# Weak nodes detection in urban transport systems: Planning for resilience in
Singapore ††thanks:  Luca Pappalardo has been partially funded by the European
project SoBigData RI (Grant Agreement 654024). Gianni Barlacchi has been
partially funded by the EIT project Cedus.

Michele Ferretti1, Gianni Barlacchi23, Luca Pappalardo4, Lorenzo Lucchini23,
and Bruno Lepri3  1Department of Geography
King’s College London, London WC2B 4BG, United Kingdom
Email<EMAIL_ADDRESS>2Department of information engineering and
computer science,
University of Trento, Trento, Italy  3Fondazione Bruno Kessler, Trento, Italy
Email<EMAIL_ADDRESS>
Email<EMAIL_ADDRESS>4ISTI-CNR, Pisa, Italy
<EMAIL_ADDRESS>
###### Abstract

The availability of massive data-sets describing human mobility offers the
possibility to design simulation tools to monitor and improve the resilience
of transport systems in response to traumatic events such as natural and man-
made disasters (e.g., floods, terrorist attacks, etc…). In this perspective we
propose Achilles, an application to models people’s movements in a given
transport mode through a multiplex network representation based on mobility
data. Achilles is a web-based application which provides an easy-to-use
interface to explore the mobility fluxes and the connectivity of every urban
zone in a city, as well as to visualize changes in the transport system
resulting from the addition or removal of transport modes, urban zones and
single stops. Notably, our application allows the user to assess the overall
resilience of the transport network by identifying its weakest node, i.e.
Urban Achilles Heel, with reference to the ancient Greek mythology. To
demonstrate the impact of Achilles for humanitarian aid we consider its
application to a real-world scenario by exploring human mobility in Singapore
in response to flood prevention.

###### Index Terms:

urban science, data science, human mobility, complex systems, network science,
multiplex networks, resilience, social good

##  I Introduction

Cities are a physical manifestation of continuous processes of human
interaction and of the weaving and knitting of social relations unfolding
through space and stratifying through time. Cities are inherently “hard” to
understand; their interlacing facets and tangled assemblages render them
difficult to frame [9, 27, 26]. In particular, the nexus of human mobility
patterns, transport planning choices and urban policies pose great
difficulties for scientific modelling. It presents, in fact, what Rittel _et
al._ [36] call a “wicked problem”, that is, a problem whose social nature and
lack of objective truth makes inherently hard to tackle with traditional
analytic tools.

Despite being still far from adequately composing these problems [9], nowadays
the variability of massive data describing human movements allows planners and
policy-makers to face relevant urban challenges through computational methods
[12, 37, 15, 6, 33, 4, 1, 32, 5]. While the observation of mobility flows
offer for instance the possibility to investigate the inner workings of urban
transport systems, this is not enough for efficient transport planning. As
Ganin _et al._ [17] note, traditional transport studies focus on efficiency
measure by observing a transport network under normal operating conditions.
This approach though fails to capture information that really matters for
real-world applications, i.e. how the network reacts when is under stress
conditions. Such is the case of traumatic events that might cause disruption
of service or worse, like in the event of natural or man-made disasters (e.g.,
floods, terrorist attacks, etc.). Thus, uncovering weak points that affect a
network’s resilience, a proven key metric for transport policies [17], is not
only a task of paramount relevance for the design of proper intervention
scenarios in case of disasters, but also an actual requirement for managing
transport services in complex global cities like Singapore. We refer to such
weak points as _Urban Achilles Heels._
111https://en.wikipedia.org/wiki/Achilles%27_heel

Starting from these considerations, we address the following questions: _(i)_
What and where are the weakest transport routes in a city, i.e., its _Urban
Achilles Heels_? _(ii)_ Given some changes in the transport system, what
scenarios are likely to occur and what is their impact on human mobility and
on the resilience of a transport system?

Here we propose Achilles, a web-based application which provides an easy-to-
use interface to explore the mobility fluxes and the connectivity of urban
zones in a city, as well as to visualizes changes in the transport system
resulting from the addition or removal of transport modes, urban zones and
single stops. Achilles is based on a multiplex network representation of
mobility data [14], where every layer describes people’s movements with a
given transport mode, e.g. buses, metros, taxis. A node in a layer represents
an urban zone of the city, edges indicate routes between zones and edge
weights indicate the amount of people moving between two nodes in a given time
window. Achilles exploits this network representation and allows the user to
visualize changes in the transport system resulting from the addition or
removal of transport modes, urban zones and single stops. Notably, Achilles
allows to compute the _Heel-ness_ of a zone, a measure introduced in this
paper which indicates the probability of a zone to disconnect the network when
edges adjacent to it are removed. The computation of the _Heel-ness_ allows
Achilles to state that the transport system has an _Urban Achilles Heel_ if
there is at least one urban zone with a non-zero _Heel-ness_.

We show how Achilles can be used to explore human mobility in Singapore and
test the resilience of its network to natural disaster, by using mobility data
from several sources. The application allows to point out weak routes among
urban areas in the city (i.e., routes where public transport does not meet the
needs of the citizens), and simulate changes in the capacity of public
transport to satisfy needs of citizens when specific events occur in the city,
e.g. closing/adding transport modes or subway/bus stops. Additionally, it can
be used by supplement humanitarian agencies for resilience preparedness and
response, and in general fragile contexts where there is a dire need of
deploying fast, scalable and effective humanitarian aid. Our approach is in
fact highly flexible since it uses only data about transport and mobility
flows. Given that many open datasets of such nature are nowadays publicly
available222https://data.gov.sg/group/transport, our approach can be
potentially applied to any other urban area to simulate traffic changes due to
specific events, such as the impact of adding or removing transport modes or
stops, the impact of closing the access to an urban area, or the organization
of city-wide public events.

##  II Related Work

Understanding human mobility is a long-standing and highly relevant challenge
for today’s increasingly urbanized world [40]. In their reconstruction of the
discipline’s evolution Barbosa-Filho _et al._ [3] highlight how through time
different scientific communities attempted to frame it using disparate data
sources; starting from geographers’ studies of spatial interaction and
regional migration [42, 21]. It is only recently though that significant
advancements having been made due to novel data sources, such as Call Detail
Records (CDRs) and GPS location data collected through mobile phone devices
[19, 33] and analyzed through the lens of network science [39, 8].

Using precisely methods from network science we can design powerful models and
simulation tools for what-if analysis of different urban planning scenarios
[30, 25]. Singapore, due to the complexity of its transport system, is
precisely a global city where such tools would be particularly valuable [24].
Network science applied to the study of transport systems has proved to be
increasingly useful in recent years. Xu and González [41] have shown that a
slight re-routing of a fraction of daily car commutes produces significant
reductions in traffic, thus alleviating the congestion state of the overall
transport system. In this perspective the development of data-driven analytic
tools can help control the transport system and improve both the customer’s
travel experience and the system’s overall efficiency. Çolak _et al._ [11]
have investigated the interplay of number of vehicles and road capacity to
determine the level of congestion in urban areas. In particular, they show
that the ratio of the road supply to the travel demand can explain the
percentage of time lost in congestion. De Domenico _et al._ [14] show that the
efficiency in exploring the transport layers depends on the layers’ topology
and the interconnection strengths.

Although these works doubtlessly shed light on interesting aspects about the
structure of urban transport, they do not provide easy-to-use tools for
exploring a city’s demand for mobility and the efficiency of the related
transport system. Giannotti _et al._ [18] partly overcome this problem by
proposing a querying and mining system (M-Atlas) for extracting mobility
patterns from GPS tracks. This system, still, does not provide a method to
assess weak points in cities, those that will collapse first in case of
traumatic events.

Wang _et al._ [22] highlight in fact many studies in resilience engineering
that try capture different facets of the problem at stake, often tackling the
most disparate domains (e.g., from spatial economics [35] to computer networks
[38] and socio-ecological systems [2]). However, as pointed out by [17], since
the majority of transport-oriented policies stress efficiency aspects measured
under optimal conditions, and neglect to consider instead stress scenarios
that happen in the real word, there is still a need to provide policy-makers
with adequate tools to actually capture a transport network resilience; here
intended once again as the network’s capacity to recover from a catastrophic
event returning to its original working state. Moreover, there is still a need
to design tools to test and prototype scenarios for rapid response under
critical conditions.

