# Public disorder and transport networks in the Latin American Context

Carlos Cartes  corresponding author<EMAIL_ADDRESS>Universidad de
los Andes, Chile.
Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas.  Toby P.
Davies  Department of Security and Crime Science, University College London,
35 Tavistock Square, London WC1H 9EZ, UK

###### Abstract

We propose an extension of the Davies et al. model, used to describe the
London riots of 2011. This addition allows us to consider long travel
distances in a city for potential rioting population. This is achieved by
introducing public transport networks, which modifies the perceived travel
distance between the population and likely targets. Using this more general
formulation, we applied the model to the typical Griffin and Ford pattern for
population distribution to describe the general features of most large Latin
American cities. The possibility of long-range traveling by part of the
general population has, for an immediate consequence, the existence of
isolated spots more prone to suffer from rioting activity, as they are easier
to reach than the rest of the city. These areas finally made it easier to
control the eventual disorder by part of police forces. The reason for this
outcome is that transport networks turn riots into highly localized and
intense events. They are attracting a large police contingent, which will
later extinguish the remaining disorder activity on the rest of the city.
Therefore, working transport networks in a city effectively reduces the number
of police force contingent required to control public disorder. This result,
we must remark, is valid only if the model requisites for order forces are
satisfied: extra police contingent can be added swiftly as required, and these
forces can move around the city with total freedom.

PACS numbers

###### pacs:

Valid PACS appear here

##  I Introduction

Public disorder, riots and civil unrest are all manifestations of collective
behaviour in which hostile action is directed towards members belonging to
other groups or authorities. If mishandled, such phenomena can sometimes
cascade into large–scale outbreaks of violent criminality salehyan2012social .
This is a significant public security problem, which has substantial
consequences in terms of both personal injuries and property damage.
Furthermore, if disorder escalates into a nation-wide episode, it can pave the
way for political instability or even the overthrow of governments, as in
Tunisia and Egypt during the Arab Spring of 2011.

The ability to understand and predict the spatial and temporal evolution of
disorder is, therefore, of huge importance for policy. As with other forms of
crime, such understanding forms a basis for preventative interventions, and
allows resources (such as policing) to be used in a way that mitigates
negative consequences most efficiently. This need is particularly acute for
rare events such as riots, since the lack of real-world examples means that
authorities have little opportunity to develop their understanding or evaluate
approaches empirically. For this reason, mathematical modelling and numerical
simulation has great potential as a means of developing insight. Not only can
they offer profound understanding of the phenomena themselves, but they can be
used as a means of exploring alternative scenarios; that is, performing ‘what
if’ experiments for riot situations. In particular, this provides a means of
testing potential control or mitigation strategies at low cost and at a high
level of granularity.

Early analyses considered collective disorder to be a return to a “natural
condition”, whereby crowds temporarily forgot civilized behavior and instead
regressed to more primitive ways. This phenomenon was referred to as the
“madness of the crowds” le2017crowd  and “mass enthusiasm”
lorenz2005aggression . The alternative suggestion that conscious and rational
decisions were involved in the participation in riots and public disorder
began with the pioneering work of Granovetter granovetter1978threshold ;
Granovetter1983 . In the models presented, potential rioters evaluate the
benefits and costs of joining public disorder, taking into account the social
influence of other individuals. Later developments, such as Epstein’s
formulation for social conflict epstein2002modeling  using Agent–Based Models
(ABM), successfully described more detailed classical patterns in public
disorder, such as the long calm periods interrupted by intermittent bursts of
collective violence. Further approaches to this problem were made by using
partial differential equations, capturing important features like the
contagious nature of rioting behavior and reflecting the observed patterns
from the French riots of 2005 berestycki2015model ;
bonnasse2018epidemiological . Also using differential equations, Davies et al.
Davies2013  successfully captured the quantitative dynamics of the London
riots of 2011. This model was based on empirical evidence from London that
rioters chose target locations on the basis of their characteristics
baudains2013target ; such behaviour is consistent with the ‘rational choice’
perspective on criminal decision-making cornish_reasoning_1986 , in which
offenders are assumed to select potential targets from a range of alternatives
in such a way that they maximise reward and minimise risk
bernasco2004residential ; ruiter_crime_2017 . In the Davies et al. model, the
participation of rioters was influenced by the volume of retail activity (i.e.
potential for looting) in possible target locations, as well as the number of
other participants already present (as a measure of ‘safety in numbers’).

