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# Generative models of scale-invariant systems of cities: Do they agree with or contradict theoretical and empirical expectations?

Cécile Tannier

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Cécile Tannier. Generative models of scale-invariant systems of cities: Do they agree with or contradict theoretical and empirical expectations?. Environment and Planning B: Urban Analytics and City Science, 2025, 10.1177/23998083251367201. hal-05221753

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Submitted on 25 Aug 2025

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# Generative models of scale-invariant systems of cities: do they agree with or contradict theoretical and empirical expectations?

**Abstract.** Cities around the world differ greatly in size and in their social and economic characteristics. Yet quantitative regularities can be observed when considering systems of cities. Quite often, rank-size distributions of cities are approximately scale-invariant and scaling laws relate the size of cities to some particular socio-economic variables. These scaling regularities are worth investigating once acknowledged their deviations to strict scale-invariance and their spatial and temporal variations. One research objective is to devise models incorporating generative processes that are thought to be crucial to the emergence of these empirical patterns, and then to test whether the models can reproduce them accurately. The aim here is to provide a critical cross-disciplinary overview of such models. Three disciplines are covered: economics, geography and physics. Models are examined and compared with respect to the explanatory processes they incorporate, with a focus on the role of space, the diversity of the entities modelled and the dynamics that the models can or cannot simulate. From this examination, recommendations are provided for building generative models of scale-invariant systems of cities that are really relevant.

**Keywords.** cities; interactions; hierarchy; scale invariance; emergence.

## 1. Introduction

Cities around the world are highly diverse in size, shape and socio-economic characteristics. However, quantitative regularities can be observed among systems of cities. In particular scale-invariant rank-size distributions of cities have been noticed very early on (Auerbach 1913; Lotka 1925; Singer 1936; Zipf 1941). Later on, authors identified scaling laws that relate population of cities to their built area (Tobler 1969) or total area (Nordbeck 1971). Then other scaling laws that relate city size to some other socio-economic variable (e.g. GDP, road-network length, economic diversity) have been identified (Kühnert et al. 2006; Pumain et al. 2006; Bettencourt et al. 2007).

Since these pioneering works, a large series of publications have shown that both rank-size distributions of cities and urban scaling laws are not universal and exhibit deviations to a strict scale-invariance. As for scaling laws, values of scaling exponents vary noticeably in both space (Meirelles et al. 2018; Ribeiro et al. 2021; Lei et al. 2022a, 2022b) and time (Pumain et al. 2006; Pumain et al. 2009; Depersin & Barthelemy 2018; Bettencourt et al. 2020; Lei et al. 2022a, 2022b). They also vary strongly according to the definition of cities that is adopted (Arcaute et al. 2015; Cottineau et al. 2017). As for rank-size distributions of cities, the validity of Zipf's law where the
scaling exponent equals 1 has been extensively criticized from both empirical and theoretical perspectives (Pumain 2004; Cottineau 2017). Moreover, rank-size distributions of cities are generally more complex than simple power laws (Verbavatz & Barthelemy 2020). Very often, the upper part of the distribution, corresponding to the largest cities, does not fit very well to any model: these cases of urban primacy were detected a long time ago (Jefferson 1939) and seem to be a generality rather than an exception.

Yet scaling regularities of city systems are worth investigating once acknowledged their deviations to strict scale-invariance and their spatial and temporal variations. Numerous models have been devised by researchers to this end. In designing the models, the idea is to incorporate processes that are assumed to be crucial to the emergence of some kind of scaling regularity and then to ascertain whether and to what extent the models can create or reproduce them. Models designed to this end are highly diverse, mainly because they fall within the scope of different academic disciplines, in particular economics, geography and physics. By the way, they are usually studied and compared separately within each scientific domain although some comparisons across two disciplines can be found (e.g. Pumain 1993; Thisse 2014; Gómez-Liévano & Fragkias 2024). The aim here is to provide a critical cross-disciplinary overview of the models with respect to the explanatory processes of scaling relations they incorporate.

