Bull Math Biol (2015) 77:390–407
DOI 10.1007/s11538-014-9949-3
O R I G I N A L A RT I C L E
Living in a Network of Scaling Cities and Finite
Resources
Murad R. Qubbaj · Shade T. Shutters ·
Rachata Muneepeerakul
Received: 2 August 2013 / Accepted: 7 March 2014 / Published online: 11 April 2014
© Society for Mathematical Biology 2014
Abstract Many urban phenomena exhibit remarkable regularity in the form of nonlinear scaling behaviors, but their implications on a system of networked cities has
never been investigated. Such knowledge is crucial for our ability to harness the complexity of urban processes to further sustainability science. In this paper, we develop a
dynamical modeling framework that embeds population–resource dynamics—a generalized Lotka–Volterra system with modifications to incorporate the urban scaling
behaviors—in complex networks in which cities may be linked to the resources of
other cities and people may migrate in pursuit of higher welfare. We find that isolated cities (i.e., no migration) are susceptible to collapse if they do not have access
to adequate resources. Links to other cities may help cities that would otherwise collapse due to insufficient resources. The effects of inter-city links, however, can vary
due to the interplay between the nonlinear scaling behaviors and network structure.
The long-term population level of a city is, in many settings, largely a function of the
city’s access to resources over which the city has little or no competition. Nonetheless, careful investigation of dynamics is required to gain mechanistic understanding
of a particular city–resource network because cities and resources may collapse and
the scaling behaviors may influence the effects of inter-city links, thereby distorting
what topological metrics really measure.
B
M.R. Qubbaj ( )
School of Sustainability, Arizona State University, Tempe, AZ 85287, USA
e-mail: [email protected]
S.T. Shutters
School of Sustainability & Center for Social Dynamics and Complexity, Arizona State University,
Tempe, AZ 85287, USA
R. Muneepeerakul
School of Sustainability & Mathematical, Computational, and Modeling Sciences Center, Arizona
State University, Tempe, AZ 85287, USA
Living in a Network of Scaling Cities and Finite Resources
391
Keywords Sustainability · Scaling · Urban networks · Population–resource
dynamics
1 Introduction
Cities host ever growing populations that collectively have already exceeded half
of the world’s population (United Nations Population Fund (UNFPA) 2007). Thus,
any actions adopted in cities will have repercussions on the global economy, the
global environment, and ultimately global sustainability (Bulkeley and Betsill 2003;
Rees and Wackernagel 1996; Yamamoto and Nadaraja 2006; Seto et al. 2010). And
many more areas in the world are yet to be urbanized—by 2030, towns and cities of
the developing world will make up 80 % of urban humanity (United Nations Population Fund (UNFPA) 2007). These new cities have great potential to contribute
to long-term sustainability; whether these potentials will be realized critically depends on decisions and policies made today. Understanding characteristics of cities
and their socio-economic dynamics is crucial for anticipating their performance,
growth trajectories, and how they may affect sustainability (Pickett et al. 2008;
Samet 2013).
Many such characteristics of cities have been documented and analyzed. In particular, many diverse urban phenomena—from physical infrastructure to energy consumption to technological innovations to crimes—exhibit scaling behaviors, or more
specifically, power-law relationships, with city population size (e.g., Batty 1995,
2008; Bettencourt et al. 2007, 2010; Lobo and Strumsky 2008; Fragkias et al. 2013;
Lobo et al. 2013; Alves et al. 2013a, 2013b; Gomez-Lievano et al. 2012). Remarkably, these seemingly disparate scaling behaviors can be synthesized under one coherent framework (Bettencourt 2013). As remarkable as they are, however, these progresses are milestones in, rather than the end of, the pursuit of a practical theory
of cities—as one cannot adequately understand the effects of cities on sustainability
without a network perspective of a system of cities (see, e.g., Neal 2013).
Through unprecedented levels of migration and trade, characteristic of today’s
era of globalization, cities form a vast network with diverse types of connections
(Castles 2002; Wolf 2007; Vanderheiden 2008; Young 2009). These connections
have profound impacts on processes important to sustainability: urbanization (Kates
and Parris 2003; Seto et al. 2010), the spread of invasive species (Trombulak and
Frissell 2000; Colunga-Garcia et al. 2009) and epidemics (Colizza et al. 2006;
Barrat et al. 2013), biodiversity loss (Lenzen et al. 2012), and sustainable development of renewable resources (van den Bergh and Verbruggen 1999; D’Odorico et al.
2010), among other things. Yet, consequences of the urban scaling behaviors highlighted above have never been systematically evaluated for a system of networked
cities. Understanding these consequences is crucial for our ability to harness the complexity of urban processes to further sustainability science.