Although spatial data has been incorporated in riot models, it has typically
been simplistic and included little information concerning urban structure. In
particular, there has been no attempt to consider the effect of transport
networks, in the form of either public transport or simply streets themselves.
This constitutes a significant shortcoming: transport networks have been shown
to have a significant effect in promoting crime in general
brantingham1991public ; swartz2000spatial ; davies_examining_2014  by either
generating new opportunities for offending or increasing the accessibility of
existing ones. Indeed, analysis of the London riots indicates a strong
correlation between the probability of a location being looted and the
proximity of underground train stations baudains2016london . In this work, we
seek to extend the previous model of Davies et al by incorporating the effect
of a transport network, in order to increase the validity and granularity of
the model. Inspired by the recent outbreaks of violence in Santiago, Chile,
and Latin American cities from Bolivia, Peru, Colombia, Equador among others
TheConversation2019 ; bjork2020mass ; WashingtonPost2020 ; TheGuardian2019 ;
TheGuardian2019-2 , we do this with particular focus on the context of Latin
American cities.

We propose to implement the typical layout of a Latin American city, as a
prototype transport network and geographic configuration, to test our
formulation. A stylised model of such cities was first proposed by Griffin and
Ford in 1980 griffin1980model . Briefly, it formulates cities as being
structured around a “spine”, where wealthy inhabitants live, surrounded by
three concentric regions in which infrastructure, public services and quality
of life in general decreases with distance to the city center. This model was
later revised by Ford in 1996 ford1996new , adding some novel elements but
maintaining its principal characteristics. Although the steady decline in
Latin American population growth since the 1980’s has led to a constant
improvement in the quality of public services and amenities for the
inhabitants of more distant zones, showing evidence that these cities are
converging towards an Anglo–American configuration borsdorf2003como , Buzai
stated recently buzai2016urban  that Ford’s 1996 model remains valid.

Of relevance for the case of the public disorder occurring in Latin American
cities, a number of recent articles have presented quantitative analysis of
the Chilean riots of 2019. For instance, Caroca et al. Caroca2020
characterized the temporal evolution of the total intensity of the events,
showing that it follows the typical curve predicted by Burbeck
burbeck1978dynamics  (rapid escalation to a pronounced peak, followed by
exponential decay). The spatial distribution of activity on the first days for
the capital city Santiago was also characterized by Cartes et. al.
Cartestransport , with the analysis showing that activity appeared to be
concentrated primarily around subway stations. Nevertheless, we are not aware
of any research concerning policies for control and mitigation applied to
those particular scenarios.

In this paper, we present the Davies et al. model used to describe the London
riots of 2011 and extend it to incorporate the effects of public transport
networks on the underlying spatial environment. These modifications mean that
riot participants are able to travel large distances (and therefore reach
distant targets) at low cost, in a way that they would not otherwise be able
to do. We then implement the model with the typical population distribution of
a Latin American city. Depending on the accessibility of the public transport
network and the police response’s effectiveness, we identify different
possible scenarios regarding the intensity of the riots and their distribution
in the city.

This document is organized in the following way. In II we introduce Davies et
al. model for riots, its extension to incorporate public transport networks,
and the considerations taken on the context of Latin American cities. In III
we present in detail the model’s results, specifically the intensity of the
riots and its spatial distribution. At IV we analyze the results presenting
our conclusions about the model, and finally, in V we show our perspectives
and future work.

##  II  The Model

###  II.1 Davies et al. (2013) formulation

The model introduced by Davies et al. is situated on a discrete spatial system
comprising two sets of locations: residential areas, indexed by $i$, and
retail sites (the targets for rioting), indexed by $j$. In essence, the model
seeks to describe the flows of rioters from their homes to retail sites, and
their interactions with police once there. The model tracks the numbers of
individuals in each location over time, with the key quantities being
$R_{j}(t)$ and $P_{j}(t)$: respectively the numbers of rioters and police in
location $j$ at time $t$. In mechanistic terms, the model has three stages,
corresponding to different phases of an individual’s involvement in rioting:

  * •

The decision to participate in rioting

  * •

The choice of site at which to offend

  * •

Possible arrest due to interaction with police

In the model, citizens pass through three possible states - inactive,
participating and arrested \- so that the overall model is analogous to the
Susceptible-Infected-Removed epidemiological approach.

For the first two of these stages, a single quantity - referred to as
attractiveness \- plays a central role. The attractiveness of a retail site
$j$, from the perspective of a citizen in residential area $i$, denoted
$W_{ij}$, represents the result of a cost/benefit calculation for individuals
considering the utility of rioting in location $j$. It takes into account
three factors: the volume of retail activity at $j$ (benefit), the distance
between $i$ and $j$ (cost) and the probability of arrest when rioting at $j$
(cost/risk). It is calculated as:

$W_{ij}(t)=Z_{j}^{\alpha_{r}}\exp\left(-\beta_{r}d_{ij}\right)\exp\left(-\left\lfloor\frac{\gamma_{r}P_{j}(t)}{R_{j}(t)}\right\rfloor\right)$
(1)

where $Z_{j}$ is the volume of retail activity at $j$, $d_{ij}$ is the
Euclidean distance between $i$ and $j$ (the minus sign reflects that it acts
as an impedance) and the third term
$\left\lfloor\frac{\gamma_{r}P_{j}(t)}{R_{j}(t)}\right\rfloor$ corresponds to
the probability of arrest at $j$ (a “deterrence”). This is proportional to the
ratio between police officers $P_{j}(t)$ and rioters $R_{j}(t)$, with the
floor function $\lfloor\rfloor$ added to account for the empirical fact that
when police are outnumbered ($\gamma_{r}P_{j}(t)<R_{j}(t)$) the disorder
becomes out of control and police are incapable of making arrests
wilensky2004netlogo . Finally the constants $\alpha_{r}$, $\beta_{r}$ and
$\gamma_{r}$ are free parameters to calibrate the model by using the available
data.

The first step in the model concerns the decision of whether to participate in
rioting. This is determined by a ‘probability of offending’, defined for each
residential area, $i$, and representing the probability that residents of that
area will choose to participate. This is modelled as a sigmoidal function,
based on the total attractiveness of all possible targets $j$, as perceived by
residents of $i$:

$P_{i}^{\rm
offend}=\rho_{i}^{\mu}\frac{\sum_{j}W_{ij}(t)}{1+\displaystyle{\sum_{j}W_{ij}(t)}}$
(2)

The multiplicative factor $\rho_{i}$ is a measure of the deprivation at $i$,
reflecting the empirical finding that higher levels of deprivation are
associated with greater participation Davies2013 ; baudains2013target . Taking
the product of this probability with the number of inactive residents of $i$,
denoted $I_{i}(t)$, gives the number of new crowd participants:

$N_{i}(t)=\eta I_{i}(t)P_{i}^{\rm offend}(t)$ (3)

where $\eta$ is a transition rate.

Once a citizen decides to participate, the next stage of the model concerns
the choice of location. This is modelled via the use of an entropy–maximising
spatial interaction model (SIM), which estimates the most probable flows
between a set of origins and destinations, given known out-flows and an
expression for the utility of each destination wilson1971family . The utility
used is the attractiveness defined in (1) above; however, it is first modified
by taking a moving average, in order to account for the delay in reaction of
participants. This represents both travel time between origins and
destinations and the time taken for information about the riot situation to
propagate through the population. This temporal average of $W_{ij}(t)$ over
the most recent $L_{r}$ time–steps gives the “effective attractiveness”
between $i$ and $j$:

$W_{ij}^{e}=\frac{1}{L_{r}}\sum_{l=0}^{L_{r}-1}W_{ij}(t-l\delta t)\,.$ (4)

Now if $A_{i}(t)$ denotes the number of residents of $i$ who are active at
time $t$, then the number of those participating in rioting at $j$ is given by
wilson2007boltzmann

$T_{ij}(t)=\frac{A_{i}(t)W_{ij}^{e}}{\displaystyle{\sum_{k}W_{ik}^{e}}},$ (5)

so that the rioters from $i$ are assigned to targets in proportion to their
attractiveness. By summing the above expression over all origin locations $i$,
the total number of rioters located at the site $j$ can be obtained:

$R_{j}(t)=\sum_{i}\frac{A_{i}(t)W_{ij}^{e}}{\displaystyle{\sum_{j}W_{ij}^{e}}}\,.$
(6)

Simultaneously with the flows of rioters, the locations of police forces are
also modelled. These are done in a similar way to the modelling of riot
behaviour, though with two differences: the number of police officers remains
constant, and an alternative utility function - not including distance as an
impedance factor - is used. This utility - referred to as demand \- depends on
the value of property at a site (again modelled as retail volume, $Z_{j}$) and
the number of rioters $R_{j}$:

$D_{j}=Z_{j}^{\alpha_{p}}\exp\left(\gamma_{p}R_{j}(t)\right)$ (7)

Here $\alpha_{p}$ and $\gamma_{p}$ are parameters used to calibrate the model.
In the same way as the attractiveness, the “effective demand” is computed by
averaging over the most recent $L_{p}$ time–steps

$D_{j}^{e}(t)=\frac{1}{L_{r}}\sum_{l=0}^{L_{p}-1}D_{j}(t-l\delta t)\,,$ (8)

again reflecting a reaction delay. Police officers are then assigned to each
location $j$ according to the proportion of total effective demand; with
$P^{\rm total}$ as the total number of officers, $P_{j}(t)$ is given by:

$P_{j}(t)=P^{\rm
total}\frac{D_{j}^{e}(t)}{\displaystyle{\sum_{k}D_{k}^{e}(t)}}\,.$ (9)