##  III Data Sources

In this work, we use different data sources from the city of Singapore with
information about (i) transport, (ii) administrative boundaries, (iii)
mobility flows and (iv) floods.

##### Transport Data

The public transport data are provided by LTA Data
Mall333https://www.mytransport.sg/content/mytransport/home/dataMall.html,
which published a wide variety of land transport-related datasets both static
and in real-time. In this work we use data about bus
lines444https://www.mytransport.sg/content/mytransport/home/dataMall.html,
which provide information to build a transport network describing the
displacements of inhabitants between different urban zones of Singapore. In
particular, for every bus line, we retrieve information about its stops and
the GPS traces describing the bus route.

##### Administrative boundaries

The urban zones areas are provided through the _shape-files_ at different
administrative division levels of the city, and are available at the website
data.gov.sg555https://data.gov.sg/dataset/master-plan-2014-subzone-boundary-
web. We use the most fine-grained division and, by applying a spatial join
operation, assign every bus stop to the corresponding urban zone. The
resulting dataset contains 323 urban zones and 4856 bus stops.

##### Mobility flows

We use data indicating both the presence of people in every urban zone and the
fluxes of people, i.e. Origin-Destination (O-D) matrices [23] between urban
zones at a given date and time. Such data were obtained from the data-as-a-
service APIs provided by
DataSparkAnalytics666https://www.datasparkanalytics.com/products/data-as-a-
service. In particular, the presence of people and the O-D matrix are
estimated starting from the mobile network data provided by different Asian
telco operators. It is worth mentioning that location signals from the telcos
are analysed in an anonymous and aggregated way. We use these data to estimate
the number of people moving by bus between two urban zones in a given time
window, since official information about the number of users traveling on the
buses is not available. We aggregated the flows by hour.

##### Flooding data

Flood prone areas forecasts, computed by the Singapore’s Public Utilities
Board (PUB), are provided as an incentive to reduce the risk of recurring
flood events and to foster proactive measures among developers and the general
public. We used such areas, retrievable at
www.pub.gov.sg777https://www.pub.gov.sg/drainage/floodmanagement to inform our
use-case with substantiated data related to a real-world problem related to a
specific setting.

We hence obtain for every urban zone $z$: (i) the bus stops in $z$, (ii) the
bus lines passing through $z$, and (iii) all the urban zones connected to $z$.
We define two urban zones $z_{1}$ and $z_{2}$ to be connected if there is at
least one bus line connecting $z_{1}$ and $z_{2}$.

It is worth to mention that despite our experiment being limited to the bus
network system, due to limited data availability, further extensions to truly
multimodal systems can be integrated with ease. This will enhance the
modelling of transport planning scenarios with increased reliance and
confidence. Limitations standing, our proposed tool can already serve for
specific use-cases, such as precisely the considered instance of Singapore’s
bus network resilience to floods.

##  IV Methodology

The use of complex networks to analyze complex systems is a powerful way to
simplify the description of a problem by means of a process of abstraction.
This process reduces the complexity of the structure to two simple elements:
nodes, i.e. the objects of the system, and edges, i.e. the relations between
these objects. However, such a representation can lead to a strong loss of
information about the real system characteristics.

In this context, an interesting novel approach was proposed by De Domenico _et
al._ in their seminal work on multilayer networks [13]. They have proposed a
new framework in which it is possible to encode information about the
different possible relations between the nodes of the system in different
layers. Each of these layers represents a specific kind of relation between a
set of nodes and can be considered as a partial representation of the whole
system without the loss of a simple structure representation: for example, a
sub-network describing a specific kind of relation between the nodes of the
network, which again reflects some of the properties or the characteristics of
the complex system. In this work we adopt a similar mathematical formulation
to describe the transport system of an urban environment. To better describe
the transport system of the city we are studying, in Section VI we
specifically model the network structure to best fit the city characteristics
following the more general formulation discussed herein.

In an urban area, two zones can be connected through several transport means
(e.g., bus, taxi, subway, etc.). As each transport mean differs from the
other, also each subway or bus line differs from the others. In fact, using
two different lines of the same mean require a transfer at some stop. In turn,
each of these transport modes may be considered as a set of different lines
following a specific path; connecting specific zones of an area; and
eventually, can be considered as an additional layer of the system. To express
this kind of information we introduce the concept of urban multiplex network
by means of the following definition:

##### Definition

An Urban Multiplex Network (UMN) is a network in which two nodes represent
zones of an urban area and can be connected, at the same time, by multiple
edges that belong to different _layers_. We model such structure with an edge-
labeled multi-graph denoted by $G=(V,E,L)$ where $V$ is a set of nodes, i.e.
the zones in which the area we are studying is divided; $L$ is a the set of
layers which encode the information about the different possible types of
connections between two nodes, e.g. the public transport mean characterization
and/or the line number; $E$ is a set of labeled edges, i.e., a set of triples
$(u,v,l)$ where $u,v\in V$ are two nodes, and $l\in L$ is a layer of the
multiplex network.

The global connectivity of two zones, which is the multidimensional
connectivity (i.e., the connectivity considering all the layers of the
network) in an urban area is a combination of two elements: (i) _connection
intensity_ and (ii) _connection redundancy_. We define the intensity of the
connection between two zones on a single layer as:

##### Definition

Connection intensity

$h_{l}(u,v)=w_{l}(u,v)\frac{|\Gamma_{l}(u)\cap\Gamma_{l}(v)|}{min(|\Gamma_{l}(u)|,|\Gamma_{l}(v)|)},$
(1)

where $w_{l}:V\times V\times L\rightarrow\mathbb{N}$ is a weight function
representing the mobility flux between two zones on layer $l$, and
$\Gamma_{l}(u)$ is the set of neighbours of the zone $u$. Connection intensity
consists hence of two factors: the first factor, $w_{l}$, indicates the
percentage of people moving between the two zones using a specific transport
layer and line $l$. The second factor,
$\frac{|\Gamma_{l}(u)\cap\Gamma_{l}(v)|}{min(|\Gamma_{l}(u)|,|\Gamma_{l}(v)|)}$,
is the percentage of common neighbours, ${|\Gamma_{l}(u)\cap|\Gamma_{l}(v)|}$,
with respect to the most selective zone,
$min(|\Gamma_{l}(u)|,|\Gamma_{l}(v)|)$. The idea is that, on each layer, the
connection intensity is influenced by the number of displacements between the
two zones, weighted by the value of selectiveness of the most selective zone,
i.e. the ratio between the neighbours shared by the two zones and the number
of neighbours of the zone with the smallest set of neighbours.

The second element on which global connectivity depends is connection
redundancy, which takes into account the relevance of a layer in the all-
layers zone’s connectivity. This quantity measures to what extent the removal
of the links belonging to a specific layer affects the capacity to reach a
neighbouring zone from the considered one. Redundancy in general is a measure
of the strength of node-node connections considering the links connecting the
two nodes over all the layers of the network. Before defining the connection
redundancy, however, we need to introduce the layer $l$ relevance, $LR(u,l)$,
for the node $u$. $LR(u,l)$ can be defined as the ratio between the number of
nodes that disconnect from node $u$ if all the links of layer $l$ are removed
(which is the same as removing an entire layer to the multiplex network) and
the number of nodes reachable before the removal of the layer[10]:

##### Definition

Layer Relevance

$LR(u,l)=1-\frac{N_{r}(u,l^{c})}{N_{r}(u,L)}$ (2)

where $N_{r}(u,L)$ is the number of reachable nodes from node $u$ using all
the multiplex network’s layers $L$, while $N_{r}(u,l^{c})$ is the number of
reachable nodes from $u$ after the removal of layer $l$, i.e. the reachable
nodes from $u$ considering only the set of layers $l^{c}=L\setminus l$.