Bettencourt et al. (2013) wrote: "In science, a model can never be proven correct, of course, but models can be excluded if they make not only empirically wrong predictions but also imply counterfactual consequences or if they contradict fundamental theoretical expectations." Indeed several authors have demonstrated that the validity of a model in the social sciences cannot be assessed merely by confronting its results against empirical data (Quine 1953). Haag (1994) emphasizes that rank-size distributions of cities obtained with models can easily be adjusted to observed distributions, which causes a problem of over-identification of models already pointed out by Quandt (1964) and Robson (1973). Moreover, equifinality of models has been recognized at least since Bertalanffy (1968): even if a model allows us to obtain results that conform with observations of the real-world, it does not mean that the processes integrated in the models are the "real-world" ones. Many different models can yield identical results. Conversely, different processes observed in the real-world can each lead to the same resulting configuration. In view of this, the purpose of this paper is not to assess whether the models can reproduce more or less correctly observed scaling regularities. It is not about the validity of the models assessed via the statistical confrontation of their results with empirical data. The purpose is to point out aspects of the models (variables, generative mechanisms) that agree with or contradict geographic and economic theory, or well-established empirical facts.
## 2. Explanatory processes integrated in generative models of scale-invariant systems of cities

Figure 1 provides a synthetic view of the general explanatory processes of city-size hierarchy and urban scaling laws most commonly found in the models. Of course, each model contains only some of these explanatory processes: all processes are never integrated in one single model.

A prominent aspect of Figure 1 is that processes operate at two levels of organization: cities and systems of cities (Berry 1964). Two categories of agent are involved in their dynamics: individuals (or households) and firms, on the one hand, and cities, on the other. The behaviours of individuals and firms is guided by *social interactions* that result from actual or virtual meetings. Meetings may be more or less random and occur during real or virtual trips that may be daily or more or less occasional. Dynamics of cities result from *spatial interactions* between them (competition and cooperation, concurrence and complementarity). Spatial interactions are intrinsically aggregated. They are real or potential flows determined jointly by the size of interacting cities, their respective locations and their activities. Flows may be migrations or exchanges (commercial exchanges, exchanges of knowledge or information, transfers or spin-offs of firms, etc.).

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Processes involved in the emergence and the persistence of scaling regularities that characterize systems of cities.</figcaption></figure>
As suggested in Figure 1, social interactions between individuals and firms as well as spatial interactions between cities are context-dependent and constrained by the geometry (shape) of networks and locations.

Apart from this, models generally introduce additional constraints (or conditions) to ensure they can create some kind of scale-invariance at the level of systems of cities. Concerning spatial interactions between cities, a usual condition is the crucial asymmetry of the interactions. Thus flows concentrate toward the largest cities via a polarization effect: polarization of villages by small towns, polarization of small towns by medium-sized towns, and polarization of all by the largest cities. In some models, the asymmetry of spatial interactions and polarization effects are created by the spatial division of activities and knowledge (i.e. economic and functional specialization of cities) and the hierarchical diffusion of innovations from the largest cities to the smallest ones. In some models, processes happen in open systems subject to demographic or economic growth or decline, or subject to contextual changes (technological changes, political changes, etc.). Asymmetry of spatial interactions, economic and functional diversification of cities (possibly increased by a hierarchical diffusion of innovations) and endogenous or exogenous demographic and economic changes are totally or partly integrated in the models. All these processes reinforce the path-dependence of the systems of cities modelled.

In models where scaling regularities are created by the action of individuals or firms, social interactions are not fundamentally asymmetrical. Their main characteristic is that the probability they will come about rises with the concentration and the diversification of individuals or firms. Moreover, social interactions occur preferentially between individuals or firms belonging to the same environment or the same social network. Over the course of time as modelled, social interactions lead to an increasing complexity of social organization that can include intra- or inter-firm divisions of labour. In some models, additional properties are attributed to social interactions: territorial unity (Bettencourt 2013; Ribeiro et al. 2017), perfect mobility of the labour force (New Economic Geographic models), or increased productivity of interactions between remote individuals or firms according to Granovetter's (1973) principle of the "strength of weak ties" (Ribeiro & Rybski 2023).