The final stage of the model concerns the arrest of rioters by police. For
each residential location $i$, the number of residents arrested in $j$ is
obtained by multiplying the flow $T_{ij}(t)$ by the arrest rate in $j$. Taking
the sum over all riot sites gives the total number of arrests of residents of
$i$:

$C_{i}(t)=\tau\sum_{j}T_{ij}(t)\left(1-\exp\left(-\left\lfloor\frac{P_{j}(t)}{R_{j}(t)}\right\rfloor\right)\right)\,.$
(10)

The above processes are combined to give a dynamical system tracking the
active and inactive populations of each residential site $i$:

$\displaystyle A_{i}(t+\delta t)$ $\displaystyle=$ $\displaystyle
A_{i}(t)+\delta t\left(N_{i}(t)-C_{i}(t)\right)$ (11) $\displaystyle
I_{i}(t+\delta t)$ $\displaystyle=$ $\displaystyle I_{i}(t)-\delta
tN_{i}(t)\,.$

####  II.1.1 Addition of transport networks

In this paper, we introduce a modification to the above model, incorporating
the effect of public transport networks. Before doing this, we first simplify
the spatial domain on which it is defined: rather than an arbitrary spatial
system, we assume that the model is defined on a regular two-dimensional grid.
This can be interpreted as a ‘rasterised’ version of a generic urban area, and
therefore represents a generalisation of the model. In this formulation, we
assume that all grid cells represent residential areas, while retail sites
(i.e. targets) are located at a subset of grid points. The model defined above
translates straightforwardly to this setting, with the distance between any
two locations defined as the Euclidean distance between the grid points.

In incorporating transport networks, the central principle is that the effect
of a transport line is to reduce the travel cost along the paths it serves. We
model a network as a set of nodes, representing stations or stops, with
connections between them representing direct transport links. The effect of
the transport line is modelled by applying a multiplicative factor $\delta<1$
to the distance between any pair of connected nodes: the link therefore acts
as a ‘shortcut’ relative to regular travel. For any two neighbouring nodes,
the distance is therefore given by

$d=\delta\sqrt{\Delta x^{2}+\Delta y^{2}}\,.$ (12)

For any pair of nodes $v$ and $w$ on the network, the travel distance
$d^{n}_{vw}$ is given by the sum of these distances over all constituent
network links; essentially the Euclidean length of the path, multiplied by
$\delta$.

Once the transport network has been implemented, the distances between all
origins and destinations are updated to take its effect into account. In
particular, for any origin $i$ and destination $j$, the distance when using
the transport network is computed on the following basis:

  1. 1.

The distance $l_{1}$ from $i$ to the nearest transport node $v$ is calculated

  2. 2.

The distance $l_{2}$ from $j$ to the nearest transport node $w$ is calculated

  3. 3.

The total effective distance is given by $d_{ij}=d^{n}_{vw}+l_{1}+l_{2}$

For the purpose of the model, the distance $d_{ij}$ is then defined as the
minimum of the Euclidean distance and transport distance.

###  II.2 Latin American Cities

We propose to implement the typical layout of a Latin American city, as a
prototype transport network and geographic distribution pattern to test our
formulation. A generic model of such cities was proposed by Griffin and Ford
in 1980 griffin1980model , consisting of a “spine” where wealthy inhabitants
live, surrounded by three concentric regions, over which infrastructure,
public services and quality of life in general decrease with distance to the
city center. The names of these regions are, from center to periphery: zone of
maturity, zone of in situ accretion and zone of peripheral squatter
settlements; see FIG. 1. This configuration is an evolution from the colonial
city gilbert1987urban  and is predominantly shaped by the migration to cities
from rural areas that occurred during the 1940s and 1950s. The fast increase
in urban population, under which new migrants located themselves at the
borders of the city, generated a variety of problems as housing, public
services and employment opportunities could not keep pace with their need.
Faced with these constraints, city planners expanded public services like
electricity, sewer systems and paved streets in only one direction, with this
preferred orientation forming the ‘spine’. Naturally, all important urban
facilities like banks, parks and restaurants - along with high–status
residential housing - concentrated around that region, in order to make use of
all the aforementioned public services. Population density within the various
regions of the city differs dramatically, with upper class groups having the
lowest density, middle class occupying the areas with higher density and lower
income groups having densities in the middle of the two amato1970comparison .
This model was later revised by Ford in 1996 ford1996new , who added some
novel elements while maintaining the principal characteristics. The steady
decline in population growth in Latin America since the 1980s has alleviated
some of the issues facing major cities, with constant improvement on the
quality of public services and amenities for the inhabitants of the further
away zones. Nevertheless, although there is some evidence that these cities
are converging toward an Anglo–American configuration borsdorf2003como ,
Ford’s 1996 model remains valid in many contexts buzai2016urban .