To address this gap in our understanding, we develop a stylized dynamical model
of populations and resources—a generalized version of the Lotka–Volterra equations
modified for scaling behavior of city attributes and network effects of interlinked
cities and resources. We then ask the following questions. What are the effects of
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the inter-city migration and trade connections? Can we derive network metrics that
predict the system’s dynamical outcomes? And are these metrics dependent on the
scaling behaviors within each city?
2 The Model
A general socio-economic model may be written as follows:
dYi
= FY (Yi ) + ΓT (Y, N, . . . ),
dt
dNj
= FN (Nj ) + ΓI (Y, N, . . . ),
dt
(1)
where Yi is a generic resource at location i, Nj the population of city j , Y = {Yi }
(i = 1, 2, . . . , S), N = {Nj } (j = 1, 2, . . . , C) with S being the total number of resource locations and C being the total number of cities. We assume that the modeled
resource is a general aggregate resource that meets the needs of cities to grow and
maintain their populations. FY and FN capture the intrinsic dynamics of resource
Yi and population Nj , respectively. ΓT and ΓI capture the movements of resources
(T for trade) and people (I for immigration/migration), i.e., the network effects.
Needless to say, the socio-economic dynamics are complicated and the general
model above can be specified in multiple ways with varying degrees of complexity. For clarity, a number of simplifying assumptions are necessary for this initial
investigation—discussion on some potential extensions is provided in Sect. 4. With
that in mind, we now conceptually specify different model components as follows
(with detailed descriptions following in subsequent paragraphs). The rates of resource harvest/extraction and the rates of resource consumption needed to maintain
existing populations are assumed to exhibit power-law relationships with population
size (Muneepeerakul and Qubbaj 2012). Members of a population can move between
cities, with the population flows being proportional to the per capita welfare in different cities—i.e., people pursue higher individual welfare. The network structure
comes into play by restricting the possible pathways of people and resource flows: a
city has access to some, but not all, resources, and people can move to some, but not
all, cities. Finally, the intrinsic dynamics of the resource is assumed to follow logistic growth, a commonly used formulation in mathematical biology. In summary, we
arrive at the following equations:
dYi
Yi
β
− Yi
= ri Yi 1 −
Hij Nj ,
dt
Ki
j
β
Nj
Mj α
dNj
Wj
=
Hij Yi −
Nj − v j Nj +
ξj k v k N k .
dt
Ej
Ej
l ξlk Wl
i
(2)
k
We now describe the model in more detail. The above system of equations has
the familiar Lotka–Volterra structure, but with key modifications, namely power-law
Living in a Network of Scaling Cities and Finite Resources
393
relationships and inter-city movement of people. For resource dynamics, the intrinsic
growth rate and carrying capacity of resource Yi are denoted by ri and Ki , respectively. The rate of extraction of resource Yi by city j is assumed to be proportional
β
to Yi Nj . A city, as noted above, may have access to more than one resource. These
city–resource links are specified in matrix H, whose element Hij represents the rate
at which city j harvests/extracts resource i. Hij = 0 if there is no connection between
resource i and city j .
The resources harvested by a city are then converted into maintaining and growing
the city’s population. The efficiency of this conversion by city j is denoted by Ej . The
rate at which resources are used to maintain a city’s existing population is assumed
to be proportional to Njα with Mj being the constant of proportionality for city j .
At each instant, a certain fraction v of people in each city is eligible to move to
other cities connected to their home city. The dispersal of these potential movers is
distributed among connected cities based on the relative per capita welfare of all cities
connected to the home city. The home city is included in this relative comparison so
that in some cases, a potential mover’s home city may already offer the highest per
capita welfare among all options and the mover may stay where it is.
A city’s welfare is assumed to be simply proportional to its resource extraction
rate, i.e., more resources to share among its population. Thus the per capita welfare
β−1
of city j , Wj , can be expressed as ( i Hij Yi )Nj . Equation (2) states that the
potential movers from city j will tend to migrate to those cities linked to city j that
have higher per capita welfare. In this sense, the migration component of this model
may be described as “welfare-modulated diffusion” (WMD). Finally, city–city links
are specified in matrix ξ , whose element ξij is 1 if cities i and j are connected, and 0
otherwise.
In our model, α and β are the same for all cities. This is consistent with findings reported by Bettencourt et al. (2007), which shows such regularity for a suite
of urban phenomena across numerous cities. In the analysis that follows, we will
focus primarily on β > 1, corresponding to the superlinear growth of the harvesting ability with increasing population size. We justify this by the superlinear scaling behavior of technologies and innovations in cities (Bettencourt et al. 2007;
Lobo and Strumsky 2008). This emphasis is also in agreement with endogenous
growth theory in economics (Romer 1994).
