allow to investigate the role played by junctions. In brief, our simulations on Central London show that our adaptation of the Metropolis algorithm for generating selforganized information networks is applicable and relevant.
To illustrate our innovative methodology, we plot the Watts–Strogatz phase diagram with scaling as rewiring parameter. The phase diagram exhibits an emergent/reduction regime followed by an uniform regime as scaling increases. That is, the smallworld and the local transitivity crossovers occur simultaneously. However only the former is significant in magnitude. The crossovers happen approximately around the scaling value of 3. More noticeably, the crossovers curve within the realistic window from 2 to 3. Thusly, as expected, our phase diagram demonstrates a smallworld crossover around the scaling value of 3. Our phase diagram also allows us to discuss the pertinence of the selffit outputs. The selffit outputs take place on average nearly the scaling value of 3/2, namely, significantly before the realistic window 2–3. They actually occur on average at the ending of the linear scaling behaviour observed at low scalings—which we may consider as degenerate. Concretely this means that selffit outputs generate on average information networks that underlie relatively large worlds, namely, that are inefficient. If the implicit belief that selforganized urban street networks have reached an optimal balance over time holds, representative information networks may rather occur within the realistic window 2–3 where their worlds are on average relatively small, namely, efficient—assuming that the smallworld effect gets counterbalanced as its effect curves. Thusly, our illustrative Watts–Strogatz phase diagram challenges the stateoftheart on generating information networks, while it indicates that selforganized information networks can undergo as scaling increases a smallworld crossover curving within the realistic window 2–3. In other words, our illustrative numerical “experiment” on Central London demonstrates that our adaptation of the Metropolis algorithm for generating selforganized information networks is indeed pertinent to gain new insights.
From a fundamental point of view, future works must focus on two points. First, we must recognize the deep origin underlying the extensive quantity associated to the scaling exponent in order to specify its very nature. Second, we must find an uncoercive way to involve junctions in order to investigate their role. From a simulational point of view, we must investigate the undergoing smallworld crossover by considering other network phenomena (Newman 2018) and a large panel of recognized selforganized urban street networks (Jacobs 1993; Crucitti et al 2006). We anticipate to observe network phenomena (Newman 2018) that counterbalance the smallworld effect within or around the realistic window 2–3 (Newman 2018). From an observational point of view, our fluctuating approach clearly challenges the current method to determine the scaling exponent of an urban street network which is based on a single arbitrary output (Porta et al 2006; Jiang et al 2008). In view to confront our simulational data against observational data, we must derive methods able to “take” the scaling exponent and to measure network measures (Newman 2018) along with their susceptibilities (Newman and Barkema 1999; Grandy 1987; Jaynes 2003). From a practical point of view, we envision that our Metropolis adaptation may initiate, alongside Monte Carlo models getting more elaborate but also more realistic, a ‘scalingdynamics’ based description of our urban street networks and, by extension, of our cities. Such a descriptive framework may provide fruitful analogies with thermodynamics and precious insights on unplanned evolution for city scientists, city designers, and decisionmakers to anticipate the evolution of our cities.
Availability of data and materials
The map of the urban street network of Central London was extracted from the Open Street Map (OSM) comprehensive archive (OpenStreetMap 2004–2020) and simplified with OSMnx \({{\mathrm{v}}0.11.4}\) (Boeing 2017). The network measures were computed with the igraph C library \({{\mathrm{v}}0.8.2}\) (Csárdi and Nepusz 2006; Csárdi et al 2020). The software used to perform the simulations is available, along with the map and generation series samples, at https://doi.org/10.5281/zenodo.3746140. The datasets generated and analysed during the current study are available from the corresponding author on reasonable request.
Notes
 1.
A state is a set of quantities completely describing a system which does not include anything about its history. Along this notion, a dynamics is a map associating to a state another state which does not depend on the past states. A Markov process is a dynamics.
 2.
A cadaster is a comprehensive land register maintained by either local or central authorities. Cadasters have been used, in some parts of the world, for levying taxes, raising armies, setting ownerships, etc.
 3.
The everybestfit join principle is almost deterministic in the sense that it resolves at random the very rare occurrences of equality between deflection angles.
 4.
We attribute the noise to the poor quality of our map data in their small streets and in their simplification of the junctions. The variations of \(\widetilde{S}_{\mu }\) and \({{\,\mathrm{SD}\,}}(\widetilde{S}_{\mu })\) remains relatively smooth because the Metropolis acceptance ratio (30) tends to smooth \(\widetilde{S}_{\mu }\) itself by rejecting the inappropriate states—among them there are the inappropriate states resulting from “corrupted” data. By contrast, the variation of \(\langle \widetilde{S}_{\mu }\rangle\) and \({{\,\mathrm{SD}\,}}(\langle \widetilde{S}_{\mu }\rangle )\) are not smoothed by the Metropolis algorithm in any manner. Furthermore, “corrupted” layout at junctions cannot be rejected because (i) our working assumptions do not take into account junctions in the computation of \(\widetilde{S}_{\mu }\) and because (ii) the singlejunctionswitch dynamics cannot break them since it only allows maximal layouts—this becomes evident as soon as the singlejunctionflip dynamics is used since then \(\langle \widetilde{S}_{\mu }\rangle\) and \({{\,\mathrm{SD}\,}}(\langle \widetilde{S}_{\mu }\rangle )\) vary almost smoothly along their respective asymptotic branch.
Abbreviations
 MaxEnt :

Maximum Entropy formalism (or principle)
 SE::

Scaling equilibrium
 TE:

Thermal equilibrium
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Acknowledgements
This work was supported by the NYUAD Center for Interacting Urban Networks (CITIES), funded by Tamkeen under the NYUAD Research Institute Award CG001 and by the Swiss Re Institute under the Quantum Cities^{TM} initiative.
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JGMB conceived and designed the study, collected and treated the map data, designed and programmed the simulation tools, performed and treated the simulations, and wrote the manuscript. SEGJ helped to shape the manuscript. Both authors read and approved the final manuscript.
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Supplementary information
Additional file 1.
Stateoftheart construction paradigm for configurations of streets (Animation A1).
Additional file 2.
Singlejunctionswitch Metropolis algorithm for configurations of streets (Animation A2).
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Benoit, J.G.M., Jabari, S.E.G. On equilibrium Metropolis simulations on selforganized urban street networks. Appl Netw Sci 6, 33 (2021). https://doi.org/10.1007/s41109021003756
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Keywords
 Urban street networks
 Selforganization
 Scalefreeness
 Metropolis algorithm
 MaxEnt
 Symmetries
 Conserved quantities
 Selfsimilarity
 Surprisal
 Graph matchings
 Ising model
 Watts–Strogatz model
 Smallworld crossover