Figure 1: Model for a Latin American City, after Ford and Griffin 1980.

It is not surprising that this population distribution inside the city tends
to produce high levels of segregation and social friction among the citizens,
since all groups are constrained to occupy a strictly bounded area, sharply
delimited by income and social status. Evidence of urban segregation is found
in Mexico schteingart2001division , Argentina torres2001cambios , Brazil
villacca1998espacco  and Chile sabatini2001segregacion . Mobility studies from
the latter country’s capital city, Santiago, found that even movements inside
the city are restricted to individuals’ designated residence areas
dannemann2018time , where they remain isolated, with the only common meeting
point in the whole urban area being the center of the city.

###  II.3 Initial population and target distribution

For our population arrangement, we used the income distribution indicated in
Ford’s model. On the other hand, our model’s population density comes from the
values mentioned above shown by Amato amato1970comparison . All our
simulations were carried out with a total population of $1.2$ million
inhabitants.

Our simplified transport network consists of only two lines intersecting at
the center of our environment, coincidentally with the city’s Central Business
District; see FIG. 2. This configuration is the simplest one that allows high
mobility for the inhabitants of the periphery, creating a direct link with the
city center. This arrangement also creates a focal location with relatively
high connectivity, easily reached from all the city’s regions.

Figure 2: Scheme for the transport network implemented in the simulations. It
bridges the suburbs with the CBD, creating a high connectivity point. Network
nodes and targets share spatial location (green dots).

The spatial distribution of targets for disorder activity was intentionally
made by sharing location with the transport network nodes, see FIG. 2. This
choice was taken because highly connected spots, like subway stations, attract
retail activity Bowes2001 ; debrezion2004impact . It was also found that
during the first four days of the Chilean riots from October 2019, in the
capital city Santiago, most of the events were concentrated around subway
stations (called Metro Network) Cartestransport . In FIG. 3 the frequency of
events, as a function of the distance to the closest subway station, is shown,
with the distribution following approximately a power law. It is evident that
most of the disorder activity was concentrated around these transport nodes:
for example, around $50\%$ of the total activity took place $1$km or less from
these well-connected spots Cartestransport . We also take into consideration
that there is a correlation between commercial and residential real estate
values schoenmaker2015real . Therefore, the perceived attractiveness of these
spots increases towards the center of the high-income region. On the other
hand, the targets decrease in value as they are located farther away from the
center, entering the peripheral low-income regions.

Figure 3: Event frequency as a function of the distance to subway stations is
shown. Around half of the total lies within $1$km or less from a subway
station.

Indeed it is possible to argue that, at least initially, disorders aren’t
motivated by a desire to make a monetary profit from the looting opportunity;
instead, they are triggered on political and social grounds. Nevertheless,
Santiago’s most affected area is effectively the most connected spot on the
whole city Cartestransport , the reasons for which are likely to be its easier
access and the awareness of its existence by the population. We consider it
essential to remark that this was not the first time this particular city spot
suffered large-scale disorders. Historically that region was the default
meeting point for any kind of massive concentration, independent of the
motivation, being it the positive result in a sport like Copa América in 2015
Emol2015  or a political one, like the student demands for improvements on
education policies in 2011 Ruz2019 .

###  II.4 Numerical implementation

For our simulation, we used a grid comprised of $64\times 64$ elements to
contain the population, and another grid, with the same dimensions, was used
to contain the targets. According to FIG 1, the initial population is
completely inactive, and its density at any location depends on the sector in
which it lies. The same distribution applies to deprivation. The actual values
are shown in TABLE 1.

Sector | Pop.$/$Element | Deprivation
---|---|---
Elite residential | 240 | 0.01
Zone of Maturity | 520 | 0.05
Zone of in situ | 400 | 0.1
accretion |  |
Zone of peripheral | 328 | 1
squatter settlements |  |
Table 1: Population and deprivation values per grid element, used on the
simulations, each sector corresponds to those shown in FIG. 1

The numerical parameters in our simulations are the same used by Davies et al.
Davies2013  as they proved to be helpful to describe an actual rioting
situation. The only exception is $dt$ and the temporal delays $L_{r}$ and
$L_{p}$, as they were increased to slow down the reaction time for both
rioters and police forces. This choice was adopted to take into account the
larger distances involved in our simulations. The actual values implemented
are shown in TABLE 2. We must remark that temporal dynamics is preserved as
long as the products $L_{r}dt$ and $L_{p}dt$ are kept constant. In this way,
we can increase $dt$ and study the long-term evolution of the system, or lower
it to examine the initial dynamics in more detail.