Within each city, the increasing complexity of its social organization results from innovations that may be technological, political, cultural or other. These innovations emerge endogenously in the city or appear through a process of hierarchical diffusion of innovations within the system of cities to which it belongs (Hägerstrand 1952; Pred 1977). In Figure 1, no arrow directly relates innovations to the behaviour of individuals and firms. This suggests that a change in behaviour
necessarily occurs through social interactions following the idea that individual behaviour and
collective references are inextricably linked (Tannier et al. 2016).

Last but not least, exogenous influences can bring about changes and cause innovations: the modification of environmental conditions (climatic optimum, drought, epidemic); the arrival of another human group via conquest or colonization; financial or political crises, etc.

The explanatory processes set out in Figure 1 are transposed into generative mechanisms in the models devised by researchers to provide a better understanding of the emergence and evolution of the scaling regularities that characterize systems of cities. As mentioned above, each model contains only some of these explanatory processes. Models primarily differ depending on whether the agents are either individuals (households) and firms, or cities themselves (Table 1).

With New Economic Geography (NEG) models and Settlement Scaling Theory (SST) models,
possible scaling regularities emerge from interacting variables that characterize the behaviours of
individuals and firms. These variables represent stylized facts or average behaviours, typically the
behaviour of *homo economicus*. Modelled social interactions can be exclusively intra-urban (as in
most SST models) or inter-urban (as in most NEG models) or both (e.g. the SST model from
Altmann (2020) and the NEG model from De Palma et al. (2019)).

In models where agents are cities, the processes modelled concern competition between cities, their complementarity and their exchanges of goods, information and population. Parsimonious models are designed to simulate the growth or decline of cities by integrating the fewest variables possible. In general, modelled processes in parsimonious models differ dramatically from processes observed in real-world systems of cities. The two other categories of models incorporate more realistic processes. Some of them integrate economic aspects (in particular, potentially diverse mechanisms of production and exchange) although the nature of the economy itself (for instance, the type of relation between the labour force and firms or the properties of the economic market) is not explicitly stated.

Among the models, some are theoretical and aim to test different hypotheses about the conditions for supra- or infra-linearity of urban scaling laws or about the conditions for hierarchical rank-size distributions of cities. Other models are calibrated with empirical data and applied to real-world systems of cities. Results of both theoretical and empirical models are often confronted to empirical rank-size distributions of cities or empirical scaling laws. However, this does not demonstrate their validity, as pointed out in the introduction. Yet in case of empirical models, the good fit of their
results with empirical distributions enable researchers to suppose that modelled processes can be considered as explanatory.

<table><caption>Table 1. Categories of model selected for the analysis and models taken as example.</caption><thead><tr><th colspan="4">Agents: individuals, households and firms</th><th colspan="4">Agents: cities</th></tr><tr><th>Category</th><th>Sample model</th><th>Discipline</th><th>Theoretical versus empirical</th><th>Category</th><th>Sample model</th><th>Discipline</th><th>Theoretical versus empirical</th></tr></thead><tbody><tr><td rowspan="2">New Economic Geography (NEG) models</td><td>Tabuchi &amp; Thisse (2011)</td><td>Economics</td><td>Theoretical</td><td rowspan="2">Parsimonious models</td><td>Manrubia, Zanette &amp; Solé (1999)</td><td>Physics</td><td>Theoretical</td></tr><tr><td>De Palma et al. (2019)</td><td>Economics</td><td>Theoretical</td><td>Gualandi &amp; Toscani (2019)</td><td>Physics</td><td>Theoretical</td></tr><tr><td rowspan="3">Settlement Scaling Theory (SST) models</td><td>Bettencourt (2013)</td><td>Physics</td><td>Theoretical</td><td rowspan="3">Models closer to concepts and theories of social sciences</td><td>SimpopLocal (Schmitt et al. 2015)</td><td>Geography</td><td>Theoretical</td></tr><tr><td>Yakubo et al. (2014)</td><td>Physics</td><td>Theoretical</td><td>LTP reinforcement model (Aoki et al. 2022)</td><td>Physics</td><td>Empirical</td></tr><tr><td>Lobo et al. (2020)</td><td>Physics</td><td>Theoretical</td><td>MARIUS (Cottineau et al. 2015a, 2015b)</td><td>Geography</td><td>Empirical</td></tr><tr><td></td><td></td><td></td><td></td><td rowspan="3">Complex models integrating numerous observed processes</td><td>Allen (1997)</td><td>Economics Geography</td><td>Empirical</td></tr><tr><td></td><td></td><td></td><td></td><td>Weidlich-Haag (1988)</td><td>Physics</td><td>Empirical</td></tr><tr><td></td><td></td><td></td><td></td><td>Simpop* (Sanders et al. 2006; Pumain et al. 2009)</td><td>Geography</td><td>Empirical</td></tr></tbody></table>