##### Definition

Connection Redundancy

$r_{l}(u,v)=(1-LR(u,l))\cdot(1-LR(v,l))\cdot\sum_{m=1}^{L}\frac{A_{u,v,m}}{L},$
(3)

where $A_{u,v}$ is a vector element of the adjacency tensor of the multiplex
that keeps track of the presence of edges between node $u$ and node $v$ over
the $l\in L$ different layers. We give a higher score to the edges that appear
in several layers, so we are interested in the complement of those values. If
the two areas are linked in more than one layer, the score raises until a
maximum of 1.

We combine connection intensity and connection redundancy taking into account
the multidimensionality of multilayer network’s connectivity: a greater number
of connections on different layers is reflected in a greater chance of having
a strong connectivity.

##### Definition

(Global) Connectivity.
Let $u,v\in V$ be two nodes and $L$ be the set of layers of an urban multiplex
network $G=(V,E,L)$. The connectivity of two urban areas $u,v$ is defined as:

$c(u,v)=\sum_{l\in L}h_{l}(u,v)(1+r_{l}(u,v)).$ (4)

It is worth noticing that the measure proposed can be used to estimate the
strength of the connectivity between nodes also in single-layer networks, in
which $r_{l}$ is one and the overall sum is $h_{l}$.

The aim of this work is to provide policy-makers, transport planners and
humanitarian agencies with a working tool that has the versatility to adapt to
different urban areas and the capacity to quantify the level of susceptibility
to disruptive events related to a transport system own structure. More
specifically, we are interested in studying the weaker ties as those
potentially responsible for the disruption of the multiplex network. In
particular, to study the weaknesses of a specific network, we introduce the
notion of Heel-ness as the relative importance of a node in the context of
network navigability.

##### Definition

Heel-ness.
Given a zone $u\in V$, the _Heel-ness_ of $u$ is defined as:

$H(u)=max\bigg{[}\frac{|V-R_{u-(u,v,l)}|}{min[c(u,v)]_{v\in
N_{u}}}\bigg{]}_{l\in L},$ (5)

where $N_{u}$ is the set of neighbors of $u$, $R_{u}$ is the set of reachable
nodes from node $u$ while $R_{u-(u,v,l)}$ is the set of reachable nodes from
node $u$ after the removal of the edge $e=(u,v,l)\in E$. As a consequence of
these definitions $|V-R_{u-(u,v,l)}|$ is the number of non-reachable nodes
after the removal of the edge $e$.

This metric quantifies this importance in terms of optimal disruptive events
classified following the Granovetter’s theory for single-layer networks [20]:
the importance of a node in preserving a network’s giant component depends on
the connectivity of the node. In particular, nodes which shares weak
connections tend to be those more important to be preserved. Heel-ness
measures the importance of a node considering the weakest connection it has,
and the number of unreachable nodes after that edge disruption. The
Granovetter’s theory was proposed in 1973 in the context of social networks.
Since then it was studied and proposed in countless different fields of
application. Following a similar approach we adapt the measures introduced in
[31] to detect important ties in transport networks. In particular we focus
our attention on the topological resilience of multilayer networks as proposed
in [14]. Topological resilience studies the robustness of a network in
resisting to topological failures, such as edge removal. Heel-ness measures
the single-event effects of the network and ranks them depending on the
greatest damage they can inflict to the network structure if one particular
edge is removed. Extending then the idea from [20] to transport networks
allows us to refer to weak ties as those edges that make the network more
fragile and susceptible to targeted edge loss. We stress further that
topological resilience takes into account only the network structure while
edges can be weighted in such a way that more important links, e.g. in a
transport network those links that sustain higher traffic flows, are naturally
considered as more crucial for the overall system connectivity.

Being able to classify nodes depending on the importance of their connection
in the network, we can classify networks based on the presence or absence of
fragile nodes. Using the definition of Heel-ness we can state that an urban
multiplex network has an _Urban Achilles Heel_ if there is at least one node
for which the _Heel-ness_ has a non-zero value.

##### Definition

Urban Achilles Heel.
We define the _Urban Achilles Heel_ of a multiplex network as the node $h\in
V$ for which

$H(h)>0\hskip 2.84544pt\text{and}\hskip 2.84544ptH(h)\geq H(u)\hskip
7.11317pt\forall u\in V.$ (6)

This definition provides a classification based on a first-order disruptive
event, i.e. an event that only causes the loss of a single edge. However, the
extension of this definition to further orders is straightforward and can be
done by sub-sequential removal of the $n$ weakest ties of the network.

Applying this definition to real transport networks can help in the
identification of important edges and nodes that are crucial for the stability
and routing of all zones in an urban area.

##  V Application Design

The application is freely available online at http://achille.fbk.eu. For
scientific dissemination and to provide access for a larger external non-
technical audience it was also complemented by a video-tutorial
http://achille.fbk.eu/about. The application design follows closely the
analytical framework described in Section I, while implementing a client-
server architecture pattern. We implemented a RESTful web service using
_Flask_ , a widely used micro web framework written in Python. In the REST
architecture the server exposes the resources to a client trough a set of
defined URLs. For example, in order to get the list of buses passing trough an
area, the endpoint is defined as
http://achille.fbk.eu/area/<id_area>/services. This API is responsible not
only for data provisioning, but also acts as the communication layer between
the user and the network models. It is worth noting that, given its
modularity, our application can be re-purposed as an agnostic provider of
services to other consumers. In particular, the exposed methods consist in:

  * •

a query endpoint returning the network features’ geometries for a given urban
zone ID, and additional information used to populate the geographic map
application;

  * •

a second query endpoint returning for each given urban zone ID the computed
network metrics, i.e., _Connection intensity, Connection Redundancy,
Connectivity_ (Section IV).

The front-end application is thus the entry-point for quickly interacting and
prototyping future transport scenarios. The User Interface (UI), visible in
Figure 1, has been developed with easiness of use and clarity of
interpretation its standard pillars. It allows a non-technical audience to
inspect the number of routes connecting an urban zone to the rest of the city
simply by clicking on an urban zone in the city panel and control them via an
interactive menu (Figure 1). Upon addition/removal of one or more routes in
the menu, the user can trigger the calculation of the network metrics (Figure
2), which are promptly displayed in three separate windows. Every window shows
a 3D map of the city, where an urban zone’s height is proportional to its
average networks value computed over the connected urban zones (Figure 3). The
menu and the bottom windows allow the user to simulate how the city’s
connectivity changes after, for example, the construction of a new route or
the temporary closing of an existing one. Further, the identification of
sensitive areas, the so-called _Urban Achilles Heel_ , can be accessed through
a revealing button. The example in Figure 2 presents four areas in the
northern part of Singapore.

Figure 1: Application User Interface (UI): Upon an area selection, the user is
presented with the 3D network metrics for the city of Singapore. The
interactive menu on the right allows then to select and deselect transport
routes. Figure 2: Metrics: The “Run” button allows to calculate the intensity,
redundancy and connectivity measures and to visualize them in the three bottom
windows. Further, every interaction with the bus lines menu will trigger the
computation of the Heel-ness metric. In the above image, four areas are
identified as such vulnerable sites.  Figure 3: Zonal Focus (example): The 3D
maps in the rightmost bottom window inform the user with additional metrics.
In the example, the height of the extrude zone is proportional to the average
value of its multidimensional connectivity, computed across all the connected
urban zones.

All of the geographic map components are fully interactive and built with the
latest web-mapping technologies taking advantage of latest WebGL standards and
3D capabilities; as such, they allow for a most fluid and seamless experience.
This is a crucial factor that allows the application to hide the complexity of
the models behind a fluid interface. It also let the technological stack move
out of the way, making room for discussions and human elements that enhance
decision-making processes.

##  VI Case study: Floods prevention in Singapore

In the following section we discuss the specific modeling adopted for the
Singapore’s transport network and the results obtained by studying the
structure of the system and its resilience to synthetic disruptive events.

