# Some recent advances in urban system science: models and data

Elsa Arcaute<EMAIL_ADDRESS>Centre for Advanced Spatial Analysis,
University College London, 90 Tottenham Court Road, London, W1T 4TJ, UK  José
J. Ramasco<EMAIL_ADDRESS>Instituto de Física Interdisciplinar y
Sistemas Complejos IFISC (CSIC-UIB), Campus UIB, 07122 Palma de Mallorca,
Spain

###### Abstract

Cities are characterized by the presence of a dense population with a high
potential for interactions between individuals of diverse backgrounds. They
appear in parallel to the Neolithic revolution a few millennia ago. The
advantages brought in terms of agglomeration for economy, innovation, social
and cultural advancements have kept them as a major landmark in recent human
history. There are many different aspects to study in urban systems from a
scientific point of view, just to name a few one can concentrate in demography
and population evolution, mobility, economic output, land use and urban
planning, home accessibility and real estate market, energy and water
consumption, waste processing, health, education, integration of minorities,
etc. In the last decade, the introduction of communication and information
technologies have enormously facilitated the collection of datasets on these
and other questions, making possible a more quantitative approach to city
science. All these topics have been addressed in many works in the literature,
and we do not intend to offer here a systematic review. Instead, we will only
provide a brief taste of some of these above-mentioned aspects, which could
serve as an introduction to a subsequent special number. Such a non-systematic
view will lead us to leave outside many relevant papers, and for this we
apologise.

## Introduction

Cities, and urban systems in general, present generic patterns despite being
the result of a diverse set of processes and constraints. In this section, we
look at some of the old and more recent attempts to encapsulate these within a
mathematical framework. For example, agglomeration effects have been observed
worldwide for over a century. This has led researchers to investigate further
the effects of city size on urban indicators. The topic of urban scaling laws
sprouting from this initial idea, has generated a lot of activity, and through
the scrutinization of the limitations of the method, new paradigms and models
have emerged. The framework on the other hand, does not consider the
heterogeneities within cities, nor the mechanisms giving rise to the observed
urban metrics. And although many of these are still an open problem, these
processes take place within the spatial fabric of the city, and their physical
embeddedness cannot be disentangled from their effect. In this sense, many of
the spatial correlations of the different processes taking place in cities,
are tightly related to the spatial distribution of functions and transport,
which are both closely linked to the morphology of cities. Such an
interdependency is yet to be understood. Advancing this field necessitates a
quantification of the _form_ of the city, and although all cities look
different, they all reveal fractal patterns. These patterns play a role in
modulating the intensity of the interactions between functions. Sometimes, the
modulating distance is not necessarily physical, and corresponds to a proxy of
higher probability of interaction due to similarity or complementarity between
the components. And it is the different intensities of interactions within
systems and across scales that gives rise to another generic pattern observed
in most complex systems: a hierarchical organisation.

## Urban scaling laws: what is missing?

Cities can be thought of structures created as a convergent solution to
sustain the many necessities of human beings. By agglomerating in an area,
individuals have been able to share resources and facilitate exchanges
allowing for better productivity. Over time, different processes, such as
trade, skill matching and specialisation, and the evolution of transport to
mention a few, have come together shaping the spatial distributions of land
uses in cities. Different models attempting to explain the observed patterns
have been proposed since the 19th century, such as von Thünen’s model of
concentric rings of land uses as detailed in his treatise of _The Isolated
State_ in 1826, Christaller’s Central Place Theory [1] aiming at explaining a
hierarchical order in the distribution of settlement sizes and their
functions, and Lösch’s location theory [2] where he emphasized that transport
cannot be disentangled to the observed agglomerations. Overall, for more than
half a century, there have been many proposals looking at cities from the
perspective of flows [3], and of complexity science through Berry’s proposal
to consider cities as “systems of systems” [4]. It is beyond the scope of this
paper to provide a comprehensive review of the different theoretical and
modelling frameworks to cities, it is nevertheless fundamental to mention two
of the pioneers and driving forces behind the development of a science of
cities within the framework of complexity science: Denise Pumain [5, 6] and
Michael Batty [7, 8].

