The Statistical Characteristics of Urban Indicators
Nathaniel Rodriguez
Santa Fe Institute, REU
University of Redlands
2010 August 13
Advisor: Luis Bettencourt
Theoretical Division, Los Alamos National Laboratory
SFI External faculty
ABSTRACT
Many characteristics of cities, such as income, patent production, GDP, and violent crime, scale
with population according to universal scaling laws. These laws are a good description of the general
trends in these urban indicators, but there are also other underlying structures in the dynamics of cities
such as urban indicators exhibiting universal statistical behavior. We have been exploring the behavior
of urban indicators and their deviation from their power laws to show that they behave similarly over
time, over multiple indicators, and over multiple nations. We also find that the indicators of these cities
follow the same types of statistical deviations from the power law and that the deviations about the
power laws of various nations for specific quantities exhibit the same distributions. These results shows
that regardless of city size, national origin, or the specific quantity, indicators share universal statistics,
even though urban centers have diverse histories and develop in very different environments.
1. INTRODUCTION
(Morris, 1902) in his comments on cities which
exemplify this outlook; one which would
influence the way policy makers and urban
theorists approach cities for many decades to
come.
Through most of this history and still
today the approach taken by city planners and
architects to 'better' urban centers has been one
based on aesthetic appeal while using
geometrically inspired designs that intuitively
could reduce or eliminate some of those
common issues that rise out of dense urban
centers such as crime, pollution, and
transportation (probably the most common in
today’s modern cities). This approach however
is based off of a very qualitative and old
understanding of urban centers which are
assumed to behave according to linear models
which ignore the inherent non-linear
agglomeration effects displayed by most urban
indicators. However, recently there has been a
paradigm shift in the way we think about cities
and how they operate. This new perspective
views urban centers as robust complex systems
Since the industrial revolution where
urbanization first took root, cities have become
a topic of great importance. During this time
cities stood at the cross roads of many social
problems including poverty, disease, crime,
social inequality, and the rising of new
ideologies. They were a boiling pot of conflict.
However, they were also the centers of
development,
economic
growth,
and
innovation. This dichotomy heavily influenced
the perspective that social scientists held toward
cities. This outlook was an emotional one as it
focused on these social problems that grew out
of cities and their relation to human wellbeing;
they chose to view cities as complicated and
chaotic entities which seemed to lack order, or
need order in stored upon them. William
Morris, a prominent architect and political
activist during the mid-nineteenth century
refers to “...the hell of London and Manchester”
and the “... wretched suburbs that sprawl
around our fairest and most ancient cities”
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(Batty, 2008), ones which we can understand in
a quantitative sense, and which may behave
according to some basic and fundamental
principles.
One of the largest problems that
research into urban systems can shed some light
on is that of sustainability. The population of
the world has exceeded 6.5 billion people, and
more than half of these now reside within urban
centers. Cities now exist as the largest
consumers of resources, and producers of waste
world-wide. This makes them a logical choice
as a place in which to solve this problem
through an intelligent enactment of policies
based on predictive models developed from
data. This problem could be practically
intractable without a firm understanding of how
cities operate and what pressures can be exerted
to incur change in the desired direction.
This leads to another important problem
that this research can help solve, which is urban
planning and development. As I had mentioned
before urban policies have been largely
influenced by outdated concepts of how cities
work. There is the lack of quantitative methods
for determining the performance of a city,
which can be used to facilitate long term policy
development and planning.
Lastly, cities represent the largest units
of human social interaction in the world, and
yet there is very little that we understand about
the mechanics of their function. We hope to
show that cities share some basic universal
properties that are not dependent on time or
nation of origin.
To be able to do these things we need
data from actual urban areas which we gain
from a number of sources. Some of the major
ones include the FBI and Bureau of Labor
Statistics (U.S.), China data online, and Urban
Audit (Europe). The types of data that are
collected include total income of households,
gross domestic product (GDP), violent crime,
patent production, energy consumption, road
and
pipe
length,
and
employment.