$\alpha_{r}=0.6$ | $\beta_{r}=0.5$ | $\gamma_{r}=0.11$ | $L_{r}=28$ | $dt=0.25$
---|---|---|---|---
$\alpha_{p}=0.65$ | $\gamma_{p}=0.012$ | $L_{p}=40$ | $\eta=0.006$ | $\tau=0.75$
Table 2: Numerical values of the parameters used for all the simulations.

###  II.5 Police forces behavior

To explain how activities of police forces are implemented, first we must
recall one of the premises of the model: because of eq. (10), every
combination on the number of rioters and police forces, at any given region in
space, can conduce to only two different and mutually excluding scenarios: i)
Police forces regain complete control of the situation and the incident is
immediately extinguished, or ii) Police forces are overwhelmed and cannot take
back control of the situation, without the addition of reinforcements. This
dichotomous behavior depends on the initial number of officers and is a
byproduct of the floor function present at the capture rate. When
$P_{j}(t)<R_{j}(t)$, police forces completely lose control of the situation,
since the floor function forbids to perform any detention, by making the
capture probability nil.

Because riot formation requires a lesser number of police forces than rioters,
at any given location, we chose that $P_{total}=\alpha_{1}R_{total}$ . Where
$R_{total}=\displaystyle{\sum_{j}R_{j}}$ is the total number of rioters and
$\alpha_{1}<1$ is a constant, smaller than one, if the total number of rioters
remains less than a threshold value, this means $R_{total}<R_{C}$, thus
promoting conditions for riot formation. Once this critical value is reached,
the total number of police forces becomes $P_{total}=\beta_{1}R_{total}$,
where $\beta_{1}>1$, allowing the possibility to regain global control of the
situation by police forces. As an additional condition, we also imposed that
when the maximum value for $P_{total}$ is reached, it is kept constant, making
the total rioting activity on the city a lot easier and faster to control. If
this last condition is not kept, control is lost for a period almost $10$
times larger than otherwise.

##  III Results

###  III.1 Rioter Distribution

The new parameters introduced in our current formulation can potentially
change the system dynamics. To explore this, we examined different
combinations of the constant values in the model set-up. By changing those
parameters, we found that the result can develop in different ways by
changing:

  * •

$\delta$: Accessibility, lower values mean equally lower travel times, $1$
represents walking to a destination.

  * •

$R_{C}$: Riot critical intensity.

  * •

$\beta_{1}$: Multiplicative factor of order forces, after the critical
intensity $R_{C}$ is reached.

All other parameters were kept constant: for example, the value
$\alpha_{1}=0.7$ was maintained because different values did not change
significantly the dynamics of the simulations, only slightly affecting the
time at which $R_{C}$ is reached (and not modifying the posterior riot
evolution).

Figure 4: Evolution of the total number of rioters $R_{\rm Total}$, as
function of time. For the parameters $\beta_{1}=1.5$, $R_{C}=1500$ and three
different values of $\delta$. Figure 5: Evolution of the total number of
rioters $R_{\rm Total}$, as function of time. For the parameters
$\beta_{1}=1.5$, $R_{C}=3000$ and three different values of $\delta$.

Further exploration of the simulation parameter space, for example $R_{C}$ and
$\beta_{1}$, indicates that changes performed on the former make an
appreciable difference only when $\delta$ is small, corresponding to a
situation in which it is easy for the active population to move around the
city; see FIGS. 4 and 5 with $\delta=0.2$. In the case when $R_{C}=1500$,
rioter accumulation has a certain “inertia”, due to the relative small police
force contingent and the proportionately high active population ready to join
the disorder. In that situation the peak value of $R_{total}$ is around
$2000$, see FIG. 4. On the other hand, when the threshold value is increased
to $R_{C}=3000$, most of the activity is immediately extinguished once that
value is reached, leaving only a scattered remaining as shown in FIG. 5.

The floor function in eq. (10) controls the influence of the second parameter,
$\beta_{1}$, on the dynamics. Different values of this parameter can therefore
only bring about different results when the increments are in the order of
$1$. In consequence, when $\beta_{1}$ lies within on the range $1<\beta_{1}<2$
all simulation results are similar, in view of the fact that the capture rate
is more or less the same. If this control parameter is further increased, with
$\beta_{1}>2$, the outcome for any value of $R_{C}$ and $\delta$ is very close
to the result shown in FIG. 5 with $\delta=0.2$. This behavior implies a quick
extinction of the intense rioting activity, leaving a scattered residual
disorder. Therefore, in a riot situation, control can be regained by order
forces when roughly $P_{total}\sim 2R_{total}$. It is a strong indication that
control can be taken back very quickly if all available order forces are
deployed as fast as possible once the threshold $R_{C}$ is reached. While the
ratio $P_{total}/R_{total}$ is the largest possible, riots will have a shorter
duration and reduced intensity.