For α, we consider cases with both α < 1 and α > 1. α < 1 reflects economies of
scale: to support the population, resources are used to build infrastructure, for which
the per capita resource requirement declines with increasing population. On the other
hand, α > 1 reflects the situation in which, as the population grows and the city
develops, people’s lifestyles becomes more extravagant and per capita resource consumption increases. We expect these two cases to interact with the scaling behavior
of harvesting ability and network structure differently, thereby representing a wider
range of plausible scenarios.
A benefit of this framework is that a multitude of different complications may be
incorporated. However, we do not wish to confound our results and their interpretation during this initial investigation of the model. To ensure that the model results
clearly capture our focus—the interplay between scaling behaviors of urban phenomena and network structure—we impose the following simplifying assumptions:
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Fig. 1 Schematic diagram of
the model
Table 1 Definitions and values
of the parameters of the model
r
Resource renewal rate
0.1
K
Resource carrying capacity
1
M
Resource usage rate for maintaining population
4 × 10−7
H
Resource harvest/extract rate
1 × 10−6
E
Resource-to-population conversion efficiency
1 × 10−4
v
Population migration rate
0.05
– The number of cities is equal to the number of resources (C = S) and each resource
belongs to and is always accessible by a specific city.
– All aggregate resources are identical (Ki = K and ri = r).
– All cities have identical harvesting ability per unit of resource (Hij = H if city j
is connected to resource i, and Hij = 0 otherwise).
– All cities are identical in how their resource requirement of population maintenance grows with population size (Mj = M).
– All cities are identical in the tendency of their populations to move in each time
step (vj = v).
With these assumptions, the effects of scaling behaviors and network structure on the
system’s dynamical outcomes can be more clearly elucidated. A schematic diagram
of the model is illustrated in Fig. 1 and parameter values used in our analysis are
summarized in Table 1.
3 Analysis and Results
The schematic diagram in Fig. 1 highlights the structural complexity of the model:
two types of nodes—city and resource—and two types of links—city–city and city–
resource. (In the following, we may simply refer to the city–resource connections
as trade and city–city connections as migration or travel to facilitate the discussion.) Furthermore, each type of node and link has its own, quite different, dynamics. Even with the simplifying assumptions above, interpretation of the results
is not straightforward. Therefore, it is constructive to gradually and systematically
investigate the model from simple to more complex configurations. In these exercises, some findings will be straightforward, while others may be subtle and counterintuitive; these would be useful in understanding the results in the more complicated
settings.
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∗ ; (b) Y ∗ /Y ∗ ; and (c) W ∗ /W ∗ : the subscript ‘1, 1’ denotes the values
Fig. 2 Contours of: (a) N ∗ /N1,1
1,1
1,1
corresponding to α = β = 1, which we use to normalize the results for clearer comparison
3.1 A Single City with Scaling Behaviors and Access to Resources
We begin by considering a single, isolated city and how its population size and per
capita welfare respond to changes in both scaling exponents and number of linked
resources. When presented with multiple resources, the single city in this scenario
has no competition for resources from other cities. We first analyze the simple system of one city and one resource to investigate the effect of scaling behaviors (see
Muneepeerakul and Qubbaj 2012). The dynamics of a one city–one resource system
can be expressed as:
dY
Y
= rY 1 −
− HY Nβ,
dt
K
(3)
H
dN
M α
β
= YN − N .
dt
E
E
The population and resource at equilibrium, denoted by N ∗ and Y ∗ , respectively, can
be found by solving:
M ∗(α−β)
N
,
H
Y ∗ 1/β
r
∗
1−
N =
.
H
K
Y∗ =
(4)
According to the above definition, the per capita welfare at equilibrium W ∗ is equal to
H Y ∗ N ∗(β−1) . Note that when β = 1, per capita welfare is independent of population
size.
We numerically calculate the Y ∗ , N ∗ , and W ∗ for ranges of α and β with results
summarized as contours in Fig. 2. Figure 2(a) shows that for a given β, N ∗ decreases
with increasing α. This is intuitive since a higher α means higher maintenance requirements for a given population, which implies that the same amount of resource
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can support fewer people. For a fixed α, however, N ∗ peaks at an intermediate β. This
results from the balance between the harvesting ability per unit of resource and the
amount of resource at equilibrium: a lower β means less harvesting ability per unit
of resource operating on more resource, whereas a higher β mean greater harvesting ability per unity of resource operating on less resource (see Figs. 2(a) and 2(b)).