As stated above, the purpose of this paper is to point out aspects of the models that agree with or contradict geographic and economic theory, or well-established empirical facts. Of course, it is not possible here to go into the details of each model. Instead we address three general questions:

1. How does space play a role in the models?

2. Do the diversity and diversification of agents take part in the models?

3. Are the models dynamic and what processes govern this dynamic?

Through the lens of these three questions, we will examine the explanatory processes of Figure 1 as represented in the models. Each question is transversal to these explanatory processes. General answers to the questions will be provided for each category of models set in Table 1 and then

<footnote>* The Simpop2 and Eurosim models.</footnote>
illustrated by taking few models as example. These models have been chosen either because they are emblematic of a particular discipline (geography, economics or physics) or because they exhibit some particularly interesting characteristics. Ultimately, answering the three questions exposed above will enable us to provide a series of recommendations for building relevant models.

# 3. How does space play a role in the emergence or persistence of scaling regularities?

## 3.1. Models with individuals (or households) and firms as agents

As underlined by De Palma et al. (2019), many even recent micro-economic models fail to integrate either the friction of distance or transport costs (Henderson 1974, 1988; Behrens et al. 2014; Davis & Dingel 2019). This contradicts theory in geography that recognises the fundamental role of these variables in the emergence and persistence of scaling regularities. In contrast, transport costs play a crucial role in New Economic Geography (NEG) models that aim to explain urban hierarchies. Under the hypothesis of a perfectly mobile labour force (workers can migrate very easily), it is transport costs that cause economic activity to concentrate. If workers can readily migrate, firms tend to concentrate in space. Conversely, if workers cannot readily migrate (e.g. because of high spatial differences in wages), firms tend to disperse. However, New Economic Geography (NEG) models generally ignore the respective locations of cities. In most cases, space is just a two-location context (two regions or a centre and its periphery), which is the standard approach in international trade theory. Consequently, such models cannot address the emergence of cities and their hierarchy. This is the case of the founding model of Krugman (1991) and its extensions. These are formally centre-periphery models that represent processes by which some activities (i.e. firms) concentrate in one place (a city, a region, a country) whereas others are in the periphery of this activity core. Congestion effects (diseconomies of agglomeration) are omitted from these models.

Nonetheless, notable exceptions can be found, including the models proposed by Tabuchi & Thisse (2011) and De Palma et al. (2019), listed in Table 1. These models are designed to simulate the emergence of cities, whose number, size and location vary with the scenario under consideration. Congestion effects are taken into account. In Tabuchi & Thisse (2011), workers are attracted by places where there are many firms because different goods are available there; but they are repelled by places where they are many workers because this depresses the local labour market. Similarly, firms are attracted by places with many consumers, who provide them with large outlets; but they are repelled by places with many firms because local competition is stiff. The friction of distance is formally represented in a very abstract way. Thus the model fails to allow for the real-world effects
of distance, contrary to spatial interaction models, for instance. In De Palma et al. (2019), spatial variables are more explicit. The utility function of individuals incorporates two terms. The first represents spatial externalities, which the authors call the “spatial interaction field”, and consists in the sum of the interactions of each agent with all others. The intensity of each interaction between two individuals decreases exponentially with the distance between them. The second term represents local congestion effects (agglomeration diseconomies), which vary with the local density of population. Nevertheless, space is isotropic in the models proposed by Tabuchi & Thisse (2011) and De Palma et al. (2019). Moreover, it is represented in a highly simplified way (an infinite straight line or the circumference of a circle) as in many micro-economic models. Consequently, some local differentiations are omitted.