Although the quest to model cities started more than a century ago, capturing
all the processes and their interdependencies continues to be a challenge. On
the other hand, emergent patterns, such as agglomeration economies, have been
identified since a century ago [9], and continue to be investigated [10, 11].
These ideas were extended beyond the realm of economics, and simplified
through the following relationship: $O\sim P^{\beta}$, where the output $O$
corresponds to an urban indicator, $P$ to the population of a city, and the
exponent $\beta$ indicates if there are effects due to concentration of
people, i.e. if $\beta>1$ the output $O$ is more than proportional to the
amount of people $P$ in the city [12]. A cautionary note on the observed
output was brought forward by Denise Pumain [13, 14]. Within the industrial
sector, the value of the exponent will not always be larger than $1$, but it
will depend on the level of maturity of the sector. For example, at the stage
when an industry is producing an innovation, the activity is mostly
concentrated in big cities, and therefore the expected agglomeration effects
with $\beta>1$ will be observed. After this initial adoption period, the
exponent will shift to $\beta\sim 1$, indicating that productivity is no
longer concentrated in big cities, but has diffused to smaller ones where
production might be cheaper due to lower rents and wages. Similar findings
related to the value of the exponent with the phase of economic growth can be
found in [15].

Scaling laws in urban systems attracted a lot of interest in many different
areas [16, 17, 18, 19, 20]. Nevertheless, some contradictions started to
emerge, touching upon important questions related to the role of city size:
are larger cities greener [21, 22, 23]? Are they more congested [24, 25]? And
are larger cities more unequal [26]? In addition, limitations of the method
were also identified: does the result depend on the definition of the system
of cities [27]? Or are these contradictions the result of a poor statistical
modelling framework [28, 29, 30, 31]? Furthermore, the reconciliation between
transversal and longitudinal scaling to better understand the evolution of
cities is yet to be discerned [32, 33]. It is also expected, that such laws
would hold for past and contemporary settlements [34]. Overall, one of the
most problematic aspects of the proposed urban scaling laws, is that they do
not consider the interaction between cities [35], nor their correlations [36].
It is well-known within the study of out of equilibrium systems that power
laws might arise from single processes leading to a homogeneous relationship
such as the simplistic equation described above [37]. Nevertheless, when many
correlated processes intervene, homogeneity is lost. In this sense, when
thinking about a system of cities, cities are not independent and isolated
systems. The correlation between the different processes inside cities needs
to be untangled [38], in addition to considering their co-evolution [14].

In a paper in this special issue [35], the authors considered proximity and
interactions between cities, and found that the value of the exponent changes
under such considerations. In [36], the authors looked at 96 countries and
found that the effect of urban scaling of GDP is correlated to the population
distribution. On the other hand, the authors in [39] showed that the observed
Zipf’s law emerges from the autocorrelation of the distribution of cities.
These latest works emphasize the increasing importance of integrating
interactions between cities and correlations into any theory or modelling of
cities. The importance of transport as an enabler of interactions, shaping the
distribution of functions and evolution of the form of the city, has been
pointed out for over a century ago. Nowadays, many of the interactions are
channelled over the internet, making the physical distance obsolete for some
of the processes. Understanding the extent of the impact of new technologies,
in particular of those as widespread as this one, is essential. The authors in
[40] contribute to this discussion, by analysing the impact of information and
communication technologies (ICT) on the agglomeration benefits, and by
pointing out its effect on the distribution of city size.