Consistently, Fig. 2(b) shows that Y ∗ is lower with increasing β (greater harvesting
ability) and decreasing α (less population maintenance requirement).
Somewhat unexpectedly, Fig. 2(c) shows that W ∗ depends primarily on α, increasing when α increases, and depends only minimally on β. This suggests the two
effects of β—harvesting ability and the resource level at equilibrium—largely cancel
each other out. On the other hand, a higher α leaves more resources to be enjoyed by
fewer people, hence greater per capita welfare.
Note that in traditional versions of our model α = β = 1 and the equilibrium
(N ∗ , Y ∗ ) (Eq. (4)) is stable (damped oscillation). Yet this is not always true for a
system with nonlinear scaling such as ours. Numerical simulations with β > 1 show
that when carrying capacity K is sufficiently large, the system can destabilize into
undamped oscillations or may even collapse (see, e.g., Muneepeerakul and Qubbaj
2012). Furthermore, the threshold of K for such destabilization decreases as α decreases.
We next assume that the isolated city is linked to S resources. Recall that all resources are identical in terms of intrinsic renewal rate r and carrying capacity K. The
dynamics of this one-city–many-resources system can be expressed as:
Yi
dYi
= rYi 1 −
− H Yi N β ,
dt
K
(5)
H
M
dN
= Nβ
Yi − N α ,
dt
E
E
i
where i = 1, 2, . . . , S. The expressions for N ∗ and Y ∗ become
1 M ∗(α−β)
N
,
SH
Y ∗ 1/β
r
1−
.
N∗ =
H
K
Y∗ =
(6)
The effects of these city–resource connections on a single, isolated city are illustrated in Fig. 3. These results suggest that, with regard to the city’s equilibrium values, a lone city connected to S resources of carrying capacity K is equivalent to the
same city being connected to one resource (as in Eq. (3)) with carrying capacity SK.
Hence, as discussed earlier, for a given superlinear β, a one-city–many-resources
system with a sublinear α needs fewer resources to destabilize than a system with a
superlinear α.
In the limit when a city is linked to an infinite number of resources (S → ∞),
Y ∗ → 0 and N ∗ approaches its maximum limit of (r/H )1/β , which is used to normalize the results in Fig. 3(a). For a larger β, this limit is smaller. Comparison across
different cases in Fig. 3(a) shows that when β > 1 and α < 1, a city linked to only
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Fig. 3 Effects of additional
city–resource links on the N ∗
and W ∗ of a city with scaling
behaviors characterized by
different sets of α’s and β’s.
W ∗ is normalized by its
counterpart value associated
with S = 1
one resource is closer to this limit than in other cases, and the benefit of additional
resources is accordingly smaller.
This limit on N ∗ and the potential destabilization discussed above are consequences of the finiteness of the resource renewal rate. Such finiteness is in contrast
to earlier modeling efforts that focus solely on scaling behavior (Bettencourt et al.
2007). There, population growth is seemingly limited only by technologies and innovations (equivalent to our harvesting ability) and never by resources. Finite renewal
rates, which we believe are more realistic in many cases, impose the “diminishing
return” on additional city–resource connections.
Finally, and perhaps counter-intuitively, more connections to resources can lower
W ∗ (Fig. 3(c), α = 0.9), though the differences are small. These differences will play
a role when people migration is included in the full model, as the migration is modulated by per capita welfare (Eq. (2)). Figure 3(c) also shows that W ∗ is independent
of S when α = 1.
3.2 All Cities Sharing all Resources
Now we consider a system of C cities and S resources, in which each city is connected to all S resources. The dynamics of such a system can be expressed as
β
Yi
dYi
= rYi 1 −
− H Yi
Nj ,
dt
K
j
(7)
dNj
M α
H β
Yi − Nj ,
= Nj
dt
E
E
i
where i = 1, 2, . . . , S and j = 1, 2, . . . , C.
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M.R. Qubbaj et al.
Fig. 4 Effects of additional
resources and competing cities
on the N ∗ , Y ∗ , and W ∗ (scaled
∗
by WC=S=1
) of an
S-city–S-resource system with
different sets of α’s and β’s
In this case, the expressions for N ∗ and Y ∗ are
1 M ∗(α−β)
,
N
SH
Y ∗ 1/β
1 r
1−
.
N∗ =
CH
K
Y∗ =
Figure 4 summarizes the results of a special case C = S; this can be thought of
a system of S cities, each with its own resource that can be accessed by all other
cities. As S increases, N ∗ decreases in all cases. Comparing these results to those
in Fig. 3 indicates that the negative effect on N ∗ of having more competing cities
is stronger than the benefit of having additional resources—another pattern useful
for our later analysis. Interestingly, W ∗ increases with S when α > 1 and decreases
when α < 1—opposite to what is observed in the case of one city linked to many
resources (Fig. 3). Furthermore, the magnitude of these changes is much greater.