Settlement Scaling Theory (SST) models differ crucially from micro-economic models by their fundamental assumption that cities consist in social interaction networks inscribed physically in space (Bettencourt 2013; Ribeiro et al. 2017; Molinero & Thurner 2021). Urban scaling laws derived from SST models are a consequence of the spatial distribution of individuals imposed by the geometry of cities and by the nature of social relations that reduce or enhance interactions between individuals (Ribeiro & Rybski 2023). Thus in the pioneering model of Bettencourt (2013), listed in Table 1, different aspects relate to the geometry of cities. In particular, each individual is characterized by an interaction surface and a distance travelled daily on the road network. However, spatial variables integrated in SST models are abstract. They can not be calculated from empirical data. Possibly they can be roughly estimated. Carbone et al. (2022) have tried to explicitly calculate the fractal dimension of real-world cities and then to estimate the scaling exponent of resulting scaling laws. However, their results are not conclusive as many other values of fractal dimension, thus values of scaling exponent, could have been calculated. Nevertheless, SST models enable testing different hypotheses that relate to the spatial conditions for supra- or infra-linearity of urban scaling laws. The SST model of Yakubo et al. (2014), for instance, offers interesting insights as regards the influence of the geometry of cities on urban scaling laws. In this model, each city is made up of a geometric network of *N* nodes, each representing an individual. This network is inscribed in a fractal space. Through the analytical study of their model and the analysis of simulation results, Yakubo et al. (2014) show that the value of the exponent of scaling laws Y ~ *N*<sup>*β*</sup> varies with the shape of the network of connected links, the spatial distribution of the attractiveness of nodes, the fractal dimension of the fractal space that contains the network, and the value of a parameter that determines the economic output from the interactions between individuals as a function of the distance between them.
In models where agents are individuals or firms, some spatial variables can be interpreted as the representation of a collective reference shared by all agents. This is the case of the interaction field in the model of De Palma et al. (2019) and in SST models. This collective reference may apply identically to all individuals or differently depending on the value of the spatial variable calculated for each individual. Nonetheless, this kind of spatial variable does not imply the inclusion in the models of a meso- or macroscopic level of organization to complete the microscopic level of individuals or firms. In contrast, in other models such as Lösch's model, total transport costs are minimized at the level of the whole system of cities (Reggiani & Nijkamp 2015). Thus they represent in a way a macroscopic collective reference, which introduces a degree of social determinism into the model.

## 3.2. Models with cities as agents

Many parsimonious models are non-spatial. This is the case of the model of Gualandi & Toscani listed in Table 1. The model of Manrubia et al. (1999) integrates an isotropic cellular space, but the cellular grid is merely a support on which adjacent populated cells form cities. Neither the distance between cities nor the transport network connecting them intervenes in these models. In contrast, in the case of SimpopLocal, the dissemination of innovations between cities is represented by gravity modelling in which a city captures more innovations from another city if the two cities under consideration are large and close together.

Models of the other two categories set in Table 1 are spatial. In the LTP model, road networks are mapped in the form of least-cost paths allowing for topography and hydrography. However, spatial interactions between cities are not modelled: only the distance between cities is involved in calculating the number of exchanges, but not their size. This contradicts theory in geography for which the size of interacting entities plays a crucial role in the volume of exchanges or migrations between them (Ullman 1954).

In contrast, spatial interactions between cities are an essential ingredient in many models such as MARIUS, Allen (1997), Weidlich & Haag (1988) and Simpop* (Table 1). Spatial interactions between cities both condition and limit city growth. Thus spatial interactions largely determine the shape of city rank-size distributions. Besides, Simpop* models explicitly differentiate among the nature of the interactions according to the type of urban functions under consideration. Some functions give rise to exchange networks represented by a gravity model. For other functions like industry and tourism, distance is not involved in the formation of exchange networks.
# 4. Role of diversity and diversification of agents