## Morphology and inequality

The previous section mentioned the importance of transport in facilitating and
enabling interactions and different types of flows between the system’s
components. In the next section we will look in more detail at some of the
models of flows. What is important to note at this stage, is that the spatial
patterns observed in cities are the outcome of years of restructuring, where
positive feedbacks might have reinforced certain paths. Such a reinforcement
is the outcome of bottom-up self-organising processes and top-down
interventions. These, however, do not necessarily lead to more resilient, nor
optimal cities. The _form_ of a city can hence be seen as the outcome of all
the above-mentioned processes, where no unique solution can be defined. The
quest of identifying the interplay between _form_ and _function_ has been
ongoing for many decades now: does form follow function or does function adapt
to form? Their co-evolution is a complex mechanism involving slow and fast
dynamics. In order to couple these systems, _form_ needs to be clearly
characterised. This has led to a wide variety of frameworks being introduced
to properly define the morphology of cities.

On the one hand, this can be thought in terms of the shape of its built
components, such as the plots and buildings constituting cities [41, 42]. This
cannot be disentangled from the _age_ of the buildings, since the introduction
of new buildings into the city is very much dependent on the probability of
buildings being demolished and plots being repurposed [43, 44]. As cities
evolve, some of the reinforced patterns correspond many times to negative
characteristics, such as segregation and poverty [45]. Understanding the
effect of the physical form on segregation is an urgent and important problem
to address. On the other hand, other reinforcement processes have left a
physical imprint corresponding to the street network. And although these are
the outcome of different intervening constraints around the world: socio-
political, historical, and geographical; the emergent pattern can universally
be recognised as a fractal [46, 47]. Furthermore, through the advancement of
network science [48, 49], street networks have been analysed through the
application and development of different centrality measures [50, 51, 52, 53,
54, 55, 56, 57]. In parallel, a whole discipline emerged from the effort of
connecting the form of cities through network-like approaches to its function:
_space syntax_ [58].

With the advent of sophisticated computational methods allowing for the
collection, manipulation, and classification of big datasets, a
characterisation of cities at a large scale has been possible [59], in
particular making use of methods from machine learning [60, 61, 62]. These
methods have been refined through the combination of different datasets,
including remote sensing data, such as LiDAR [63, 64, 65].

As paths got reinforced and cities evolved, the process took place in a non-
uniform way, leading to different growth rates in the system. This produced a
multifractal organisation of the street network [66, 67], and of urban systems
in general [68]. The self-organised processes, together with top-down
interventions are responsible of the observed morphologies of cities. London
for example, evolved from a multifractal to a mono-fractal, following the
introduction of a greenbelt around the city constraining its growth [66]. In
more extreme cases, interventions such as the one undertaken by Hausmann in
Paris in the late 19th century, have seen large parts of the system destroyed
to reconstruct a new order.

The morphology of a city constraints the spatial distribution of functions.
Multifractal methods can be used to understand both, the spatial disparities,
and the skewness of the distribution [69, 70] leading to inequality. On the
other hand, there are proposals on reducing inequality through the
redistribution of flows [71]. The topics of inequality and segregation could
have their own special issue in the journal, and it is not our intention to
propose a review of the topics here. It is important nevertheless, to mention
their embeddedness within many different layers of the city, including
mobility patterns [72, 73]. This allows us to further motivate the role of
flows within urban systems as an essential component. For example, these play
a central role in the characterisation of spaces [74, 75], and in defining
their importance within the system.

## Hierarchical organisation

The previous section revealed that the morphology observed in cities is a
consequence of the reinforcement of the connectivity between places occurring
in a non-homogeneous way. The heterogeneous interactions take place at
different scales. For example, at a very granular level, denser parts of the
city have a higher probability of encounters, and these could be identified as
neighbourhoods. At the next level, cities can be represented in terms of their
neighbourhoods as nodes, which are many times defined in terms of
administrative boundaries, such as boroughs or census tracts. Neighbourhoods
also present different degrees of interaction between them, which can be
characterised in terms of commuting flows [76], or any other type of
interaction proxy [77]. This generates a nested structure which encodes a
hierarchical structure where different parts of the system are more connected
than others, generating feedback loops across scales. Such a structure is not
confined to cities; interactions can be defined between cities, regions,
countries, or between the different scales [78]. Furthermore, hierarchical
organisation is a commonly observed pattern in complex systems [79, 80].