Again, since people will be assumed to move in pursuit of better per capita welfare in
the full model, these trends can create strong gradients that influence those migration
patterns.
3.3 General Cases
Now, to complete—and inevitably complicate—our model, we allow people to migrate among cities. We consider general cases in which only fractions of all possible
city–city and city–resource links are realized. With the simplifying assumptions dis-
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cussed above, Eq. (2) becomes
dYi
Yi
β
= rYi 1 −
− Yi
Hij Nj ,
dt
K
j
dNj
Wj
M α
Hij Yi − Nj + v
ξj k N k − Nj .
=
dt
E
E
l ξlk Wl
β
Nj
i
(8)
k
As stated previously, the above system of equations captures dynamics that take
place in a complex network with different types of links and nodes. From a network
perspective, we seek topological attributes of this city–resource network that are useful measures or predictors of the dynamical outcomes. Therefore, we proceed with a
detailed look at simple networks consisting of only a few cities and resources, still assuming C = S. These simple cases highlight the richness of the population–resource
dynamics in this type of network. They also highlight the difficulty of interpreting
equilibrium results.
One potentially confounding factor is the sensitivity to initial conditions in some
settings. This sensitivity is pronounced when city–city connections (i.e., migration)
are nearly absent. Cases that interest us the most will involve some city-city connections so that this is not a serious concern for our analysis. Nonetheless, to remove this
confounding factor, we assume that the initial population and resource of each city
are those given by Eq. (4). In other words, we assume all cities have been in isolation
long enough to have reached their steady states before any connections to resources
or other cities are formed. With this assumption, the model’s dynamical outcomes are
attributable solely to network topology.
Figure 5 illustrates our first simple system—a system of 3 cities and 3 resources
with a specific city–resource configuration. For α = 0.9, and in the absence of any
city-city links (Fig. 5A), City 1, with access to two resources, has a larger population
than the isolated City 3 but with slightly less per capita welfare. At the same time, City
2, forced to compete for its only linked resource with City 1, collapses (top table in
panel A). The possibility of such a collapse is important to our search for meaningful
topological metrics: when a city collapses, all links connected to it effectively stop
functioning as links, essentially altering the network’s topology and distorting the
network metrics we capture.
We now explore the effects of introducing a single city–city link. Figures 5B and
5C show that a link from City 2 to either City 1 or City 3 rescues City 2 from collapse (see top tables for α = 0.9), although with much fewer people and less per
capita welfare relative to an isolated city (i.e., City 3 in panels A and B). Interestingly, City 1 in Fig. 5B supports more people with better per capita welfare than
an isolated city. Figure 5D shows very small redistribution of people and welfare—
there are little differences in both even without the link (i.e., Fig. 5A). These patterns
can be understood in terms of the balance between the two underlying forces in our
‘welfare-modulated diffusion’ mechanism: gradients of people and welfare. People
move according to two gradients here: population size and per capita welfare. With
per capita welfare being equal, people move from crowded cities to less populated
ones; with population sizes being equal, people move toward cities with greater per
capita welfare.
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Fig. 5 Illustration of the effect of adding a city–city link to a simple network of three cities and three
resources with a specific city–resource configuration. The top table in each panel is for α = 0.9 and the
lower table is for α = 1.1. For all cases, β = 1.2. The per capita welfare is scaled by the constant resource
harvesting rate H , i.e., w̃∗ = W ∗ /H
A similar trend can be observed for α = 1.1 (lower tables of panels in Fig. 5): some
migration via a city–city link can rescue the otherwise collapsing City 2 (Figs. 5B
and 5C). However, when α = 1.1 the magnitude of migration is greater and results
in more even distributions of population size and per capita welfare. Consider, for
example, Fig. 5B: when α = 0.9, per capita welfare in City 1 is 24 times greater than
that in City 2, while it is only 6 times greater when α = 1.1.
A city–city link can have other subtler effects. To illustrate, consider a slightly
more complicated system of 4 cities and 4 resources with the particular city–resource
configuration in Fig. 6. With Fig. 6A as the baseline, we examine Figs. 6C through 6F,
with a focus on Cities 3 and 4. In Fig. 6A, Cities 3 and 4 are identical in terms of N ∗
and W ∗ due to the symmetry in their local network structure—no travel allowed, one
link to a good resource, and another to a depleted one—and their initial conditions
(i.e., those of an isolated city as in Eq. (4)). In Figs. 6C through 6F, a city–city link
breaks this symmetry—but depending on α, the symmetry can be broken in opposite
ways.