###  VI-A Network structure

When dealing with real-world problems a case-dependent modelling usually
ensures better performances and results in terms of actionable solutions. In
the advanced case study, we deal with Singapore’s transport network with the
aim to specifically uncover possible solutions to disruptive events (e.g.
intense rainfall and consequent floods…) which might negatively impact the
road network. We apply the methodology presented in section IV considering the
reduced transport network composed by more than 300 different scheduled bus
lines. This operation enable us to scope down the issues and focus solely on
those related to the road network.

In our framework, each bus line represents a layer of the network; each layer
consists of all the Singapore’s administrative areas (according to a
government sanctioned classification available at
data.gov.sg888https://data.gov.sg/dataset/master-plan-2014-subzone-boundary-
web); each zone is a node of the multiplex. Further, each bus line fully
connects to every zone in which a bus line has a stop with all the others
zones in which the same bus line has a stop. In this configuration, disruptive
events that involves only limited zones, which may not necessarily disconnect
the entire line if a reroute is prompted, are taken into account. The model is
indeed built in such a way that edges can be removed gradually depending on
the nature of the disruptive event we want to simulate. In figure 4 we
schematically represent the network structure used to model and analyze the
Singapore’s bus transport network. As depicted, each layer share with the
others the same nodes but changes the connections between the zones depending
on the bus line it represents.

Figure 4: Schematic of the structure of the multiplex network adopted to model
the road public transport services for Singapore. Each layer represents a
different bus line. Each of these fully connects different zones of the city.
This structure naturally takes into account possible rerouting strategies that
can maintain the layer partially connected.

Even in this specific network configuration, the metrics introduced in section
IV maintain their significance. The relevance of each layer depends, in fact,
on the relative number of neighbours in a layer with respect to the multiple
degree of the node. In particular, the shape of the connectivity, implemented
from the definition proposed in [10] ensures that the Heel-ness of a node
consistently depend on the node’s layer relevance and the node’s connection
redundancy instead that only on the connection intensity. We stress that, if
this would have been the case, fully connected layers would have brought to an
initial connectivity measure that would have encoded only operational
information, i.e. information about the mobility flow passing trough links,
for all the nodes that share the same bus lines independently on the possible
different size of the disconnected component they might cause after a
disruptive event involving one or all of their connections. This highlight the
importance of a combined approach that considers both the structure of the
multilayer network, i.e. relevance and redundancy of the connections, and the
network usage, i.e. the information about mobility flows over different links.

###  VI-B Experiments

Moving from of the aforementioned considerations we conducted an extensive
connectivity assessment by observing Singapore’s urban network resilience to
the removal of high and low connectivity links separately. We use this case
study to test the behavior of multilayer node connectivity, as defined in IV,
on a real transport network. To this extent we compute the connectivity for
every node in the network and we sort them accordingly to the obtained score.
In figure 5 we show the connectivity for the network in its initial
configuration. We can see that there are two dominant behaviors.

The left side of the histogram consists of areas with a low connectivity
score. These are the areas dominated by the contributions given by the number
of layers who share a connection showing different peaks. Each peak
corresponds to the different number of shared layers. The right side consists
of those urban zones that are geographically central and functionally
important urban nodes; the number of layers that two such nodes share is no
more dominating the contribution to the connectivity score, while its
increasing importance is given by the length of the different bus lines and
number of nodes they are connected with.

As illustrated in Figure 5, connectivity can be used to classify and sort
nodes depending on the strength and importance of their ties. This allows us
to test the weak ties theory for transport networks.

We start by removing the weakest tie in the network, and the subsequent in
decreasing order of weakness, iterating the computation after each removal.
The expected outcome is a faster network percolation than achieved adopting an
opposite strategy, i.e. removing the stronger ties of the network first. The
result of this experiment are presented in Figure 6. The relative size of the
largest network component changes with the removal of a given percentage of
links. The values are sorted by increasing (red solid line) or decreasing
(blue dashed line) connectivity-score order. The red line shows the different
speed of percolation faced by the same system under targeted attacks with
different strategies.

Additionally, it is worth noting how the network faces a first significant
disruption after the removal of more than 50% of total number of edges. This
finding holds even with the adoption of a more effective disruptive strategy.
Such result highlights the high degree of resilience exhibited by Singapore’s
bus network, which is consistent with the city’s highly efficient transport
network [7]. Further studies might benefit thus from replicating the above
experiments in other locales with different transport systems, to check
whether a similar pattern is present and how it behaves to disruptive events.

Figure 5: Connectivity value density within the network’s initial
configuration state. Connectivity is computed for each couple of nodes, which
are then are sorted depending on the computed score.

##  VII Discussion and Implications

Our simulated experiment has been conducted, due to data availability, only on
the bus transport layer. However, it is straightforward to extend our
framework to multiple transportation systems (e.g., metro and train). Testing
the network resilience to the aforementioned stress conditions in a truly
multimodal perspective represents thus an important tool at disposal of
transport planners to assess (i) the current state of the transport network;
(ii) to plan future operations; (iii) and to keep running the system at its
overall optimal capacity.

In particular, we found that Singapore has a resilient urban network, as all
the nodes are still reachable when removing up to 30% of the links. Moreover,
the deletion of links in decreasing order of connectivity affect less the
network’s global connectivity, as more than 97% of the urban zones are still
reachable after the removal of 90% of the links (Figure 6, blue dashed line).
In contrast, when deleting the links in increasing order of connectivity the
network crumbles faster, as almost 20% of urban zones become unreachable after
the removal of 90% of the links (Figure 6, red solid line). These results
suggest that the proposed connectivity and risk metrics can profitably be
deployed to discover those urban connections whose existence is crucial for
the network resilience. Moreover, those metrics can also help to evaluate the
impact of changes and intervention in the city’s mobility (e.g., the closure
of a bus line). Figure 6 highlights the points where the accelerated network
disassembling commences, i.e., the _Urban Achilles Heel_ : up to those values
the transport system exhibits a fair resilience, but surpassing such
thresholds provokes a rapid network fall-out.

Independently on the specific results obtained for the Singapore’s transport
network, we believe that our approach and our application can be used in
several situations, both from a geographical and structural point of view.

#### Application deployment

Considering for instance the level of complexity and the structural
organization of Singapore [7], one of the most developed cities in the eastern
hemisphere, our model simulation showcases a relatively sound resilience to
potential catastrophic events. While this is an interesting result in itself,
another valuable aspect of our application is its ease of reproducibility in
urban contexts characterized by more fragile infrastructural systems. In
cities and urban areas with newer or less developed transport systems the
application of the proposed methodology might in fact yield higher impact
results.

From from a data availability standpoint our application can easily be
deployed to other contexts. Out of the four major sources of information
presented in section III, namely _a) Transport Data, b) Administrative
Boundaries, c) Mobility Flows and d) Flooding Data_ only the very last one is
specifically tied to the Singapore’s context and the use-case introduced.
Other data sources, such as _Administrative Boundaries_ can in fact be
commonly found either through local authorities official data portals, or via
open data alternatives such as OpenStreetMap999https://www.openstreetmap.org/.
The pivotal source, i.e. _Transport Data_ , can additionally be found across
different urban settings also through open data standards such as the _General
Transit Feed Specification (GTFS)_ , which _”…allows public transportation
agencies to provide real-time updates about their fleet to application
developers”_. GTFS is _de facto_ the standard for exposing and consuming
public transit data; can be globally found and integrated through _ad hoc_
search engines such as TransitLand101010https://transit.land/; and come with a
robust track-record of academic research and industry applications. [29, 16,
28].

Lastly, in a final effort to extend the scalability of our application to
different contexts and use-cases, we devise possible implications of our tool
by distinguishing different typologies of users, namely urban planners and
policy makers, and humanitarian agencies.

#### Urban planners and policy-makers

Current approaches in transport network resilience focus on developing
comprehensive frameworks or policies that largely prioritize efficiency and
reducing delays, rather than considering the impact of stressful events on the
network [17]. While these approaches are perfectly understandable from a
policy’s stand-point, they might often leave real-world transport systems
vulnerable to changing conditions. These conditions might also be exacerbated
in complex urban environments such as Singapore. Thus, the insights provided
by testing such scenario might prove extremely useful instead in preventing
catastrophic events and improving the overall resilience of the system.