At this point, it is important to recall that many of the observed processes
are the outcome of interactions which have been modulated by the their spatial
distributions and the speed of transportation [81]. Hence, the co-evolution of
different systems within cities is coupled to technological advancements. And
although many of the interactions take place in the cyberspace, mobility in
cities also shows a strong hierarchical organisation, see for instance [82,
83, 84]. Such organisation mimics Christaller’s idea in the central place
theory for the hierarchy between cities, with some areas being of higher
category than others and attracting or emitting most of the trips (hotspots)
[85, 86]. This is a mesoscopic level of description of the city’s
organisation, since it requires the analysis of aggregated mobility and of the
structure of the hotspots and their levels in space. However, as was shown in
[82], the fact that a city is widespread or more compact in terms of its
centres of activity can be connected to indicators of the quality of life such
as the transport modes use to travel to work, the levels of pollution and
public health records.

## Urban mobility

The role of mobility in cities is to interconnect the different zones
constituting an urban area. There are plenty of reasons why people residing at
a certain neighbourhood need to travel like, for instance, to work, to find
goods and services, for leisure purposes, etc. As shown by the last results
discussed on the hierarchical organisation of cities, both morphological
aspects and mobility are strongly entangled. Nevertheless, these two questions
have been addressed in the literature as two separate issues. The next
important quest is to bring these two fields together. In the meantime, let us
look at the work done on mobility, in particular at the data needed to
characterise it and at the models proposed to explain it at different scales.

Traditionally, census surveys collect the present residence location of
citizens and, in many countries they include a question on the place of
residence in the previous census. Since the period between surveys is around
10 years, this information provided a basis for the very first analyses on
migration flows [87]. Much later, already in the 2000s for the US for example,
the surveys incorporated the question of county of work. In this way, it was
possible to outline the commuting mobility flows at country scale. In terms of
surveys, there have been as well a tradition on performing local travel
surveys in several cities. The final goal is to improve the management of the
public transport system, but since the questions asked are more specific these
are very valuable information sources. Unfortunately, in most of the cases
transport surveys are not standardized across cities and they have an
eminently local character.

For digital data sources, one of the first works was related to the ”where is
George?” experiment [88] in which individuals introduced in a web the code of
the bank notes across the US. Following the notes through their locations, it
was possible to gain insights on the potential travels. However, this could
only provide a type of proxy of mobility. Mobile phone records introduced a
much more direct way to measure it. The first results focused on land use in
different areas of the cities [89, 90, 91, 92], while a later work by M.
González et al. analysed individual mobility patterns [93]. After that, there
has been plenty of work with this type of data, including the analysis of the
social network in space, mobility in cities [94], etc (see, for instance, [95]
for a review). Other digital sources of data include online social networks as
Twitter or Facebook.

## Models of mobility

Theoretical models must adjust to the scales that they intend to reproduce.
Mobility can be seen as a personal phenomenon in which individuals or agents
crawl across the city, or as an aggregated entity connecting city zones with
flows of people travelling between them. The approaches used to address these
two scales are based on different levels of knowledge on the system and input
data details. While individual-based models attempt to reproduce trajectories,
something that requires rules on people decision-making and data on their trip
demands, aggregated models need only parameters accounting for the city-zones
properties as sources or sinks for travels.

## Individual-based models

The original individual models were based on concepts developed for random
walks [96] and Lévy flights (see for instance [97] for a recent review). The
main variable is the position $X_{i}(t)$ of every agent $i$ at time $t$ and
the metrics are built out of it as, for instance, the mean square displacement
per agent or the mean (median) radius of gyration [93]. The basic versions of
these models lead to diffusion of different types depending on the particular
statistics of the trip-lengths or jumps. Nevertheless, the population spreads
over all the space and the final state tends to have uniform density, which is
not a very realistic feature in urban systems.

