A synergetic perspective on urban scaling, urban regulatory focus and their interrelations

1.1. Urban allometry

The notion of urban allometry suggests that, similar to natural-organic complex systems, in cities too it can be shown statistically ‘that important demographic, socioeconomic, and behavioural urban indicators are, on average, scaling functions of city size that are quantitatively consistent across different nations and times’ [1]. Based on data from the USA, Germany and China, Bettencourt et al. [1] have proposed, firstly, that most urban indicators can be determined in terms of the following scaling law:

The title of Bettencourt et al.’s paper [1], ‘Growth, innovation, scaling, and the pace of life in cities', indicates that cities differ in their ‘pace of life’, namely, that there are fast- and slow-paced cities—a view that goes back to the classical urban theories of Simmel, Wirth, Milgram and others [2–5]. Bettencourt et al.’s allometric approach attempted to quantify this view, relate it to city size, and show statistically that (contrary to organic systems) ‘the pace of urban life is predicted to increase with [city] size’. As they demonstrated, this property shows itself in the various urban indicators that scale superlinearly (β > 1) with city size, ranging from innovation and wealth creation to crime rates, rates of spread of infectious diseases such as AIDS, and even pedestrian walking speed.

In a subsequent study, Bettencourt [6] rederived the scaling laws in a direct way. Based on a set of four assumptions regarding resources accessible to citizens, infrastructure, human mentality and social interaction, he predicted ‘scaling exponents for a wide variety of urban indicators, from patterns of human behaviour and properties of infrastructure to the price of land’ [6, p. 1439]. By this, he suggested, his modelling approach ‘ties together the most microscopic needs and behaviours of individuals anywhere to the most macroscopic aspects of the urban infrastructure’ [6, (S19)].

When scrutinizing Bettencourt's recent studies, we arrived at the conclusion that they rest on (at least) two preconditions: cities are self-similar fractal structures and they are subject to laws of large numbers; the implication: the approach applies to large enough cities, but not to small city/town sizes.

An alternative interpretation of the relations between the macroscopic properties of urban scaling and the microscopic aspects of urban agents' behaviour was recently suggested by Ross & Portugali [7]. Their paper explains ‘why’ and ‘how’ the size of cities has an effect on their inhabitants' and users’ behaviour. Commencing bottom-up from humans' basic cognitive capabilities, they answer the ‘why’ and ‘how’ by extending the principles of Higgins' [8] regulatory focus theory (RFT) regarding humans’ motivational system, to the context of cities. RFT and its extension to cities as urban regulatory focus (URF) are described next.

1.2. Regulatory focus theory

Higgins’ RFT regarding humans' motivational system demonstrates that individuals’ goal-directed behaviour is regulated by two distinct, and independently operating, motivational systems—promotion and prevention [8]. It further shows that while every person is driven by both promotion and prevention, promotion-oriented individuals are assertive, focus on winning and tend to take risks in order to achieve their goals, whereas prevention-oriented individuals are non-assertive, tend to avoid risks and to focus on not losing.

Higgins’ RFT with its promotion–prevention tendencies refers to individuals' basic personal character and it is thus also termed chronic regulatory focus. Subsequent studies demonstrated that a person's chronic regulatory focus is context dependent, namely, that groups have an effect on the regulatory focus strategies of their members. In particular, Faddegon et al. [9] demonstrated empirically that the likelihood of one's behaving in a promotion or prevention way depends not only on one's personal (chronic) regulatory focus, but also on the atmosphere of the group one belongs to. That is, an assertive high-tech company (e.g. Apple company with its ‘think different’ slogan) and a conservative insurance company (whose possible slogan might be ‘better safe than sorry’) affect differently the personal regulatory focus of their employees. Based on these findings, Faddegon et al. [9] have suggested that promotion and prevention can characterize whole groups, thus giving rise to a collective regulatory focus. This effect was found in the context of small groups where face-to-face interaction prevails as well as in the case of larger groups where face-to-face interaction is rare.

1.3. Urban regulatory focus

Inspired by the above research, Ross & Portugali [7] theorized that regulatory focus processes occur at the urban level and suggested a link between RFT and the statistical findings of urban allometry: as noted above, according to urban allometry studies [1,6], large cities are usually characterized by fast-paced and competitive urban dynamics, compared to small cities which are usually relaxed and slow-paced with a focus on safety, stability and security.