Let us first consider α = 0.9. In Figs. 6C through 6F (top tables), by adding more
people to one of the cities (3 or 4) during the initial stage, the link to either Cities 1
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Fig. 6 Illustration of the effect of adding a city–city link to a simple network of four cities and four
resources that has a specific city–resource configuration. The top table of each panel is for α = 0.9 and the
lower one is for α = 1.1. For both cases, β = 1.2
or 2 increases the harvesting ability of that city during the initial dynamics—a small
advantage that accumulates and eventually diverts that city to dominance, and another
to a collapse. (This is a somewhat troublesome finding: it implies that a city can be
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adversely affected by externalities it cannot control—things like other cities deciding
to connect to each other.)
The effect of non-symmetry is less clear when α = 1.1: a link to City 1 benefits
the long-term dominance and survival (Figs. 6C and 6D, bottom tables), but a link to
City 2 does not (Figs. 6E and 6F, bottom tables). A closer look at simulation results
reveals that the early benefit of additional people and harvest ability is still present
in all these cases, but it is short-lived. After the initial redistribution of people, per
capita welfare in City 2 decreases but remains sufficiently high to ensure the city’s
long-term persistence. This small decrease in W ∗ , coupled with a steep population
gradient, results in the subsequent flow of people back to City 2, thereby diminishing
the harvesting ability of the connected city and causing its eventual collapse.
Finally, all panels in Fig. 6 show the essentially depleted Resource 2 (asymptotically zero). Similar to a collapsed city, when a resource is depleted all links to it are
effectively severed, altering the structure of the network and distorting our regressions
of network metrics.
All these effects make more challenging our search for meaningful topological
metrics of the overall network. It implies that local network structure may, in potentially many cases, be insufficient to explain the dynamic outcome of a city. For
instance, City 3 in Fig. 6(A, B, D, and F) has the same local network structure in
each case. Yet City 3 has very different steady-state outcomes depending on the local
structure of other cities and on the collapse of some cities and resources.
3.4 Local Topological Metrics as Predictors of Long-Term Outcomes
As the number of cities and resources increases, the possible number of network
configurations grows beyond the scope of this project to analyze formally. This is
particularly problematic as the previous section demonstrates that the addition of
just a single link can make the difference between long-term existence and collapse
of a city. To understand which topological characteristics of the network may best
predict a city’s long-term population, resource, and welfare, we take a more statistical
approach.
In particular, we randomly generate network configurations, using an Erdős–Rényi
type algorithm, for a system of 15 cities and 15 resources on which to run the dynamical model. It is important to note that our networks can be decomposed into two
distinct networks for purposes of analysis. The first is a city–city network, consisting
only of cities and their links to each other. The second is a city–resource network,
a bipartite network consisting of two types of nodes—cities and resources. Links in
this second network exist only between cities and resources. By default every city is
linked to one (its own) resource. We denote the total number of links in the city–city
network as ΛCC and the total number in the city–resource network as ΛCR .
To determine which network attributes best predict long-term outcomes, we
regress equilibrium values of each city’s population size and per capita welfare
against several node-level topological metrics (Table 2) of the overall network linking
cities to each other and to resources. These metrics include well-established network
metrics (e.g., Wasserman and Faust 1994) as well as novel metrics developed specifically for this study. These metrics are derived strictly from the topology of the static
network and are not affected by dynamics or equilibrium values.
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Table 2 Network metrics collected in this study. Note that closeness and betweenness centrality were not
collected because we did not require our stochastically generated networks to be connected, which is a
requirement of these standard metrics
Standard network metrics collected
kiR
Number of resources to which i is linked (i can only be a city)
kiC
Number of cities to which i is linked (i can be either a city or a resource)
k̄CC
Mean number of cities each neighbor is linked to (Average degree of neighbors in the city–city
network)
Ci
City i’s clustering coefficient within the city–city network
Metrics developed for this study
kiR
C
j =1 kj
Mi
Number of cities competing with city i =
Rie
City i’s ‘effective’ number of resources =
Die
Distributed effective resources = product of the migration adjacency matrix ξ —with 1’s added
to the diagonal—and R e . This metric is meant to capture the accumulation of resources
represented in Rie , which is then converted to people, who are then distributed by the migration
network
Ri+
Number of cities connected to city i whose R e ≥ Rie
Ri−
RM
kiR
C
j =1 1/kj
Number of cities connected to city i whose R e ≤ Rie
Modified effective resources = R e × k C /M
We sample from possible network configurations having low, medium, and high
numbers of total links in both the city-city network (ΛCC = 15, 50, and 75 total
links) and the city–resource network (ΛCR = 15, 100, and 150 total links). This leads
to nine network categories for each of which we randomly generate 25 networks
and then collect individual city and resource equilibrium values at both α = 0.9 and
α = 1.1. In Table 3, we present both the most highly correlated network metric (the
top metric) and the metric that best explains the residuals from its regression (the
bottom metric). We also report the correlation coefficients when only the best and
both metrics are used in the regression (in parentheses).