Unfortunately data of this kind has only
become readily available over the last decade,
though some quantities go back about half a
century (such as income). For the most part
governments have only recently become
interested in collecting extensive amounts of
data and making it publicly available. Our
research focuses on income, GDP, patents, and
violent crime, but we plan to include additional
indicators in the future.
It is our goal in this project to use this data
to show that there are universal properties that
all cities share regardless of time, national
origin, or urban indicator.
2. RESULTS
Power Law Scaling
It has been shown in previous work (L.M.A
Bettencourt, et al., 2007) that many urban
indicators scale based on population according
to power law relations. A power law relation is
something that has become more common to
observe in natural systems, in particular in
biology with the scaling of many facets of the
organism including metabolic rate (Geoffrey
West, et al., 2010). Due in part to this, it has
started to become common to consider cities as
organisms (L.M.A Bettencourt, et al., 2007)
(Samaniego, 2008), as you can construct many
parallels between the function of a city and that
of a biological organism (in resource
consumption and waste production, in road
networks, and in many other ways), which is an
interesting concept to behold. These scaling
relations suggest that cities are self-similar
across population scales. So we can introduce
the concept of a ‘perfect’ city in the sense that
its urban indicators follow perfectly their power
laws with no deviation. If all cities behaved this
way we could say that they are all perfectly
scaled version of each other; Flagstaff with a
population of about 50,000 would be identical
to New York with a population of over 20
million if you scaled it to that size. Of course,
actual cities do deviate from the power, but
what we will find later is that these deviations
exhibit some very interesting properties.
The power law itself is defined
according to the following expression:
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𝑌(𝑡) = 𝑌𝑜 𝑁(𝑡)𝛽
increasing returns.
Increasing returns of scale is
characterized by a 𝛽 > 1 (generally between
1.1 and 1.3), which means that a larger city will
posses or produce more of the particular
quantity per person than a smaller city. This
includes quantities like crime, GDP, patents,
and income.
These three groups tell us in general that
it is not surprising that cities exist, as you can
consider them as entities that can create wealth
(GDP) and produce innovation (patents) at a
much higher rate than individuals or small
groups could, while at the same time being far
more efficient with its use of resources than
those same individuals or small groups. Seen in
this light their existence, while may not be
inevitable, is certainly favorable.
What we aim to do in addition to this
previous work is to compare urban indicators
over time, over multiple nations, and in relation
to other indicators. By doing this we can
determine if cities share these properties
universally, which gives us insight into their
nature and robustness. This is an important
statement to make because for the longest time
cities have been viewed as these complicated
and chaotic entities that are a mosaic of
competing and conflicting forces. Each has its
own history of development in a different
environment, and are each inhabited by people
of many different cultural backgrounds and
political institutions. Yet regardless of these
things, cities remain inherently the same worldwide, which challenges what we traditionally
think about the nature of these systems.
We can compare indicators by
collapsing the power law relation. I will explain
how we do this first for indicators over time,
and then for multiple indicators (and nations).
According to our power law relation (1) we can
write can the linear form as:
(1)
Where Y(t) is the quantity being measured at
some time t, 𝑌𝑜 is some normalization constant,
N(t) is the population at the same time t, and β
is the scaling factor by which the quantity
increases in relation to population. Power laws
can be identified by plotting quantities on a loglog scale and observing that their relation
becomes linear with some slope β, which is the
scaling exponent (of course the process of
identifying an actual power law relation and its
scaling exponent is much more rigorous than
this – see Clauset, et al., 2009). Some urban
indicators that exhibit power law scaling and
their respective scaling exponents are listed in
previous work done by Luis Bettencourt, et al.
(2007). The scaling exponents are broken down
into three separate categories: economies of
scale, linear relations, and increasing returns of
scale.