More elaborated theoretical frameworks add aspects such as continuous time
movements to the random walks, so that agents can travel at any time and do
not do it in a synchronised way, or fractional random walks in which the next
jump is a product of a process with long memory with respect to the previous
displacements. As occurs with Lévy flights, it is possible to consider
ambivalent models in which both the space jumps and the waiting times follow
power-laws. These ideas are motivated by empirical observations in different
datasets [88, 93, 98, 99].

One issue important to mention is that individuals typically return to one or
several locations, e.g. home or work. Taking as basis the previous random-
walk-like frameworks, models have been proposed to include return to previous
visited locations [100, 99]. Usually, the probability of returning is
proportional to the number of visits that an individual has paid to a place,
which defines the so-called preferential return [99]. The rest of the time the
agents explore new environments. It has been also proposed that individuals’
interest in a place may also decline with time in some cases. This mechanism
helps to modify the center of mass of each agent’s movements after a medium-
long period of time and goes under the name of recency [101]. Other studies
have shown that people visit regularly a finite number of places, and if a new
place becomes frequent, another one is abandoned [102].

Humans are a social animal, and, therefore, realistic models need to
contemplate the effect of social interactions on mobility [103, 104, 105].
There are plenty of circumstances in which this becomes an important question:
people may travel in family or in other generic groups; additionally, they may
share a common destination synchronising their trajectories to meet somewhere.
From a data point of view, traces of these behaviours can be observed using
online social networks [106, 107, 108], cell phone records [109, 110, 111] and
surveys [112, 113]. This phenomenon can help to improve individual model
predictions on trajectories thanks to the correlations in displacements with
the social contacts [114]. The effect of group mobility has been considered in
transportation microsimulations since a little more than a decade ago [103,
104, 105, 115, 116, 117, 118, 119, 120]. Interestingly, the relation between
social network and mobility is bidirectional. Our friends determine some of
our mobility patterns, but we establish as well new social relations with
people with meet in the visited locations. Traces of the organisation of
mobility can be detected in the social network, which allowed for the
definition of models characterising this interplay [121, 122, 123].

## Modelling aggregated mobility flows

Passing now to the aggregated models, generally trip flows are characterized
as origin-destination (OD) matrices. Every element of the matrix at row $i$
and column $j$ conveys the information on the number of trips between areas
$i$ (origin) and $j$ (destination). An OD matrix can also be represented as a
directed weighted network, with links pointing from the area of origin to the
one of destination and the weight standing for the number of trips. These
matrices have been used traditionally to express the trip demand between zones
of a city and are, therefore, an essential tool for infrastructure planning.
Finding models able to infer the OD matrices from non-mobility variables is
thus a question of great relevance. Traditionally, two family of approaches
have dominated these endeavours: the intervening opportunities and the gravity
models.

Intervening opportunities models were initially introduced in 1940 [124]. The
idea guiding these models is that the population behaves as a source of trips
and the destination depends on the number of opportunities an agent sees
around her/his residential area. The probability of the agent to select one of
these opportunities and, therefore, to set one destination for her trips
relies on different functions that try to quantify the quality of the
opportunity. These models have been profusely studied for a long time, see for
example Refs. [125, 126, 127, 128, 129, 130]. Recently, a self-consistent
version called radiation model has been introduced [131]. The radiation model
considers the ”quality” of opportunities as a random variable and, as a
consequence, the one to be selected should correspond to the largest quality
value. The selection of extremes under these circumstances generates a few
families of universal distributions, depending on the nature of the original
random variable and its moments. This universality allows thus to close the
expressions and find an equation for the flows of trips between areas. Later,
other versions of the model have been considered to improve the treatment of
the spatial scales and the nonlinear relation between opportunities quality
and zone attractiveness [132, 133, 134].