In the overwhelming majority of treatments the number of effective resources
available to a city Re is the best predictor of both the city’s equilibrium population
and its per capita welfare. More links to resources, coupled with less competition for
those resources, translates into higher sustained populations with higher individual
welfare. The predictive power of Re depends on the structure of our model, particularly the dynamics governing the migration between cities. It is worth noting that
this measure is closely related to the measure of recursive power discussed in Neal
(2011), which better differentiates between power and centrality in city–city networks
and is likewise a better predictor of system outcomes than traditional network metrics. Note also the pattern of the best predictor of the regression residuals: at a low
ΛCR , the number of competing cities M tends to best explain these residuals, but as
more cities are connected to more resources (i.e., higher ΛCR ), other metrics take its
place.
In the case of α = 0.9 and a high number of city–resource links (ΛCR = 150), no
network metrics are meaningfully correlated with equilibrium outcomes (R 2 < 0.09
404
M.R. Qubbaj et al.
Table 3 Node-level metrics most highly correlated with both equilibrium population and per-capita welfare. Correlation coefficients are shown in parenthesis. β = 1.2 in all cases. “**” indicates that no factors
have a correlation larger than 0.29 (meaning the highest R 2 < 0.09)
α = 0.9
ΛCR = 50
α = 1.1
ΛCR = 100
ΛCR = 150
ΛCR = 50
ΛCR = 100
ΛCR = 150
1. R e (0.71)
1. R e (0.53)
1. R e (0.55)
2. M(0.78)
2. D e (0.58)
2. k C (0.70)
1. R e (0.79)
1. R e (0.66)
1. R M (0.64)
2. M(0.78)
2. k C (0.73)
2. R e (0.70)
Best correlation with a city’s equilibrium population
ΛCC = 15
ΛCC = 50
ΛCC = 75
1. R e (0.66)
1. R e (0.48)
2. M(0.77)
2. k C (0.56)
1. R e (0.70)
1. R e (0.67)
2. M(0.84)
2. k R (0.71)
1. R e (0.75)
1. R e (0.64)
2. M(0.85)
2. D e (0.69)
**
**
**
1. R − (0.75)
1. R e (0.67)
1. R − (0.71)
2. M(0.83)
2. D e (0.70)
2. D e (0.72)
1. R e (0.70)
1. R e (0.62)
1. R e (0.57)
2. M(0.77)
2. k R (0.63)
2. k C (0.61)
1. R e (0.77)
1. R e (0.73)
1. R e (0.68)
2. M(0.85)
2. R M (0.76)
2. R M (0.69)
1. R e (0.74)
1. R e (0.69)
1. R e (0.69)
2. M(0.85)
2. k R (0.73)
2. k R (0.71)
Best correlation with a city’s equilibrium per-capita welfare
ΛCC = 15
ΛCC = 50
ΛCC = 75
1. R e (0.63)
1. R e (0.50)
2. M(0.74)
2. k C (0.54)
1. R e (0.68)
1. R e (0.69)
2. M(0.78)
2. R M (0.72)
1. R e (0.73)
1. R e (0.67)
2. M(0.83)
2. R M (0.69)
**
**
**
for every independent variable). This is largely due to the high frequency of cities
collapsing under these conditions. Interestingly, this high frequency of collapse did
not occur when α = 1.1 and metrics related to resource availability are again strong
predictors. This may appear puzzling at first, but recall that a lower α means that less
resources are required to maintain a given population size. Or, alternatively, a larger
population may be maintained with a given amount of resources. Thus decreasing
α is analogous to increasing the efficiency with which resources are converted into
population. This leads to a higher population when the cities are isolated, i.e., before
any connections are made. Once high levels of city–resource connectivity is in place,
these large populations exert unsustainable pressure on the supporting resource pool.
The resulting fierce competition then leads to the collapse of many resources and,
consequently, the collapse of dependent cities.