To test the above theorization, Ross & Portugali [7] have conducted a set of laboratory experiments in which each subject was, firstly, tested for his/her chronic regulatory focus (by means of Higgins et al.'s [10] questionnaire); secondly, was shown an (image of) urban scene of a fast-paced or a slow-paced city; and thirdly, was once again tested for his/her regulatory focus. This second test was implemented by means of the recognition memory task (a variation of a ‘signal detection task’), which together with the response bias measure has been successfully used in the past in the context of RFT studies [9,11]. In the present set of experiments, the dependent variable was the participants' response bias that measured the extent to which fast-paced cities versus slow-paced cities affected the personal (chronic) regulatory focus of the subjects. In the model we develop below (§4), we denote the response bias as b. As noted above, while every person is driven by both promotion and prevention, some people are promotion oriented while others prevention oriented. Higher values of the response bias measure (e.g. higher b) indicate an increase in the personal promotion-oriented component due to the urban effect, while lower values (lower b), a decrease of the promotion component and an increase of the prevention component due to the urban effect.

Using the above methodological apparatus in their series of laboratory experiments, Ross & Portugali [7] have demonstrated that the promotion urban context of fast-paced cities (which are usually large cities) tend to intensify both promotion- and prevention-focused behaviours, thus motivating individuals to behave in extreme and polarized ways, whereas slow-paced cities (which are usually the small-sized urban environments), tend to encourage relatively more moderate and less polarized behaviour.

Note that we use and emphasize the expression ‘usually’ in the above paragraphs. This is to indicate that there are exceptions—small but fast-paced cities (e.g. Oxford, Silicon valley, …). Urban scaling studies, such as Bettencout et al.'s, refer to generic cases and general statistical regularities, to which there are always exceptions. Ross & Portugali [7], and thus our present study, refer essentially to slow versus fast-paced cities irrespective of their size, when the link to city size is due to [1,6].

1.4. Aims

The situation so far is as follows: from urban scaling studies we learn that urban allometry with its linear, superlinear and sublinear statistical regularities is a basic characteristic of cities as complex adaptive systems [1]. From Bettencourt's [6] subsequent study, as well as from other studies [12,13], we further learn that the above statistical regularities can be explicitly derived out of the property that cities as complex systems are self-similar fractal structures. This rederivation, however, is implicitly pre-conditioned by the laws of large numbers with the implication that the solution excludes the very small cities and towns in the urban system. Finally, from Ross & Portugali [7] study about URF, we learn that large and small cities affect the prevention–promotion tendencies of their inhabitants and users in different ways as above. What still remains an open question following these studies, however, is the way (or the extent to which) these motivational–behavioural reactions are related to the dynamics of cities as complex, adaptive, self-organization systems. How the dynamic of cities of different sizes is affected by, and is affecting, the promotion and prevention tendencies of their inhabitants and users?

Our aim in this paper is to ‘close the circle’ and answer this open question. We do so from the theoretical perspective of synergetics—Haken's [14] theory of complex self-organization systems—and its application to the domain of complexity theories of cities (CTC) by means of the notions of synergetic inter-representation networks (SIRN), information adaptation (IA) and their conjunction (SIRNIA). According to these theoretical perspectives, and as has been demonstrated in some details [15,16], urban dynamics is characterized by an ongoing interaction between external information/data conveyed by the urban environment and internal information that originates in urban agents' mind/brain—a process captured by the notion of SIRN; in this process each urban agent adapts the incoming information/data from the city, by means of information previously constructed in the agent's mind/brain, as well as by the agent's chronic regulatory focus (i.e. its cognitive motivational inclination)—a process captured by the notion of IA.

Applied to the context of the present study, the external data/information is the size and properties of cities of various sizes as found by urban allometry studies, while the internal information is the promotion–prevention tendencies of the urban agents. The process of SIRNIA here refers to the way urban agents adapt to the information conveyed by the various urban environments, the effects of this adaptation process on their motivation, action and decision-making in the city and the resultant changes that take place in the city.

Our discussion below starts (in §2) with a short reminder of synergetics, SIRN, IA and their conjunction, first in general and then in the context of the dynamics of a city. Section 3 that follows suggests an SIRNIA perspective on individual URF and on its role in the dynamics of cities. In §4, we introduce a mathematical model constructed, on the one hand, on the basis of the theory developed in §3, while on the other, on the empirical findings of Ross & Portugali [7]. The paper concludes with suggestions for further research.