## 4.1. Models with individuals (or households) and firms as agents

In micro-economic models, utility functions are generally identical for all individuals. However, a string of publications in economics emphasizes the necessity to consider heterogeneous agents (e.g. Behrens & Robert Nicoud 2015; Arthur, 2021). Actually the most elaborate versions of the models of Tabuchi & Thisse (2011) and De Palma et al. (2019) integrate a diversity of agents with different rules of behaviour. In the case of De Palma et al. (2019), cities are all the same size at the end of a simulation in which all individuals are identical. In contrast, a hierarchical system of cities emerges when the model includes two interacting populations of individuals with different utility functions. In this case, the hierarchy of cities with respect to their size and their respective locations varies according to the intensity of both local congestion effects and global spatial interactions. In Tabuchi & Thisse (2011), agents are more diverse than in De Palma et al. (2019): the model encompasses a range of industries, a variety of goods produced, and two types of individual, namely workers (mobile) and farmers (immobile). Simulation results reveal that the equilibrium pattern may involve more than two sizes of city when industries are heterogeneous enough. Besides, in the case of heterogeneous industries, the number of goods and the number of varieties supplied in a city increase with its size, which is reminiscent of Christaller’s urban hierarchy principle.

In SST models, the rules that determine the number of social interactions of each individual and the economic output from these interactions are usually identical for all individuals. However, some researchers put forward the fact that economic growth, increasing returns and agglomeration economies are higher in large cities not only because social interactions are more numerous but also because they are more diversified. “(...) it is not just size but also that it is the diversity of specific intra-city organization of economic and social activity (...) that should be used to understand urban scaling behaviours” (Sarkar et al. 2020). Arvidsson et al. (2023) have also showed that the extreme inequalities that exist within urban populations account for a major part of the inequality between cities, in contradiction with the assumption of city homogeneity. Interestingly, some SST models introduce a diversity of activities within cities. For instance, Lobo et al. (2020) (listed in Table 1) propose an SST model in which the diversity of economic activities (i.e. the number of different activities) is a function of the number of inhabitants. More specifically, the diversity of activities in a city corresponds to the product of the diversity of activities of each individual and the connectivity (number of social interactions) between individuals. The idea of the model is that larger cities offer more possibilities for social interactions because of the increased connectivity between individuals. This increased connectivity favours individual specialization, and thus favours the diversity of
activities. Through their model, Lobo et al. (2020) implicitly assume that more social interactions
enable more innovation resulting in a greater social division of labour. However, they fail to
consider the distinction between the intra- and inter-firm division of labour that is crucial in
economy. In the model of Lobo et al. (2020), the division of labour corresponds actually to the
individual specialization of Smith (1776). Yet Marshall (1890) and Young (1928) later developed
the idea that the division of labour within firms (individual specialization) leads to the division of
labour across firms (individual versatility). Besides, the model of Lobo et al. (2020) disregards the
fact that innovations can create negative externalities and eventually a fall in population.

## 4.2. Models with cities as agents

Some characteristics of cities (e.g. under- or over-representation of some social classes) or some
types of activity are only observed in cities of a certain size or benefiting from a specific site or
situation. Yet the qualitative change in cities according to their size is not necessarily admitted by all
researchers. In a recent paper, Ortman et al. (2020) assert: "Since these can be of any scale,
boundaries between traditional categories of city, town and village are somewhat illusory, and the
terms city and settlement can be used interchangeably." Nonetheless, many researchers
acknowledge that "as objects scale, they change qualitatively and this is particularly true of cities"
(Batty 2020). In other words, a large city is not just an enlargement of a small city: their
characteristics, their functions and their ways of operating are different.

The fact that the socio-economic diversity of cities contributes to the more or less marked hierarchy
of systems of cities is taken into account in Allen (1997), Weidlich & Haag (1988) and Simpop*.
Besides, the joint change of their size and their socio-economic characteristics, evidenced by
geographers and economists (Pumain & Saint-Julien 1978; Duranton & Puga 2000), is represented
in Simpop* by the possibility for each city to acquire supplementary urban functions along with its
growth in population. This principle is also introduced by Allen (1997): external demand is used to
represent global functional changes; then the model distributes these global changes in terms of
activities and population in each place. The other models listed in Table 1 do not integrate this
principle.