Economies of scale are characterized
by a 𝛽 < 1, which means that a larger city will
posses or require fewer of the particular
quantity per person than a smaller city. This
includes mostly infrastructural quantities such
as road surface, electrical cable length, and the
number of gasoline stations within the urban
center. This sort of relation can be thought of as
an increasing capacity in larger cities to be able
to efficiently allocate and use infrastructural
related resources.
Linear relations are characterized by a
𝛽 ≈ 1, which mean that the quantity scales
exactly proportionally with population. These
include: total housing, total employment, and
water consumption. That these quantities
exhibit this sort of scaling make qualitative
sense in that if we had a population of 1000 in
some urban center and we increased the
population by 1000, we would expect that we
would need approximately as many addition
homes as we had needed with our initial 1000.
This is in contrast to the other two scaling
relations where we would need either fewer
homes for the next 1000 (and the next 1000
after that even fewer) in the case of economies
of scale, or more homes in the case of
log(𝑌) = log(𝑌𝑜 ) + 𝛽log⁡(N) (2)
The intercept log⁡(𝑌𝑜 ) represents a correction
value that takes into account any biases that
may arise in the data from year to year (such as
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inflation, or differences in currency from one
nation to another). We then fitting a β to all the
years using (where we know that β does not
vary much from year to year):
follow the scaling law as they grow. The case of
crime is similar, though the data is more
discrete, causing a falling off effect from the
power law as the city population becomes small
and you begin to have cases that exhibit no or
very little violent crime for a particular year.
This sort of effect seen at the tail is common
amongst power laws in general (Clauset, et al.,
2009). For urban centers this effect is only
apparent for the more discrete indicators such
as violent crime and patents, but there is no
reason to expect that it should not be true of
income or GDP for sufficiently small
populations, however urban centers are defined
in the U.S. as having populations of 50,000 or
greater, and this definition is similar in Europe
and other countries (save China), so there is a
limit in size for what can be considered an
actual urban system. What these two graphs
exemplify is that cities share these properties
for many decades, and so strongly suggests
universality over time.
The approach we took for collapsing the
data for multiple quantities and nations was
different because the scaling exponents can
differ by a larger amount than for a single
quantity year to year. As a result large errors
log⁡(𝑌𝑖 ) − 〈log⁡(𝑌)〉 = 𝛽[log(𝑁𝑖 ) − 〈log(𝑁)〉]
(3)
Where the ‘‹›’ brackets represent the mean
value. Then we fit the intercept, log⁡(𝑌𝑜 ), for
each year so we are able to determine what the
correction factor should be. Using this we then
subtract the correction term from the linear
form to get:
log⁡(𝑌⁄𝑌𝑜 ) = 𝛽log⁡(𝑁)
(4)
Where we then define the respective axes of our
graph according to 𝑓 = log⁡(𝑌⁄𝑌𝑜 ) and 𝑥 =
log⁡(𝑁).
Figures 1a and b show the results of
collapsing U.S. income and crime for about 360
cities each year. For the case of income the
points for each year fall exactly on the same
line. The trails left by some individual cities as
their population changed from year to year are
visible and show how those cities closely
Figure 1. These figures show the collapsed power laws for U.S. income and violent crime in a loglog plot where the slope of the line represents the scaling exponent. From year to year the slopes of
the indicators change only marginally. (a) The income graph shows how well the data can fall
together on a single line. Trails of the cities can be made out in the graph as they follow the power
law. The two most obvious cases of this are New York (far top right) and Los Angeles (just below
New York), where you can see that they are fairly average cities that do not deviate much from the
power law. (b) The violent crime graph exhibits a tail that falls off of the power law. This is due to
the discreteness of the data; in small cities the number of violent crime can drop to zero in some
years.
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would propagate from the origin where the
intercept is fit to the various quantities, whose
values may differ by many magnitudes. For
each urban indicator we use the same technique
as described previously to collapse the data
over multiple years, we then determine the
mean values of the combined data for the
indicator, which are expressed as 𝑌̅, the mean
̅ , the mean
of the quantity being observed, and 𝑁
of the population. By subtracting the means
from their respective indicators the graphs are
centered at the origin and problem of error
magnification is reduced as all quantities are
essentially scaled relatively to their means.