#### Humanitarian Organizations

The possibility to prototype scenarios and test the mutable conditions of a
system has direct implications for other cities; other kinds of disruptive
events; and other sectors, such as the humanitarian response. The latter is in
fact an area of application where the ability to prevent damaging phenomena by
containing negative impacts and providing an adequate rapid response is
absolutely crucial [34]. By providing a clean interface and a tool able to
scale over a wide portfolio of settings, our application is suited to address
challenges where timing is the key factor, such as _pre-event_ preparedness
prevention or _post-event_ rescue team deployment.

##  VIII Conclusion

Transport planning is a difficult arena to operate in. This is due to the
inherently ”wicked nature” of the issues at stake coupled with the complexity
of urban environments [9, 36]. Advancements in understanding human mobility
have recently been made through the use of computational methodologies (see
for instance [11, 33, 4, 1]). However, the need to operationalize and distill
such methods into actionable tools tailored to real-world problems is still
present. In particular, the humanitarian sector might benefit from the
application of models and methods coming from the new science of data. Our
proposed Achilles application is a step forward in this direction, providing
an easy-to-use interface on top of complex models that allow non-expert users,
such as non-profit organizations, to rapidly prototype different stress
scenarios. It represents thus a tentative example of how data science can be
put to good use to tackle issues of social relevance. In the presented
experiments we tested the case of Singapore, particularly suited due to its
natural geography and the frequent disruption thereby caused on the bus
transport network system by floods. Since the application is built deploying
data that are not site-specific (e.g. O-D matrices), it is scalable to
different environments, making it suitable to be applied even in fragile
contexts. Further, due to its flexible architecture Achilles accommodates new
data sources with ease, adapting its modelling capacity to meet local needs.
As unpredictable stress events might present themselves in the future,
transport planners need to be vary of mutable conditions. This requires not
only a close monitoring of transport networks under normal operational
settings, but the ability to quickly respond to new challenges [22]. The
solution we advance lies precisely in this direction.

Figure 6: Stability of the urban network to links removal. The x axis shows
the percentage of removed links. The y axis shows the size of the greatest
network component.

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# Existence of dicritical singularities of Levi-flat hypersurfaces and
holomorphic foliations

Andres Beltran  Dpto. Ciencias - Sección Matemáticas, Pontificia Universidad
Católica del Perú. Av. Universitaria 1801, San Miguel, Lima 32, Peru
<EMAIL_ADDRESS>,  Arturo Fernández-Pérez  Departamento de Matemática -
ICEX, Universidade Federal de Minas Gerais, UFMG Av. Antônio Carlos 6627,
31270-901, Belo Horizonte-MG, Brasil<EMAIL_ADDRESS>and  Hernan
Neciosup  Dpto. Ciencias - Sección Matemáticas, Pontificia Universidad
Católica del Perú. Av. Universitaria 1801, San Miguel, Lima 32, Peru
<EMAIL_ADDRESS>
###### Abstract.

We study holomorphic foliations tangent to real-analytic Levi-flat
hypersurfaces in compact complex surfaces. We give some hypotheses to
guarantee the existence of dicritical singularities of these objects. As
consequence, we give some applications to holomorphic foliations tangent to
real-analytic Levi-flat hypersurfaces with singularities on $\mathbb{P}^{2}$.

###### Key words and phrases:

Levi-flat hypersurfaces - Holomorphic foliations

###### 2010 Mathematics Subject Classification:

Primary 32V40 - 32S65

This work is supported by the Pontificia Universidad Católica del Perú project
VRI-DGI 2016-1-0018. Second author is partially supported by CNPq grant number
301635/2013-7

##  1\. Introduction

In this paper we study holomorphic foliations tangent to real-analytic Levi-
flat hypersurfaces on compact complex surfaces with emphasis in the type of
singularities of them. Singular Levi-flat hypersurfaces on complex manifolds
appear in many contexts, for example the zero set of the real-part of a
holomorphic function or as an invariant set of a holomorphic foliation. While
singular Levi-flat hypersurfaces have many properties of complex subvarieties,
they have a more complicated geometry and inherit many pathologies from
general real-analytic subvarities. The interconnection between singular Levi-
flat hypersurfaces and holomorphic foliations have been studied by many
authors, see for example [2], [4], [9], [10], [11], [12], [13], [14], [17],
[18] and [23].

Let $M$ be a real-analytic hypersurface (with singularities) in a compact
complex surface $X$. We emphasize that $M$ is a codimension one (real) closed
subvariety. Also, unless specifically stated, subvarieties are analytic, not
necessarily algebraic. Let us denote $M^{*}$ the set of points of $M$ near
which $M$ is a nonsingular real-analytic hypersurface. $M$ is said to be Levi-
flat if the codimension one distribution on $M^{*}$

$T^{\mathbb{C}}M^{*}=TM^{*}\cap J(TM^{*})\subset TM^{*}$

is integrable, in Frobenius sense. It follows that $M^{*}$ is smoothly
foliated by immersed Riemann surfaces, the foliation defined by
$T^{\mathbb{C}}M^{*}$ is called the Levi-foliation and will be denoted by
$\mathcal{L}$.

A holomorphic foliation $\mathcal{F}$ on $X$ can be described by a section
with isolated singularities $\omega\in H^{0}(X,\Omega^{1}_{X}\otimes
N_{\mathcal{F}})$ for some line bundle $N_{\mathcal{F}}$ called the normal
bundle of $\mathcal{F}$. The zero locus of the section $\omega$ is a finite
set $\textsf{Sing}(\mathcal{F})$ called the singular set of $\mathcal{F}$. A
point $q\not\in\textsf{Sing}(\mathcal{F})$ is said to be regular. We shall say
that a holomorphic foliation $\mathcal{F}$ on $X$ is tangent to $M$ if the
leaves of the Levi foliation $\mathcal{L}$ on $M$ are also leaves of
$\mathcal{F}$. A singular point $p\in M$ is called dicritical if infinitely
many leaves of the Levi-foliation on $M$ have $p$ in their closure. Recently
dicritical singularities of real-analytic Levi-flat hypersurfaces have been
characterized in term of the Segre varieties, see for instance [22]. Using
this characterization, the notion of dicritical singularity in the theory of
holomorphic foliations coincide with the notion of Segre-degenerate
singularity of a real-analytic Levi-flat hypersurface and therefore, we can
use results of residue-type indices associated to singular points of
holomorphic foliations [5], [24] to prove the following result.

######  Theorem 1.

Let $\mathcal{F}$ be a holomorphic foliation on a compact surface $X$ tangent
to an irreducible real-analytic Levi-flat hypersurface $M\subset X$ such that
$\textsf{Sing}(\mathcal{F})\subset M$. Suppose $c^{2}_{1}(N_{\mathcal{F}})>0$.
Then there exists a point $p\in\textsf{Sing}(\mathcal{F})$ such that
$\mathcal{F}$ has a non-constant meromorphic first integral in a neighborhood
of $p$. In particular, $p$ is a dicritical singularity of $M$.

We emphasize that the existence of dicritical singularities of $M$ depends of
the condition $c^{2}_{1}(N_{\mathcal{F}})>0$, because otherwise the result is
false. In section 6 we give some examples that show the importance of the
condition $c^{2}_{1}(N_{\mathcal{F}})>0$. The condition
$\textsf{Sing}(\mathcal{F})\subset M$ is used in the proof of Theorem 1, we
unknown if this condition can be removed.

In the sequel we apply Theorem 1 to non-dicritical projective foliations.

######  Corollary 1.

A non-dicritical holomorphic foliation $\mathcal{F}$ on $\mathbb{P}^{2}$ does
not leaves invariant a real-analytic Levi-flat hypersurface passing by its
singularities.