2.1. Synergetics

This interdisciplinary field of research studies how the interaction between many individuals (participants, components, etc.) can produce specific macroscopic structures obeying their own laws [14]. This means that these structures act in specific ways to determine the behaviour of the individuals who (or which) contribute by their behaviour to that of the whole system—i.e. circular causality. As has been shown by synergetics, the whole process can be dealt with by means of two basic concepts and their mathematical formalization: (1) order parameter(s) (OP), (2) slaving principle. An OP (or very few of them) is an observable, in many cases measurable, quantity that describes the state of the total system. Simultaneously, the OP prescribes the behaviour of the individuals (‘slaving principle’—a terminus technicus that has nothing to do with slaving!). A third important concept is the control parameter (CP). Similar to the OP, it is an observable, in many cases measurable, quantity, which, when it grows beyond a certain threshold, has the potential to destabilize the OP and thus leads to a phase transition and to the emergence of a new OP. An efficient way to exemplify the dynamics of synergetics is by means of the laser paradigm that is described graphically in figure 1.

• Download figure
• Open in new tab
• Download PowerPoint

Explicit mathematical applications of the laser paradigm to urban context were suggested by Weidlich [18] who employed the original mathematical formalism of synergetics in studies on public opinion and inter- and intra-urban migrations. Portugali [15,19] studied socio-cultural spatial segregation in cities by means of cellular automata and agent-based urban simulation models, employing synergetics and its laser paradigm as the interpretive theory. He has further suggested the latter interpretive theory in the study of specific urban phenomena such as the New York lofts and Tel-Aviv closed balconies [16].

Some concrete examples may help us to illuminate the range of applications: local concentrations of professions such as tailors in one street; restaurants at a specific square; artists living in the same quarter. Wealthy people build their houses in the same quarter, poor people in others. Racial discrimination leads to segregation. In all these cases, the relevant OP is the number of people having the same attitude or particular feature that distinguishes them from all other people. In this context, the slaving principle means that it is advantageous (in a material or even immaterial context (reputation)) to follow the main stream of the ‘group’ one belongs (or feels) to. In some cases, we may interpret the slaving principle as a social pressure. The control parameters are quantifiable ‘features’ such as land prices, rental rates, to mention but few. Quite remarkably, based on the laser paradigm, equations for the order parameters can be (and have been) established (see above). This kind of approach is continued and developed further in §4 below.

Beyond the above studies, examples of urban analogies are plentiful and exist on many time and length scales, ranging from spontaneously arising, self-regulated human behaviour in the form of applause after a concert when the irregular clapping of hands gives way to a synchronized clapping; or the la-ola wave in a stadium; through traffic on a city highway with speed limit, where at low vehicle density one observes uncorrelated motion, at medium density, uniform motion, while at elevated density, density waves and traffic jam; to cases of Matthew effects where the rich become richer (see [20] in this connection); to the phenomenon of latent demand [21] referring, for example, to the urban reality in which traffic congestion cannot be solved by introducing new roads, because putting in a new road just brings to the front latent demand that is activated by the new road link. (We are thankful to an anonymous reviewer for suggesting this example.)

2.2. Synergetic inter-representation networks and information adaptation

Cities are dually complex in that the city as a whole and each of its parts—the urban agents—is a complex system [15,16]. Looked upon from the perspective of the above laser paradigm, we have two self-organization processes: one at the scale of the city as a whole, in which the agents, by means of their interaction, give rise to the city in a bottom-up manner, and the city acting as OP enslaves the behaviour of its agents in a top-down manner; and so on in circular causality. The other self-organization process takes place in each individual agent, but constrained by rules, laws, etc. of the city. The SIRN process (figure 2a) captures this process.

• Download figure
• Open in new tab
• Download PowerPoint

Cities are also hybrid complex systems in that they are composed of artefacts (buildings, roads, neighbourhoods, cities, etc.) which are simple systems, and human agents, each of which, as just noted, is itself a complex system. Urban agents thus interact not only among themselves but also with the urban artefacts. The process of IA (figure 2b) employs information theory in order to capture the information exchange between urban artefacts and the human agents. The notions of SIRN and IA were introduced in some details in the past [15,16,22,23]; here, we focus on their conjunction (SIRNIA) as it is associated with urban dynamics.

In the context of urban dynamics, the notion of SIRNIA refers to the interaction between urban agents and their city. It suggests that an urban agent is ongoingly subject to two flows (figure 3): a flow of data that comes from the city and a flow of information that originates in, and comes from, the agent's mind/brain/memory. By means of the latter, and in a bottom-up manner, the agent's brain first transforms the incoming data flow into a quantitative, ‘syntactic’, Shannonian information (SHI); this syntactic SHI triggers a top-down process that transforms the quantitative SHI into a qualitative semantic or pragmatic forms of information (SI and PI, respectively). SI refers to the meaning per se (i.e. ‘this is a chair’), whereas PI to the action possibilities afforded by an entity, in a way similar to Gibson's [24] notion of affordances (i.e. ‘this object affords seating’). For a detailed discussion and bibliography on the relations between SHI, semantic information and pragmatic information, see [22,23].