Figure 2 shows the results of collapsing
multiple quantities. Binned versions of crime
and patents were used to reduce the effect of
the discreteness of the data, though one could
also simply cut off the tails of the power laws,
which is a common practice, though a sensitive
one as you lose data for use in the fitting of the
power law exponent to the data. If too much
data is cut off you begin to lose confidence in
the accuracy of the exponent. If this is done the
bias produced in the scaling exponent due to the
non-Gaussian nature of the distribution of the
data is reduced. Still, using binned quantities
still shows that each quantity falls closely
together on the same line, with the minor
exception of patents who have a slightly higher
scaling exponent than the other quantities,
though it is not obvious that this is a result of
the actual way in which patents scale or
whether it is a relic from the biased distribution
at its tail. Regardless, this plot shows that there
is a surprisingly close connection between
indicators of increasing returns. This goes to
strongly suggest that there exists universality
between cities for arbitrary indicators. Now it
may be argued that there could be correlative
connections between the indicators themselves,
but it has been shown in recent research by
Bettencourt and others (Bettencourt, et. al.,
2010), that there exists only weak or no
correlations between various indicators such as
income and patents, or income and violent
crime, to have some examples. To be clear,
these are measuring the relative correlations
between the quantities independent of
population size. These results mean that
Figure 2. A graph of the multiple collapsed quantities on a log-log plot.
Both crime and patent data have been binned in this case. The quantities
fall closely on the same line, though patents has a slightly larger slope
than the other indicators.
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indicators can be considered to be essentially
independent of each other to some degree.
Lastly for power laws, there is figure 3
which shows the result of collapsing multiple
nations. The GDP for a number of countries
including the United States, China, and some
European nations are included in the plot. The
plot shows a very strong connection between
the GDP’s of different nations, as they all have
close to the same scaling exponent and they all
fall along the same line. China of course stands
out at being somewhat unique in comparison to
the other countries. This is probably due to the
incredible economic growth that the nation has
been experiencing over the last decade. In
contrast, the much more stable economies of
the United States and Europe show much
narrower distributions from the power law than
in China. What is also interesting is the strange
‘hook’ that appears to separate itself from the
rest of the distribution. This also has a physical
explanation. China has a number of special
economic zones which are made to be attractive
for foreign investors and allows more
capitalistic practices. One of these, Shenzhen is
arguably the most successful and prosperous
city in China and it belongs to one of these
special economic zones. In the 70’s Shenzhen
was only a small fishing village, but now it is
home to over 9 million people. The ‘hook’ in
the data represents the city of Shenzhen. Its trail
shows how over time it still follows the same
power law scaling even though it is far removed
from the rest of the distribution.
From these three cases we can argue in
favor of the idea that there exist universally
shared properties between cities regardless of
time, quantity, or nation of origin with respect
to scaling. What about the deviations from the
power laws? As we have seen in the previous
plots there are some interesting cases (China in
particular) where the data deviates from the
power law. Eying the plots qualitatively does
not lend one to think that there could be a single
Figure 3. A graph of GDP collapsed across multiple nations. The data
show a striking similarity in their slope. The more economically stable
nations in Europe and the United States show smaller deviation from
their power laws. However, China has a much larger spread in distri-bution. The ‘hook’ that is present in the Chinese GDP is represented
by Shenzhen, the most prosperous city within the special economic
zones of China.
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the power law, but rather exhibit a ‘long term
memory’ as Luis Bettencourt describes it in his
recent research (Bettencourt, et. al., 2010). Take
for example San Jose, which is now known as
Silicon Valley. It over performs relative to the
power law in various quantities such as income
and patents. If in 1969 San Jose is found to be
over performing relative to the average, then it
will continue to over perform for many
decades. This effect is best shown in a plot
borrowed from Luis’ paper (figure 4). The
figure shows how the residuals for income
remain relatively steady over many years.