Now we apply Theorem 1 for holomorphic foliations of degree 2 with a unique
singularity on $\mathbb{P}^{2}$.

######  Corollary 2.

Let $\mathcal{F}$ be a holomorphic foliation of degree 2 on $\mathbb{P}^{2}$
with a unique singular point $p$. Suppose that $\mathcal{F}$ is tangent to a
real-analytic Levi-flat hypersurface $M\subset\mathbb{P}^{2}$ and $p\in M$.
Then, up to automorphism, $\mathcal{F}$ is given in affine coordinates by the
1-form

$\omega=x^{2}dx+y^{2}(xdy-ydx).$

Moreover, $R(x,y,z)=\frac{y^{3}-3x^{2}z}{3x^{3}}$ is a rational first integral
for $\mathcal{F}$.

We say that a singularity $p$ of a germ of a real-analytic Levi-flat
hypersurface $M$ is said to be semialgebraic, if the germ of $M$ at $p$ is
bihomorphic to a semialgebraic Levi-flat hypersurface. We recall that a Levi-
flat hypersurface is said to be semialgebraic, if it is contained in an
algebraic Levi-flat hypersurface. To continue we apply Theorem 1 to find
semialgebraic singularities of real-analytic Levi-flat hypersurfaces which are
tangent to singular foliations in complex surfaces. Similarly results of
algebraization of singularities of holomorphic foliations can be found in [15]
and [7].

######  Corollary 3.

Let $\mathcal{F}$ be a holomorphic foliation on a compact surface $X$ tangent
to an irreducible real-analytic Levi-flat hypersurface $M\subset X$ such that
$\textsf{Sing}(\mathcal{F})\subset M$. Suppose that
$c^{2}_{1}(N_{\mathcal{F}})>0$ and $\mathcal{F}$ has only a unique simple
dicritical singularity $p\in M$. Then there exists an algebraic surface $V$, a
rational function $H$ on $V$ and a point $q\in V$ such that the germ of $M$ at
$p$ is biholomorphic to a semialgebraic Levi-flat hypersurface
$M^{\prime}\subset V$ in a neighborhood of $q$.

Finally we state a result that gives conditions for to guarantee the existence
of dicritical singularities of $M$ in presence of invariant compact curves by
$\mathcal{F}$.

######  Theorem 2.

Let $\mathcal{F}$ be a holomorphic foliation on a compac surface $X$ tangent
to an irreducible real-analytic Levi-flat hypersurface $M\subset X$. Suppose
that $C\cdot C>0$, where $C\subset M$ is an irreducible compact curve
invariant by $\mathcal{F}$, then there exists a point
$p\in\textsf{Sing}(\mathcal{F})\cap C$ such that $\mathcal{F}$ has a
meromorphic first integral in a neighborhood of $p$.

To prove Theorem 2 we use the Camacho-Sad index (cf. [6]) and a result of
Cerveau-Lins Neto (see Theorem 3.1).

We organize the paper as follows: in section 2 we review some definitions and
results about indices of holomorphic foliations at singular points. In section
3 we give the proof of Theorem 1. Section 4 is devoted to show some
applications of Theorem 1. In section 5 we provide the proof of Theorem 2.
Finally, in section 6 we show some examples which show the importance of the
hypotheses in theorems 1 and 2.

##  2\. Indices of holomorphic foliations

In this section we state two important results of indices of holomorphic
foliations: Camacho-Sad index [6] and the Baum-Bott index [1]. The first one
concerning the computation of $C\cdot C$, where $C\subset X$ is an invariant
compact curve by $\mathcal{F}$ and the third one concerning the computation of
$c^{2}_{1}(N_{\mathcal{F}})$. More references for these index theorems can be
found in [24, Chapter V], see also [3] and [5].

First, we recall basic definitions about holomorphic foliations on complex
surfaces. Let $\mathcal{F}$ be a germ of holomorphic foliation at
$0\in\mathbb{C}^{2}$. Recall that a germ of foliation $\mathcal{F}$ of
$0\in\mathbb{C}^{2}$ is dicritical if it has infinitely many analytic
separatrices through the origin. Otherwise it is called non-dicritical.

######  Definition 2.1.

We say that a foliation $\mathcal{F}$ on a compact surface $X$ is non-
dicritical, if the singular set $\textsf{Sing}(\mathcal{F})$ of $\mathcal{F}$
consists only of non-dicritical singularities in $X$.

######  Definition 2.2.

A meromorphic (holomorphic) function $h$ is called a meromorphic (holomorphic)
first integral for $\mathcal{F}$ if its indeterminacy (zeros) set is contained
in $\textsf{Sing}(\mathcal{F})$ and its level hypersurfaces contain the leaves
of $\mathcal{F}$.

###  2.1. Camacho-Sad index

Let us consider a separatrix $C$ at $p\in X$. Let $f$ be a holomorphic
function on a neighborhood of $p$ and defining $C=\\{f=0\\}$. We may assume
that $f$ is reduced, i.e. $df\neq 0$ outside $p$. Then [20], [24] there are
functions $g$, $k$ and a 1-form $\eta$ on a neighborhood of $p$ such that

$g\omega=kdf+f\eta$

and moreover $k$ and $f$ are prime, i.e. $k\neq 0$ on
$C^{*}=C\setminus\\{p\\}$. The Camacho-Sad index [6] is defined as

$\text{CS}(\mathcal{F},C,p)=-\frac{1}{2\pi
i}\int_{\partial{C}}\frac{1}{k}\eta.$

If $C\subset X$ is a compact holomorphic curve invariant by $\mathcal{F}$, one
has the formula due by Camacho-Sad.

######  Theorem 2.3.

[6, Camacho-Sad]

$\sum_{p\in\textsf{Sing}(\mathcal{F})\cap C}\text{CS}(\mathcal{F},C,p)=C\cdot
C.$

###  2.2. Baum-Bott index

Let $\mathcal{F}$ be a holomorphic foliation with isolated singularities on a
complex surface $X$. Let $p\in X$ be a singular point of $\mathcal{F}$; near
$p$ the foliation is given by a holomorphic vector field

$v=F(x,y)\frac{\partial}{\partial{x}}+G(x,y)\frac{\partial}{\partial{y}}$

or by a holomorphic 1-form

$\omega=F(x,y)dy-G(x,y)dx.$

Let $J(x,y)$ be the Jacobian matrix of $(F,G)$ at $(x,y)$, then following [1]
one can defined the Baum-Bott index at $p\in\textsf{Sing}(\mathcal{F})$ as

$\text{BB}(\mathcal{F},p)=\text{Res}_{p}\Big{\\{}\frac{(\text{Tr}J)^{2}}{F\cdot
G}dx\wedge dy\Big{\\}}.$

The Baum-Bott index depended only on the conjugacy class of the germ of
$\mathcal{F}$ at $p$. For example when the singularity $p$ is non-degenerated
then

$\text{BB}(\mathcal{F},p)=\frac{(\text{Tr}J(p))^{2}}{\det
J(p)}=\frac{\lambda_{1}}{\lambda_{2}}+\frac{\lambda_{2}}{\lambda_{1}}+2,$

where $\lambda_{1}$ and $\lambda_{2}$ are the eigenvalues of the linear part
$Dv(p)$ of $v$ at $p$. The set of separatrices $S$ through $p$ is formed by
two transversal branches, both of them analytic. We note also that

$\text{CS}(\mathcal{F},S,p)=\frac{\lambda_{1}}{\lambda_{2}}+\frac{\lambda_{2}}{\lambda_{1}}+2,$

and thus $\text{BB}(\mathcal{F},p)=\text{CS}(\mathcal{F},S,p)$. Of course this
remains valid for non-dicritical generalized curve foliations.