####  III.1.1 Low accessibility results $\delta=1$

Here we will describe the spatial distribution of disorder activity when there
is low accessibility. In this case, there is no available public transport
network, and therefore any movements by the active population or potential
rioters are made by walking. In consequence, any rioting activity will tend to
be within a close distance of the participant’s residence.

Because of these mobility restrictions, rioting activity is generated and
remains confined to the city’s peripheral zones. It seems that the origin of
the disorder is located at the boundary between the two outer regions, which
are also the most deprived areas of our configuration. The fact that a
division between two regions increases the possibility of conflict is in
agreement with rutherford2014good , where these particular spots, localized at
the boundaries, generally have a historical record of public disorder and
civil violence, as shown in FIGS. 6 and 7.

On the opposite side of the city, high-income areas are virtually conflict-
free, as should be expected, because of their low levels of deprivation. The
scarce activity is concentrated at the center of our configuration, where the
frontier with other regions occurs.

On the other hand, at the peripheral regions, riots are widespread, summoning
large fractions of the population overall the surface surrounding the target
region. We also found a high asymmetry on the vertical distribution of
rioters, shown in FIG. 7; in fact, the southern half of the city has a riot
activity that lasts almost two times more than the northern counterpart. The
difference is a byproduct of a slight asymmetry in the population and
deprivation initial distributions, being the north half marginally more
significant than the south. The relative difference lies in the order of
$1/32$.

The result indicates that small and almost negligible differences in the
initial conditions have as in consequence large deviations in the disorders’
long-term behavior. In this particular case the total duration of the activity
in the southern region has half the duration of that within the northern
region. Furthermore, it is more intense, recruiting a larger number of rioters
in less time. This localized increase in the number of rioters will attract
more police forces, leading to a faster extinction of the activity once the
threshold $R_{C}$ is reached. This excess of forces will be displaced
subsequently to the remaining areas with active conflicts. This movement can
be very slow, only depending on the equations’ delay term, as shown in eq.
(8).

Figure 6: Horizontal rioter distribution for $\beta_{1}=1.5$, $R_{C}=3000$ and
$\delta=1.0$. Figure 7: Vertical rioter distribution for $\beta_{1}=1.5$,
$R_{C}=3000$ and $\delta=1.0$.

####  III.1.2 High accessibility results $\delta=0.2$

In this case, the origin of rioter activity is the intersection point of the
transport network. From there, the disorder expands towards the peripheral
regions of the city. This expansion has the shape of travelling waves, similar
to those found by Berestycki berestycki2015model ; bonnasse2018epidemiological
; berestycki2020modeling ; unlike in that case, however, these waves are not
characterized by a constant speed, but instead the “disorder front” slows down
as the riot reaches the periphery of the city, see FIGS. 8 and 9. The
extinction of rioter activity, a consequence of police action, also shows a
traveling wave pattern with a lower speed.

It is important to remark that the peak of rioter activity of the horizontal
distribution is not exactly located at the intersection. Still, it is shifted
towards the elite residential sector. This phenomenon is explained by the
balance between more attractive targets for the population and the traveling
distance. Another high activity spot is located in the opposite direction but
still close to the Central Business District. Nevertheless, the larger portion
of rioting activity lasts, comparatively, for a very short time. There are
still some residual rioter groups that stay active for a longer time, but they
are scattered widely across the domain in a way that is homogeneous over the
vertical part of the domain and highly asymmetric on the horizontal segment;
see FIGS. 8 and 9. Therefore in this case some activity is preserved, on the
long term, at the most deprived regions of the city. Nevertheless, peak
rioting activity is at least five times lower. Its duration is eight times
shorter than the case without a transport network, comparing FIGS. 6 and 8
intensity and time scales between.

Figure 8: Horizontal rioter distribution for $\beta_{1}=1.5$, $R_{C}=3000$ and
$\delta=0.2$. Figure 9: Vertical rioter distribution for $\beta_{1}=1.5$,
$R_{C}=3000$ and $\delta=1.0$.

##  IV Analysis and Conclusions

We extended the Davies et al. model by incorporating transport networks to
consider population displacement over long distances in the city, and applied
this extended model to a typical Latin American city configuration. To choose
the targets and regions more prone to be attacked, we based our model on the
first days of rioting activity from Santiago, Chile, during October 2019. Most
of the disorders and criminal activity were highly concentrated around subway
stations, and the most active spot was the best–connected region in the city.