Functional diversification of cities in the course of time can also result from the emergence of new
functions or new activities created by innovations. This process is at the heart of SimpopLocal:
cities acquire innovations endogenously and by hierarchical diffusion. Innovations also appear in
Simpop* caused by exogenous inputs. Then the diffusion of innovations is hierarchical as the
largest cities have the largest and most diversified exchange networks, enabling them to acquire the
most recent functions sooner. In contrast, models based on differential equations as with Allen
(1997) and Weidlich & Haag (1988) cannot simulate evolutions created by novelties or innovations in the system modelled. This may explain why the long-term dynamics of Allen's and Weidlich & Haag's models are mainly induced by improvements in transport and communications whereas the long-term changes simulated by Simpop* are mainly induced by the emergence of new urban functions.

# 5. Modelled dynamics

## 5.1. Models with individuals (or households) and firms as agents

Micro-economic models are static in the sense that their results are analysed in a state of equilibrium. Thus when De Palma et al. (2019) evoke the effects of population growth on the simulated number and size of cities, they do not introduce demographic growth per se: they allow for variations in the total population for different simulation scenarios. Similarly, transport costs or the efficiency of transport and communications do not change over the course of a given simulation in the models of Tabuchi & Thisse (2011) and De Palma et al. (2019). The number, size and location of cities are observed in the equilibrium state, considering a fixed value of transport costs or efficiency of transport and communication technologies.

SST models are also static and result in scaling laws at a time t. Under the hypothesis that city systems are ergodic (i. e. exponents of scaling laws estimated for cross-sectional and longitudinal series of data are equivalent) inequalities between cities observed at one point in time are similar to differences in states observed for a single city over the course of time. However, this hypothesis is clearly disproved (Pumain (2012; Depersin & Barthelemy 2018; Xu et al. 2019). Besides, some authors of SST models assert that fundamental components of scale invariance are identical in ancient and contemporary systems of cities, but this claim is highly arguable. Considering for instance urban density, Lobo et al. (2020) claim that “across cultures and history, human settlements become denser, on average and at predictable rates, with population”. However, this claim contradicts the observed decrease in urban density since several decades. Indeed the rate of urban expansion is worldwide notably faster than that of population growth until the 1970s (Liu et al. 2020; Seto et al. 2011). In large European cities the decrease in density began in the most central neighbourhoods (from the late eighteenth century in Paris) before spreading to adjacent neighbourhoods (from 1970–1980 in Paris) and then to the inner suburbs (Pumain 2021).

Finally, NEG and SST models enable researchers to better understand the conditions for the existence but not the evolution of hierarchical systems of cities and the characteristics of such systems.
## 5.2. Models with cities as agents

Most models with cities as agents are dynamic. This is the case of all the models listed in Table 1 except for the LTP reinforcement model. In this model, the total population of the city system modelled is introduced at the end of the simulation and distributed among the cities in proportion to their centrality score. The fact that demographic and economic evolutions are absent from the LTP model is highly problematic because such evolutions strongly influence empirical scaling regularities, namely the exponent of urban scaling laws (Pumain *et al.* 2006) and the shape of rank-size distribution of cities (Pumain 2004).

Basically, the increase or decrease in population or economic activity in a given city may result from *in situ* growth processes: endogenous economic growth (creation of new firms or new outlets, increased output from existing firms, etc.) or demographic growth resulting from natural increase. These endogenous demographic and economic dynamics may differ from one city to another. Such endogenous dynamics are represented more or less explicitly in some models. In Manrubia et al. (1999), a reaction process increases the population of each cell with a probability *p*, or decreases it with a probability $1-p$. In SimpopLocal, innovations acquired by cities increase their economic output and their population. In a specific version of MARIUS models, the wealth (and consequently the population) of cities increases partly as a function of their capacity to take advantage of local resources (coal, gas, oil).