######  Theorem 2.4.

[5, Brunella] Let $\mathcal{F}$ be a non-dicritical germ of holomorphic
foliation at $(\mathbb{C}^{2},0)$ and let $S$ denote the union of all its
separatrices. If $\mathcal{F}$ is a generalized curve foliation, then

$\displaystyle\text{BB}(\mathcal{F},0)$ $\displaystyle=$
$\displaystyle\text{CS}(\mathcal{F},S,0).$

We recall that the foliation, induced by $v$, is said to be generalized curve
at $0\in\mathbb{C}^{2}$ if there are no saddle-nodes in its reduction of
singularities. It is easily seen that Theorem 2.4 is false for saddle-nodes
singularities.

The following formula will used in the proof of Theorem 1.

######  Theorem 2.5.

[1, Baum-Bott]

$\sum_{p\in\textsf{Sing}(\mathcal{F})\cap
X}\text{BB}(\mathcal{F},p)=c^{2}_{1}(N_{\mathcal{F}}).$

##  3\. Existence of dicritical singularities of Levi-flat hypersurfaces

In order to prove the Theorem 1 we need the following results.

######  Theorem 3.1 (Cerveau-Lins Neto [9]).

Let $\mathcal{F}$ be a germ at $0\in\mathbb{C}^{n}$, $n\geq{2}$, of
holomorphic codimension one foliation tangent to a germ of an irreducible real
analytic hypersurface $M$. Then $\mathcal{F}$ has a non-constant meromorphic
first integral. In the case of dimension two we can precise more:

  1. (1)

If $\mathcal{F}$ is dicritical then it has a non-constant meromorphic first
integral.

  2. (2)

If $\mathcal{F}$ is non-dicritical then it has a non-constant holomorphic
first integral.

Now we prove the following lemma.

######  Lemma 3.2.

Let $\mathcal{F}$ be a germ of a non-dicritical holomorphic foliation at
$0\in\mathbb{C}^{2},$ tangent to a germ of real-analytic Levi-flat
hypersurface $M$ at $0\in\mathbb{C}^{2}$. Then the Baum-Bott index satisfies
$\text{BB}(\mathcal{F},0)\leq 0.$

###### Proof.

Since $\mathcal{F}$ is non-dicritical at $0\in\mathbb{C}^{2}$, we have
$\mathcal{F}$ has a non-constant holomorphic first integral
$g\in\mathcal{O}_{2}$ by Theorem 3.1. In particular, $\mathcal{F}$ is a curve
generalized foliation and Theorem 2.4 implies that

$\text{BB}(\mathcal{F},0)=\text{CS}(\mathcal{F},S,0),$

where $S$ is the union of all separatrices of $\mathcal{F}$ at
$0\in\mathbb{C}^{2}$. To prove the lemma we need calculated
$CS(\mathcal{F},S,0)$. In fact, as $dg\wedge\omega=0$, then $dg=h\omega$,
where $h\in\mathcal{O}_{2}$. Moreover, if $g=g^{\ell_{1}}_{1}\cdots
g^{\ell_{k}}_{k}$, we get $S=\displaystyle\bigcup^{k}_{j=1}C_{j}$ with
$C_{j}=\\{g_{j}=0\\}$ and

$\sum^{k}_{j=1}\ell_{j}g_{1}\cdots\widehat{g_{j}}\cdots
g_{k}dg_{j}=h_{1}\omega,$

where $h_{1}=\frac{h}{g^{\ell_{1}-1}_{1}\cdots g^{\ell_{k}-1}_{k}}$. Hence

$\text{BB}(\mathcal{F},0)=\text{CS}(\mathcal{F},S,0)=-\sum_{1\leq i<j\leq
k}\frac{(\ell_{i}-\ell_{j})^{2}}{\ell_{i}\ell_{j}}[g_{i},g_{j}]_{0}\leq 0,$

here $[g_{i},g_{j}]_{0}$ denotes the number of intersection between $C_{i}$
and $C_{j}$ at $0\in\mathbb{C}^{2}$. ∎

To continue we prove Theorem 1.

###  3.1. Proof of Theorem 1

We use the Theorem 2.5 and Theorem 3.1 to prove that $\mathcal{F}$ has a
dicritical singularity in $X$. In fact, suppose by contradiction that
$\textsf{Sing}(\mathcal{F})$ consists only of non-dicritical singularities.
Take any point $p\in\textsf{Sing}(\mathcal{F})$ and let $U$ be a small
neighborhood of $p$ in $X$ such that $\mathcal{F}$ is represented by a
holomorphic 1-form $\omega$ on $U$ and $p$ is an isolated singularity of
$\omega$. Since $\mathcal{F}$ and $M$ are tangent in $U$ and $p\in M$, we have
$\mathcal{F}|_{U}$ admits a holomorphic first integral $g\in\mathcal{O}(U)$,
that is, $\omega\wedge dg=0$ on $U$, by Theorem 3.1.

Applying Lemma 3.2, we get $\text{BB}(\mathcal{F},p)\leq 0$, for any
$p\in\textsf{Sing}(\mathcal{F})$. But Baum-Bott’s formula implies that

$\sum_{p\in\textsf{Sing}(\mathcal{F})}\text{BB}(\mathcal{F},p)=c^{2}_{1}(N_{\mathcal{F}})>0,$

which is absurd. Therefore, there exists a dicritical singularity $p$ of
$\mathcal{F}$. Applying again Theorem 3.1, we obtain a non-constant
meromorphic first integral for $\mathcal{F}$ in a neighborhood of $p$.

##  4\. Applications of Theorem 1

First we apply Theorem 1 to non-dicritical holomorphic foliations of
$\mathbb{P}^{2}$.

######  Corollary 4.1.

A non-dicritical holomorphic foliation $\mathcal{F}$ on $\mathbb{P}^{2}$ does
not leaves invariant a real-analytic Levi-flat hypersurface passing by its
singularities.

###### Proof.

Suppose by contradiction that there exists a non-dicritical foliation
$\mathcal{F}$ of $\mathbb{P}^{2}$ which admits a real-analytic Levi-flat
hypersurface $M\subset\mathbb{P}^{2}$ invariant by $\mathcal{F}$ such that
$\textsf{Sing}(\mathcal{F})\subset M$. Since
$N_{\mathcal{F}}=\mathcal{O}_{\mathbb{P}^{2}}(d+2)$, where $d$ is a positive
integer, one has $c^{2}_{1}(N_{\mathcal{F}})=(d+2)^{2}>0$. Therefore
$\mathcal{F}$ has a dicritical singularity by Theorem 1. But it is an absurd
with the assumption. ∎

The Jouanolou foliation $\mathcal{J}_{d}$ of degree $d$ on $\mathbb{P}^{2}$,
is given in affine coordinates by

$\omega_{d}=(y^{d}-x^{d+1})dy-(1-x^{d}y)dx.$

It is well known that $\mathcal{J}_{d}$ belongs to a holomorphic foliations
class of $\mathbb{P}^{2}$ without algebraic solutions, this means,
$\mathcal{J}_{d}$ does not admits invariant algebraic curves, see for instance
[20].

On the other hand, we know that $\mathcal{J}_{d}$ is a non-dicritical
foliation of $\mathbb{P}^{2}$, because $\mathcal{J}_{d}$ has $d^{2}+d+1$
singularities and for each singularity passing only two analytic separatrices.
Hence the above corollary shows that $\mathcal{J}_{d}$ does not leaves
invariant real-analytic Levi-flat hypersurfaces of $\mathbb{P}^{2}$ passing by
its singularities.

Now we apply Theorem 1 to holomorphic foliations of degree 2 on
$\mathbb{P}^{2}$ with only one singularity.

######  Corollary 4.2.

Let $\mathcal{F}$ be a holomorphic foliation of degree 2 on $\mathbb{P}^{2}$
with a unique singular point $p$. Suppose that $\mathcal{F}$ is tangent to a
real-analytic Levi-flat hypersurface $M\subset\mathbb{P}^{2}$ and $p\in M$.
Then, up to automorphism, $\mathcal{F}$ is given in affine coordinates by the
1-form

$\omega=x^{2}dx+y^{2}(xdy-ydx).$

Moreover, $R(x,y,z)=\frac{y^{3}-3x^{2}z}{3x^{3}}$ is a rational first integral
for $\mathcal{F}$.