Cities that underperform continue to
underperform, such as Brownsville, over many
years. This is regardless of the many policies
this city has no doubt enacted in an attempt to
pull them out of relative poverty. This shows
how ineffective current policies are at actually
influencing long term growth or at changing the
behavior of the city. Also indicated in the plot
are gray bars which represent national periods
kind of distribution that would explain the
behavior of all the residuals.
Residual Distributions
Now we will consider the deviations from the
power laws. We define the residual as the
difference between the power law and the data
at any particular population size. The exact
expression is:
𝛽
𝜉𝑖 = log(𝑌𝑖 ) − log⁡(𝑌𝑜 𝑁𝑖 )
(5)
Where ξ is the residual, 𝑌𝑖 is the observed
quantity for a particular city, and the last term is
the power law expressed for the population of
that city. Essentially the residuals define the
uniqueness of a city, as each city has its own
deviation from the power law for each of its
urban indicators, and this combination of
residuals is specific to each city. These
residuals have some interesting properties.
They do not have a random distribution about
Figure 4. A plot of the residuals for income in the United states from 1969 through
2006. The gray bars represent times of national recession. The plot shows the
tendency of the residuals to remain steady over many decades, as well as their
robustness in being unaffected by economic downturns. An special case is San Jose,
whose residual spikes during the internet boom of the later 90’s and then falls to
its previous levels following the respective bust in the early 2000’s.
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of recession. Even during these periods the
residuals remain incredibly robust and appear
not to be affected at all by these economic
down turns. San Jose, however, is an interesting
case as you can see how the internet boom and
bust affect the city’s income. In this case the
effects of the upward and downward market
forces are clearly observed in the residuals, but
even after the bust the residuals return to their
pre-internet boom values! Bridgeport could also
be considered another specific case as it is a
banking town. So during times of recession
when banks are affected by the economic
downturn this urban system is more expressive
of those changes.
Due to this nature of residuals a city
taxonomy can be created (Bettencourt, et. al.,
2010) which matches the performances of
various urban centers independent of
population size so you can gain a true relative
performance indicator for an individual city
with respect to others in the nation.
It is reasonable to ask whether similar
universalities can be found as those discovered
within power laws given this knowledge of the
systematic behavior of residuals? To answer
this question we look at the residuals in a
collapsed form so that they can be presented
together and so that we can draw out any
similarities or differences in the distributions.
This is done by taking the probability density
function in terms of the particular quantity and
eliminating parameters:
ln⁡(𝑌⁄𝑌𝑜 𝑁𝛽 )
𝑁
𝑃(𝑌) = 𝑠𝑌𝑙𝑛(10) 𝑒
−|
ln⁡(10)
𝑃
𝑃(𝑥) = 𝑁𝑒 −|𝑥|
data (7).
Figure 5 shows an example fit for a
quantity collapsed over multiple years, in this
case the GDP of the United States. This shows a
very good correlation between the distributions
of the data over several years, which suggests
that there are indeed universal properties that
hold from year. This similarity from year to
year is expressed in the other quantities we
looked at as well.
Figure 5b shows our collapse of the
GDP for multiple nations. This uses the same
data that we used in our plot of the collapsed
GDP power laws, so this gives us a look at how
the distributions relate. What is quite
extraordinary is that China shares the same
distribution as all the other nations, even though
its cities had a much greater spread about the
power law. This is an encouraging result as it
helps support that even across nations this
universality in distributions holds true.
We are currently working on collapsing
the distributions for multiple quantities where
we hope to discover the same sort of
universality and complete this analysis.