Our model’s simulation results were quite surprising because the assumed
number of police forces was directly dependent on the total number of rioters.
This behavior leads to a counterintuitive outcome. When there is a working
transport network, it facilitates the formation of disorder areas, reaching
very swiftly a high number of rioters located at precise spots on a relatively
small region in the city. On the other hand, if the transport network is not
working, the rioting activity takes more time to develop a critical mass, and
is extended over a larger area of the city. Contrary to what intuition could
indicate, however, it is this second case where the produced disorder has a
more considerable intensity, lasting longer and requiring a much larger
deployment of police forces. We found that the necessary number is
approximately two times larger than for the same situation, with working
transport networks.

This counterintuitive behavior is a consequence of the police requirement,
which appears in eq.(7). This order forces behavior means that a fast rioter
concentration over a small region in space stimulates an equally quick
increment on police forces on those regions mentioned above. Once those
initial disorders are controlled, and because the total number of police
forces cannot decrease, those forces will start to quickly move around the
city, as they are required. This strength of resources allows the police
forces to effectively regain control in any other area where disorder is
formed. On the other hand, when there is no working transport network, this
accumulation of forces is slower and more spatially extended, meaning that
they take more time to regain control.

An interesting characteristic of the spatial propagation of rioting activity
is that it moves, forming well-defined intensity fronts of traveling waves.
This kind of behavior has previous been predicted by Berestyki
berestycki2015model ; bonnasse2018epidemiological ; berestycki2020modeling .
When transport networks are present, those waves’ origin is located at the
intersection of the transport lines (the most connected spot on the city) and
later moves towards the peripheral and more deprived regions, as shown in
FIGS. 8 and 9. This predicted behavior was observed during the riots of
Santiago during October 18th 2020, when it was commemorated the first year of
the riots of October 2019 Emol2020-1 ; Emol2020-2 .

At this point it is important to recall that a fast and efficient disorder
extinction is based on two basic premises:

  * i

Police forces can increment their numbers as the situation requires, with no
limit and as fast as needed.

  * ii

Those forces can move freely all over the city. The displacement times do not
depend on the distances involved.

When those two conditions are not fulfilled, transport networks’ presence only
increases the speed at which rioters accumulate at the target points,
conducing to a much larger and extended disorder activity. Therefore, without
this increased accessibility, most of the population is confined to choose to
attack only between those targets located nearby they reside in.

There are no quantitative studies on the effects of geographical segregation
over crowd behavior in Latin American cities, as far as we are aware.
Nevertheless, we can note the work of Lim et al. lim2007global  and Rutherford
et al. rutherford2014good . On those studies, it is shown that boundaries
between large-scale segregated areas are historically sources for localized
spots of high social tension and civil violence. In addition to the
geographical components, there are also social factors; for example, it was
found by Bonnasse–Gahot et al bonnasse2018epidemiological , for the French
riots of 2005, that the most relevant sociological variable used to predict
civil violence was the population of young males aged 16–24 with incomplete
high–school education. However, a previous formulation proposed by Pires and
Crooks pires2017modeling , using an agent-based model and incorporating social
network analysis and geographical information, identified that the buildup of
tension, resulting in the provably civil disorder, is mainly decreased by
improving employment possibilities for the population. However, if only
education is increased, the result is a more tense and unstable climate. The
previous conclusion could be of particular interest for the case of Chile
because there, historically, most riots are concentrated at specific spots of
cities and seem to be more the result of organized activity than a spontaneous
reaction to deprived conditions govea1981riot . If Pires and Crooks’ model is
correct, these environmental conditions look like an explanation for the civil
disorder organization in Chile somma2017discontent . This seems to be a
compounded effect of a significant increase in the number of higher education
students OECD2012 , and the economy deceleration over the last decade
bcentral2019 , creating a build–up of social tension.

##  V Perspectives and Future Work

For this work, we used a radically simplified population distribution pattern
for our model, which captures only the most basic features of a generic Latin
American city. A natural extension for our model would be using real
geographical data from a major Latin American city. In this way, the approach
demonstrated here could be used to make specific predictions. In particular,
it could be of practical utility to anticipate the potential level of riot
activity at a given location and the number of police forces needed to control
any eventual disorder.

Another model characteristic that could be improved is police behavior. In the
present model, police forces can move instantaneously all over the city
without any distance penalization. A possibility to overcome this issue could
be to use an agent-based model, via which differential travel times could be
included, potentially leading to a more realistic outcome from their
interaction with the civil population.

Finally, another interesting configuration to study could be government
intervention over the communication networks, as occurred during the Arab
Spring in 2011 hassanpour2014media . In this case, temporal disruption of the
internet and telephone lines appeared to amplify the disorder, increasing
citizen participation, by fragmenting the global communication network and
allowing the formation of numerous smaller and independent groups. In future
work, therefore, we could study the effects of information control on the
population.

## Acknowledgments

C.C. wishes to acknowledge the support of FONDECYT (CL), No. 1200357 and
Universidad de los Andes (CL) through FAI initiatives.

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