• Download figure
• Open in new tab
• Download PowerPoint

4.1. Impact of U on P

This impact can be taken care of by means of equations (4.4) and (4.5). U_M acts as additional ‘force’ F(t) on $\overline{b}$ in (4.4), so that we identify

There is also an impact of U on P via equation (4.5), or written as a formula

We will solve equation (4.3) with equations (4.8) and (4.12) in §4.3.

4.2. Impact of P on U

As mentioned above, U_M and U_s must be functional of P(b;t) which can be expressed by $\overline{b}$ and S. In this way, U_M and U_s become functionals of $\overline{b}$ and S at several times, i.e. of b(t), b(t’) etc. (In fact, an individual adapts more quickly than a whole city).

We now make the assumption that the citizens adapt quickly so that we may neglect time delays such as t–t′, etc. In other words, U_M and U_s become simple functions of $\overline{b}$(t) S(t), e.g.

Under the reasonable assumption that the impact of $\overline{b}(t),S(t)$, is not too large, we may approximate U_M and U_s by low-order polynomials of $\overline{b},\hspace{0.17em}S$, where the impact of the size of $\overline{b}$ is far more dominant than that of S: an ‘observer’ (new citizen) is far more impressed with the strength of activities (e.g. speed) rather than with their variety. Consequently,

When we insert equation (4.14) in equations (4.4) and (4.8), we observe that the term a₀ $\overline{b}$ of equation (4.14) can be combined with the term $-\gamma \overline{b}$ in equation (4.4) so that, de facto, nothing has changed. Thus, we may drop a₀ $\overline{b}$ or just include it in $-\gamma \overline{b}$. All in all we arrive at our first fundamental model equation

We turn to U_s which is a measure of the restlessness, fluctuations, noise—in our case of a city/village. Here again, the size of $\overline{b}$ is the dominant cause, rather than the variation S of the distribution of b. Thus we arrive at our second fundamental model equation

4.3. Solution to the fundamental equations

Equations (4.15) and (4.16) can be solved in two steps. First we solve equation (4.15) exactly. Then we insert the result in equation (4.16), which can be solved exactly (though perhaps (?) not in closed form). For our present purpose, it suffices to treat the steady state solution to the equations (4.15) and (4.16), i.e.

The quadratic equation (4.17) possesses two solutions (bifurcation!):

Concomitant with these solutions are S₁, S₂. By inserting ${\overline{b}}_1,{\overline{b}}_2$ into equation (4.18), we obtain

And, similarly

4.4. Making contact with observed data

Our model contains five parameters, γ, $b_0,a,Q,c$. As it transpires from the form of equations (4.15) and (4.16), γ plays the role of a relaxation constant, or in other words, γ is the inverse of an adaptation time, τ. Since this is much shorter than the lifespan of an individual or even a city, we may assume that γ is large. Furthermore, we may measure the constants a, C, Q in terms of γ. This entails that we may formally put

So we have to relate $b_0,a,Q,\hspace{0.17em}c$ to observed data. We denote them by ${\overline{b}}_1(0),S_1(0),{\overline{b}}_2(0),S_2(0)$. Using equations (4.19–4.23), we readily obtain

It remains to discuss how to obtain the data of the mean and standard deviation σ (or equivalently $S={\sigma }^2$) of the response bias.

• (a) By means of field experiments with populations of large/small cities.

• (b) As studies in Ross & Portugali [7] paper by ‘laboratory’ experiments.

We observe that the relations (4.24–4.27) allow us to calculate the model parameters a, b, c, Q from measured data, while, conversely, the relations (4.19–4.22) allow us to ‘predict’ the observed data on the basis of the model parameters. To illustrate our further procedure, we analyse ‘the effect of urban context on promotion-focused participants' [7]. We denote this case by 1a) promotion low.

According to [7]

We denote the mean of small city by ${\overline{b}}_1$ and of big city by ${\overline{b}}_2$. We use the variance $S={\sigma }^2$ with indices 1, 2 corresponding to ${\overline{b}}_1,\hspace{0.17em}{\overline{b}}_2$. Thus

Using equations (2.24)–(2.27), we obtain

We apply this procedure to all cases reported by Ross & Portugali [7]:

• 1a) promotion focus low, 1b) high

• 2a) prevention focus low, 2b) high

• 3a) dominant focus: promotion, 3b) prevention.