The gravity model [135] takes the population of the origin as the source for
the trips, so the trip number is proportional to it. The attractiveness of the
destination area is related as well to its population, the relation can be
linear in the simplest form of the model or, more generically, nonlinear. But
the main question differentiating the gravity model from the intervening
opportunities ones is that the flow of trips decays with the distance between
origin and destination with a deterrence function. Most commonly, such a
function can be an exponential or a power-law and its form may depend on the
geographical scale considered, the purpose of the mobility or the
transportation mode [136]. The simplicity of this law has made it very popular
for applications, for instance, in transport infrastructure planning [137,
138], geography [128] and spatial economics [139, 140]. The gravity model can
be deduced from a maximum entropy principle [141]. It is also important to
mention that the basic equation for the gravity model is unconstrained: given
the population and distance between areas, one obtains directly an estimation
of the flows. This is not the case for the radiation model, which is origin
contained with the number of outgoing trips per area given as an input.
Constrained can also be considered in gravity models, these can be at the
origin, at the destination or at both [142, 143]. Assessing the effectiveness
of these models is ongoing research [144], in some cases the scale of the
system plays an important role [145], while in some others, the results from
the radiation and gravity models are in agreement when assessing interventions
[146].

## Comparison between models and future trends

In the case of commuting, there have been several works to compare the
performance of both families of models. For example, the flows predicted by
models with different levels of constraints are directly compared with the
empirical values in Ref. [142]. The results seem to favour the exponential
gravity model, even though by a narrow margin. More recently, a field
theoretical framework for mobility has been proposed [147]. In this case, the
average mobility of the flows out of each area are vectorially averaged and
the different models have been used to explain the empirical patterns. Again,
for the case of commuting the winner was a gravity model with exponential
deterrence function, and in this case the margin with the other models was
much wider.

Hierarchy in space also emerges naturally if one thinks about how the areas
are embedded into one another: neighbourhoods form part of a city, cities lie
within regions, regions constitute countries, etc. Recently, a model to
exploit this hierarchical organisation has been proposed, where mobility at
different scales gives rise to a nested structure of containers [83]. Finally,
and recalling the Song’s model of preferential return, another model has been
introduced exploring the role of the frequency of visits of individuals and
mobility patterns, giving rise to a scaling law between the number of visitors
and the product of the visiting frequency and the travel distance [148].

## Conclusions

In the quest to model processes within cities, whole disciplines have emerged
over the last century, from economics to transport modelling. Although each of
these has produced great advancements, the time has come to couple them.
Firstly, we discussed the need to include interactions and correlations within
and between cities when accounting for agglomeration effects. We pointed out
at the role of transport as an enabler of the interactions driving many of the
observed processes. These take place in space, and the morphology of a city
has proven to play a substantial role in reinforcing certain patterns
determining the location of functions. Furthermore, such patterns present a
hierarchical organisation.

After this, we have discussed mobility, a very interrelated phenomenon with
morphology, and that encompasses the interactions between places. We
introduced the two main modelling approaches to mobility: individual-based
models, and models of aggregated mobility flows. The past decade has seen a
surge in important works on mobility, driven by an increased availability of
mobility data through mobile phones and other social media data, which has
helped fine tune the models.

Agglomeration effects cannot be disconnected from the location of functions,
nor from the differentiated opportunities given by transport, which are
manifested through the mobility patterns left by people in a city. In this
sense, the time is now ripe to try to integrate all these different components
towards a better understanding of cities.

## Acknowledgments

E.A. acknowledges funding from the UK Engineering and Physical Sciences
Research Council (EPSRC) (Grant No. EP/M023583/1). J.J.R. acknowledges funding
from the project PACSS (RTI2018-093732-B-C22) of the
MCIN/AEI/10.13039/501100011033/ and by EU through FEDER funds (A way to make
Europe), and also from the Maria de Maeztu program MDM-2017-0711 of the
MCIN/AEI/10.13039/501100011033/.

## Author contributions

E.A. and J.J.R. contributed equally in conceiving and writing the paper.

## Competing Interests statement

The authors declare no conflict of interest.

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