The increase or decrease of population or economic activity in a given city may also result from the arrival or departure of households or firms (migrations, relocations) from or to other cities of the city system under consideration. In Gualandi & Toscani (2019), the amount of population entering or leaving a city depends on the difference between the size of the city and an ideal size fixed a priori. In MARIUS models, a city's wealth increases with the volume of economic exchanges with other cities. This volume is calculated by spatial interaction modelling. City wealth is then converted into population. In Simpop\* models, city wealth (and so population) also increases with the amount of economic exchanges with other cities. In Allen (1997), the growth of population and each type of economic activity of each city is modelled via growth rates common to all cities coupled with random fluctuations. In Weidlich-Haag (1988), incoming and outgoing migration flows between cities produce the global dynamics while asymmetries (or discrepancies) with respect to this global dynamics are modelled econometrically and integrate different city attributes.

Last but not least, the rise or fall in a city's population or economic activity may result from the households or firms arriving from or departing to locations outside the city system under consideration (urban-rural migrations, international migrations, offshoring). In Gualandi & Toscani
(2019), the growth of cities results in part from the arrival of population from their rural environment. A specific version of MARIUS models integrates an urban transition process in which cities form to attract population from their rural environment. The rural-urban migration potential varies spatially. In other models, an exogenous demographic trend raises or lowers the total population of the city system under consideration. In SimpopLocal, the demographic trend represents a monotonic increase in the total population of the system of cities. In Simpop*, the demographic trend applies uniformly directly to each city.

Another fundamental aspects of the dynamics of systems of cities is the change in the range or speed of exchanges or trips over the course of time (Bretagnolle et al. 2000; Bretagnolle & Pumain 2010). This fundamental change is integrated in some models (e. g. Allen, Weidlich-Haag, Simpop*, LTP). However, the co-evolution of city systems and transport networks is never represented in models listed in Table 1. Such a co-evolutionary process is at the heart of the SimpopNet model (Schmitt 2014) and the models of Raimbault (2020, 2021).

# 6. Conclusion

The time-frames of social and spatial interactions differs dramatically (Pumain & Sanders 2013). Social interactions are observed mainly on a daily or weekly time scale whereas interactions between cities are observed over several years or even decades or centuries. More specifically, competition (especially economic competition) between cities (residential migrations, change of activity locations) occurs on the time scale of a few years. The economic specialization of cities that relates to innovation cycles (change in exponent values of urban scaling laws) occurs on the time scale of a few decades. And changes in urban hierarchy (especially the value exponent of rank-size distributions of cities) occurs on a time scale of centuries. Accordingly, models that aim at reproducing a given phenomenon on an inappropriate time scale should be considered irrelevant.

Conversely, variables required in the models vary depending on the time scale under consideration. In particular, models addressing changes that occur on a time scale of a century or more should explicitly or implicitly integrate changes in the economy itself. On the middle and long term, demographic changes as well as changes in the range or speed of exchanges or trips should been taken into account as they are major drivers of the emergence and evolution of scaling regularities. Models addressing dynamics on the time scale of a few decades should integrate changes in the activities of agents thus the diversity of their activities. The necessity to introduce a diversity of agents is not so clear for models addressing changes on narrow time scales (a few years or less) or for static models. Nevertheless, several modelling results show that scaling regularities that emerge
differ crucially if this diversity is taken into account or not. Some results even show that no scaling regularity emerge without agent diversity.

Social interactions between individuals or firms as well as spatial interactions between cities involve size effects that should be incorporated in the models. Both social interactions and spatial interactions are intrinsically spatial (Grasland 2009). Thus space should be introduced in all models. It can be introduced via variables that describe either physical or functional distances between places or the membership to specific spatial entities (e. g. economic networks, political entities or a communities that favours the exchanges between individuals, firms or cities). In case of theoretical models built to explore conditions for the emergence of scaling regularities, it is possible to consider an isotropic space without any spatial differentiation. But in this case, the confrontation to empirical scaling laws or rank-size distributions of cities may be irrelevant. Indeed, empirical laws or distributions deviate to perfectly scale-invariant distributions partly because of spatial differentiations in terms of distances or membership to specific entities. In any case, models can not be validated solely by the statistical confrontation of their results to empirical data because of over-identification and equifinality issues mentioned in the introduction.

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