While the results on $M,\sigma$ (or in our notation $\overline{b},S$) were discussed by Ross & Portugali, scrutinizing table 1 leads us to some new insights based on the parameters $b_0,a,c$. First of all, c > 0 means that larger bias b leads to higher fluctuations in accordance with our above hypothesis. The result that in all cases ${\overline{b}}_1>b_0,\hspace{0.17em}{\overline{b}}_2>b_0$ means that living in a community enhances the personal bias; however, differently depending on his/her regulatory focus. Concerning b₀ and a, we note a remarkable symmetry between 1a) promotion focus low, and 2b) prevention focus high. This means, that low promotion focus and high prevention focus have the same effect on ${\overline{b}}_1,{\overline{b}}_2$. On the other hand, the fluctuations of S₁ are smaller in case 1a) than in 2b).

Approximately the same symmetry holds in the cases 1b), 2a), but this time including the fluctuations. In the case 3a), b₀ is still somewhat larger than that of case 1b) as can be expected. We may interpret $a\hspace{0.17em}{b}_0$ as ‘effective’ social interaction ‘constant’. And, in fact, in all cases 1a) till 3b), this product yields practically the same value! While our model captures qualitatively and quantitively how the city context changes the mean bias of persons and the variance/standard deviation, we are aware that it was based on experimental data of 201 graduate students from the University of Tel Aviv. Surely, more data and studies are thus needed to validate the model and to draw more general conclusions about the behaviour of persons living in large and small cities.

4.5. Stability of solutions

To bring out the essentials, we consider equation (4.15) for the special case b₀ = 0, i.e.

The steady state solutions are

To check their linear stability, we insert ${\overline{b}}_2=(\gamma /a)+\epsilon ,\hspace{0.17em}{\overline{b}}_1=\eta$ in equation (4.30) and obtain (in linear approximation)

Figure 4 shows a plot of V. In terms of figure 4, the position of the ball on the slope of a hill symbolizes the size of $\overline{b}$ and its further development. Evidently the ball on the right slope of figure 4a indicates that $\overline{b}$ will increase indefinitely, i.e. that the average response bias increases forever—surely in contrast to human nature. Thus, in order to apply our model to a real situation, we must invoke a barrier such as drawn in figure 4b. Mathematically, the barrier can be taken care of by an additional term in equation (4.33), whose explicit form is not important for our discussion. As can be seen, ${\overline{b}}_2$ is a lower limit for people who want to belong to the ‘large city’ group.

• Download figure
• Open in new tab
• Download PowerPoint

Note, firstly, that in the light of these considerations and the results of Ross & Portugali's [7] paper, a high enough promotion focus seems to be a necessary prerequisite for the formation of the ‘large city’ group. Secondly, that the just-mentioned behaviour of $\overline{b}$, which can be read from figure 4 left and right, is fully substantiated by the exact time-dependent solutions to equation (4.15) (including $b_0\ne 0$).

4.6. Summary of the model

In our above model, we have dealt with the following phenomenon that results, as noted above, from Ross & Portugali study: urban context of large, fast-paced cities encourages promotion-focused behaviour; adding to this empirical finding the logic of synergetics' circular causality process, it can be said that the encouraged promotion focus behaviour intensifies the fast-paced dynamics of the city, which in turn reinforces and further strengthens a pattern of promotion-focused behaviour, and so on. Thus, we are confronted with a problem of circular causality in a system with many participants, a problem at the core of Haken's synergetics [6]. In the parlance of synergetics [14], $\overline{b}$ plays the role of an OP and equation (4.15) is the OP equation. To tackle this specific problem, following a suggestion made in Ross & Portugali [7], we have chosen the ‘response bias’ of people, b, as the characteristic variable. Our central task has been to determine the equation for the probability distribution function P(b;t) that develops in the course of time, t, because of the circular process, and solve this equation. To this end, we have performed the following steps:

• 1. Since we aim at a comparison with measured data (cf. [7]) that are the mean value M and standard deviation σ and are compared to the complete distribution function P(b;t) sparse data, we invoked a method that allows us to make the best guess on an equation for P(b;t) under the conditions of given $\overline{b}$ and σ or S [17] (for a short summary cf. the appendix). This leads us to a Fokker–Planck–$\hat{\mathrm{I}}$to equation.

• 2. Instead of solving this partial differential equation, we derive from it equations for $\overline{b}$ and S, first for people not in urban context.

• 3. We extend the $\overline{b}$ and S equation by taking the urban context into account.