3. DISCUSSION
What we have found from this research is that
cities exhibit a large amount of universality
across time, quantity, and origin, for both the
power law relations and for deviations from the
power law. This is quite an extraordinary result
as it means that cities are robust systems that
behave according to some very basic rules
which are fundamental to the system and fairly
independent of the environment. We can look at
urban centers anywhere in the world and they
will behave in approximately the same way as
they are the same system. This raises some new
and important questions like: What are the
processes that give rise to these scaling laws
and distributions? Why do these processes
behave in the way that they do? And what are
the rules that govern the development of these
urban centers? Statistical analysis is not
sufficient to answer all of these questions, but it
has shed light on the empirical nature of cities.
𝑃
−𝜇| ⁄𝑠
⁡ (6)
(7)
In the first expression (6) there is the
probability density of the quantity Y, where s is
an estimator, μ is the mean of the residuals, and
P is the exponent of the distribution. The other
parameters are absorbed into x and eliminated
through a change of variables. This leaves only
the parameter P which we can then fit to the
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Rodriguez
Figure 5. (a) A plot of the collapsed distributions of the GDP for the United States from 20012006. The distributions for each year are identical and fit very well to a curve that has an exponent
around 1.6, which is in between the exponential and normal distributions. (b) A plot of the
collapsed distributions of the GDP for multiple nations and years. The GDP from various nations
follows the same distribution which is again in between an exponential and normal. This plot uses
the same data from figure 3 in the collapsing of the power laws.
China represents a very important source of
data to help us answer some of these deeper
questions. The data for the United States only
reliably goes back to 1969, which is not far
enough for us to determine the origins and
reasons why cities like San Jose became over
performing in the first place, or why places like
Brownsville struggle in relative poverty. Due to
China’s fairly recent and extraordinary
economic growth, new urban centers are being
born with a high frequency, and data is
available for us to look into the origins of cities
like Shenzhen in a quantitative sense. This sort
of analysis can help us better understand what
underlying rules might govern the development
of each city and how similar cities differentiate
over time.
The Chinese data though can be difficult
to work with as china does not have a definition
for a metropolitan area in the sense that the
U.S. and Europe do. Generally statistics are
gathered based off of districts where data for
the major city in that district is lumped together
with the surrounding regions (ideally suburbs)
under that cities administration. This is the
closest to a metropolitan definition that China
has at the moment. In addition to this the
population of cities are determined from
residency papers that are issued from the
government and which allow a person so live in
certain area, however these tend to lead to
underestimates of the actual population as there
are many individuals that reside in these urban
areas but do not have residency in them. For a
large city such as Beijing, the official
population and the actual population may differ
by several million.
We plan on collapsing the distributions
for multiple quantities soon; however we found
some unsuspected distributions in the data for
the United States related to income. We found
that there is an asymmetry in the distribution of
the
residuals
where
there
is
an
underrepresentation of very poor cities, while
there is an overrepresentation of moderately
underperforming cities. As we do not have
income data for other nations we cannot
determine yet if this is unique to the United
States or if this is something general to other
cities as well.
We have also found that a statistical
approach using fitting and relying on the best
R-squared value of the fit is insufficient to
determine the kind of distribution the data
follow, as our fitness landscapes are very
shallow. So the difference in R-squared
between a P value of 1 (exponential
distribution) and a P value of 2 (normal
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distribution) is not very large. We aim to solve
this by creating a stochastic model that will
help determine the origin of the distribution.
4.
REFERENCES
Aaron Clauset , C. R. Shalizi, M. E. J.
Newman, 2009, SIAM Review 51, 661-703,
2009
W. Morris, Architecture, Industry and
Wealth: Collected Papers (Longmans, Green,
and Co., London, 1902).
Michael Batty, Science vol 319, February
2008
L.M.A Bettencourt, et al., PNAS vol. 104
no. 17, April 24, 2007
L.M.A Bettencourt, et. al., in review PNAS,
2010
Horacio Samaniego, Melanie E. Moses,
Journal of Transport and Land Use 1:1
(Summer 2008) pp. 21–39
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