4 Conclusions and Future Directions
In this paper, we develop a dynamical modeling framework that embeds population–
resource dynamics—with the nonlinear scaling behaviors characteristic of many urban phenomena and the finite renewal rates of resources—in complex networks in
which cities may be linked to each other and to the resources of other cities. We
then simulate dynamics of this intricate web of cities and coupled resources to examine how the scaling behaviors and network structure affect the sustainability of
Living in a Network of Scaling Cities and Finite Resources
405
individual cities and resources. We find that isolated cities (i.e., no migration) are
susceptible to collapse if they do not have access to adequate resources. By linking
to other cities, however, they may persist at some positive population level. However,
these effects of inter-city links can change due to the interplay between the nonlinear
scaling behaviors and network structure. A city’s long-term population level is, in
many cases, largely a function of the city’s access to resources that have little or no
competition associated with them. But this is only a statistical trend: since cities and
resources may collapse and scaling behaviors may influence the effects of inter-city
links, careful investigation of dynamics is required to gain mechanistic understanding
of a particular city–resource network.
The incorporation of various factors in this model, such as network connections,
nonlinear dynamics, and scaling behaviors, greatly increases the complexity of the
system’s dynamics. Thus for this initial exploration, several simplifying assumptions
are necessary to facilitate a clear interpretation of results. Our treatments in this paper
only scratch the surface of the numerous ways in which this model may be extended
and explored. Moving forward, we can explore the richness and insights that this
model has to offer by carefully relaxing specific assumptions. We discuss some of
these possibilities in the following paragraphs.
In this paper, we focus on the deterministic dynamics to obtain clear insights
as a first exploration of the dynamics of the coupled city–resource network. There
is certainly fluctuation around the scaling laws employed here; indeed, it has
been shown that some of these fluctuations are log-normally distributed, and some
stochastic modeling frameworks have been proposed (Gomez-Lievano et al. 2012;
Alves et al. 2013a). Incorporating such effects is a potential extension of this work.
Many of the assumed homogeneities can be replaced with more realistic heterogeneities: resources may have different r and K, and they may be harvested at different rates by different cities. There can also be more than one type of resource
required to support populations. Indeed, such multiplicity of resources is necessary
if our model is to be used to study the implications of those urban scaling behaviors
on globalization. With different types of resources and a range of harvesting ability,
one would then be able to incorporate such concepts as comparative advantage and
substitutability—critical features of globalization.
Our model currently includes only a single, generic type of resource that supports
all cities. It does not allow alternative resources which might become more attractive when competition for the single resource in this model becomes too fierce. In
other words, we do not allow cities in this model to develop a comparative advantage
and switch to a substitute resource in the face of increased competition. Therefore,
another future direction for this model is to allow multiple resource types to understand how increasing connectivity among cities leads to specialization with respect
to resource extraction.
In its current form, the model makes use of only a limited set of the empirical
scaling behaviors of urban phenomena, namely those captured by α and β. Other urban phenomena, which also exhibit scaling behaviors, can certainly be incorporated.
For example, crime, which affects welfare and wellbeing, also increases superlinearly with population size (e.g., Bettencourt et al. 2007; Gomez-Lievano et al. 2012;
Alves et al. 2013b). Incorporating such scaling behaviors will therefore alter the mi-
406
M.R. Qubbaj et al.
gration dynamics and the eventual outcomes, thus presenting yet another interesting
extension.
The generation of the random network in this work is restricted to a simple Erdős–
Rényi algorithm. More sophisticated and realistic networks may be generated such
as scale-free networks generated through a preferential attachment or a small-world
network generated through a simple rewiring algorithm. Importantly, real world networks are constantly changing: the static nature of our random network can thus be
replaced with a more dynamical, evolutionary one. That is, the network structure can
evolve based on feedbacks such as the benefits that a given link provides to a city. In
addition, cities may form links before their isolated systems reach steady states; these
would effectively set and continually change “initial conditions” of the population–
resource dynamics. Experiments with evolving networks may include subjecting the
network to stochastic shocks and allowing it to rearrange afterwards.
In addition to relaxing assumptions as suggested above, the analysis may be expanded to focus on the network-level outcomes. While this paper has focused on the
long-term sustainability of individual cities and resources, it is important to think,
too, of the sustainability of the entire network, and many standard network-level metrics exist to aid in this effort. If one city survives at the cost of others, is the system as
a whole in some sense sustainable? What is the optimal network structure for some
global metrics of wellbeing? What are the characteristics of the resulting network if
it is allowed to undergo certain evolutionary dynamics? These questions enrich an
assessment of the sustainability of networks of cities. Indeed, the research directions
discussed in this section exemplify the many promising applications of complex network theory that await contributing to sustainability science.
Acknowledgements The authors wish to thank two reviewers for their useful comments. M.R.Q. and
R.M. also acknowledge the support from NSF grant GEO-1115054.
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