###### Proof.

Let us assume that $\mathcal{F}$ has a unique singularity, say $p$. Applying
Theorem 1 we have $\mathcal{F}$ has a non-constant meromorphic first integral
$f/g$ in a neighborhood of $p$. Now since $\mathcal{F}$ has degree 2, one can
apply the main theorem of [8] which asserts that, up automorphism,
$\mathcal{F}$ is given in affine coordinated by one of the following types:

  1. (1)

$\omega_{1}=x^{2}dx+y^{2}(xdy-ydx)$;

  2. (2)

$\omega_{2}=x^{2}dx+(x+y^{2})(xdy-ydx)$;

  3. (3)

$\omega_{3}=xydx+(x^{2}+y^{2})(xdy-ydx)$;

  4. (4)

$\omega_{4}=(x+y^{2}-x^{2}y)dy+x(x+y^{2})dx$.

The foliations given in (2) and (3) has a ‘transcendental” first integral
respectively

$\left(2+\frac{1}{x}+2\left(\frac{y}{x}\right)+\left(\frac{y}{x}\right)^{2}\right)\exp\left(-\frac{y}{x}\right)\,\,\,\text{and}\,\,\,\,\left(\frac{y}{x}\right)\exp\left(\frac{1}{2}\left(\frac{y}{x}\right)^{2}-\frac{1}{x}\right),$

and the foliation in (4) has no a meromorphic first integral, see [8]. Then
$\mathcal{F}$ only can be the foliation induced by

$\omega_{1}=x^{2}dx+y^{2}(xdy-ydx).$

It is easy to check that $R(x,y,z)=\frac{y^{3}-3x^{2}z}{3x^{3}}$ is a rational
first integral for $\mathcal{F}$. ∎

An algebraizable singularity is a germ of a singular holomorphic foliation
which can be defined in some appropriated local chart by a differential
equation with algebraic coefficients. It is proved in [15] the existence of
countable many classes of saddle-node singularities which are not
algebraizable. Recently Casale [7] considered simple dicritical singularities,
these singularities corresponds to singularities of holomorphic foliations
that after one blow-up appear a unique leaf tangent to the exceptional divisor
with tangent order one. He show that a simple dicritical singularity with
meromorphic first integral is algebraizable (cf. [7, Theorem 1]).

######  Theorem 4.3 (Casale).

If $\mathcal{F}$ is a simple dicritical foliation of $(\mathbb{C}^{2},0)$ with
a meromorphic first integral then there exist an algebraic surface $S$, a
rational function $H$ on $S$ and a point $p\in S$ such that $\mathcal{F}$ is
biholomorphic to the foliation given level curves of $H$ in a neighborhood of
$p$.

Similarly, our aim is give a result of algebraization of singularities for
real-analytic Levi-flat hypersurfaces in complex surfaces.

######  Corollary 4.4.

Let $\mathcal{F}$ be a holomorphic foliation on a compact surface $X$ tangent
to an irreducible real-analytic Levi-flat hypersurface $M\subset X$ such that
$\textsf{Sing}(\mathcal{F})\subset M$. Suppose that
$c^{2}_{1}(N_{\mathcal{F}})>0$ and $\mathcal{F}$ has only a unique simple
dicritical singularity $p\in X$. Then there exists an algebraic surface $V$, a
rational function $H$ on $V$ and a point $q\in V$ such that the germ of $M$ at
$p$ is biholomorphic to a semi-algebraic Levi-flat hypersurface
$M^{\prime}\subset V$ in a neighborhood of $q$.

###### Proof.

Since $p\in X$ is the unique singularity of $\mathcal{F}$, we have
$\mathcal{F}$ has a non-constant meromorphic first integral in a neighborhood
of $p$ by Theorem 1. Then it is follows from Theorem 4.3 that there exist an
algebraic surface $V$, a rational function $H$ on $V$ and a point $q\in V$
such that $\mathcal{F}$ is biholomorphic to the foliation given level curves
of $H$ in a neighborhood of $q$. According to [17, Lemma 5.2], there exists
real-algebraic Levi-flat hypersurface $N\subset V$ such that $M$ is
biholomorphic conjugated to a subset $M^{\prime}\subset N$. Thus $M^{\prime}$
is semi-algebraic. ∎

##  5\. Dicritical singularities of Levi-flat hypersurfaces in presence of
compact leaves

We use Camacho-Sad’s formula (Theorem 2.3) and Theorem 3.1 to prove the
following result.

######  Theorem 5.1.

Let $\mathcal{F}$ be a holomorphic foliation on a compac surface $X$ tangent
to an irreducible real-analytic Levi-flat hypersurface $M\subset X$. Suppose
that $C\cdot C>0$, where $C\subset M$ is an irreducible compact curve
invariant by $\mathcal{F}$, then there exists a point
$p\in\textsf{Sing}(\mathcal{F})\cap C$ such that $\mathcal{F}$ has a
meromorphic first integral in a neighborhood of $p$.

###### Proof.

Suppose by contradiction that all the singularities of $\mathcal{F}$ over $C$
are non-dicritical. Take any point $q\in\textsf{Sing}(\mathcal{F})\cap C$ and
let $U$ be a neighborhood of $q$ in $X$ such that $C\cap U=\\{f=0\\}$,
$\mathcal{F}$ is represented by a holomorphic 1-form $\omega$ on $U$ and $q$
is an isolated singularity of $\omega$. Since $\mathcal{F}$ and $M$ are
tangent in $U$, we have $\mathcal{F}|_{U}$ admits a holomorphic first integral
$g\in\mathcal{O}(U)$, that is, $\omega\wedge dg=0$ on $U$, by Theorem 3.1.
Then $dg=h\omega$, where $h\in\mathcal{O}(U)$. If $g=g^{\ell_{1}}_{1}\cdots
g^{\ell_{k}}_{k}$, we get $f=g_{i}$ for some $i$ and

$\sum^{k}_{j=1}\ell_{j}g_{1}\cdots\widehat{g_{j}}\cdots
g_{k}dg_{j}=h_{1}\omega,$

where $h_{1}=\frac{h}{g^{\ell_{1}-1}_{1}\cdots g^{\ell_{k}-1}_{k}}$. It
follows that,

$CS(\mathcal{F},C,q)=-\sum_{i\neq
j}\frac{\ell_{j}}{\ell_{i}}[g_{i},g_{j}]_{q}$

where $[g_{i},g_{j}]_{q}$ denotes the intersection number between the curves
$\\{g_{i}=0\\}$ and $\\{g_{j}=0\\}$ at $q$. In particular,

$CS(\mathcal{F},C,q)\leq 0\,\,\,\,\,\text{for
any}\,\,q\in\textsf{Sing}(\mathcal{F})\cap C.$

It follows from Theorem 2.3 that

$\sum_{q\in\textsf{Sing}{\mathcal{F}}\cap C}CS(\mathcal{F},C,q)=C\cdot C.$

But this is a contradiction with $C\cdot C>0.$ Now since $p\in M$, we can
apply again Theorem 3.1 to find a meromorphic first integral for $\mathcal{F}$
in a neighborhood of $p$. ∎

##  6\. Examples

First we give an example where the hypotheses of Theorem 1 are satisfied.

######  Example 6.1.

The canonical local example of a real-analytic Levi-flat hypersurface in
$\mathbb{C}^{2}$ is given by $\text{Im}z_{1}=0$. This hypersurface can be
extended to all $\mathbb{P}^{2}$ given by

$M:=\\{[Z_{1}:Z_{2}:Z_{3}]\in\mathbb{P}^{2}:Z_{1}\bar{Z}_{3}-\bar{Z}_{1}Z_{3}=0\\}.$

Moreover, this hypersurface is tangent to holomorphic foliation $\mathcal{F}$
given by the levels of rational function $Z_{1}/Z_{3}$. Note also that
$\mathcal{F}$ has degree $0$ and therefore
$c^{2}_{1}(N_{\mathcal{F}})=c^{2}_{1}(\mathcal{O}_{\mathbb{P}^{2}}(2))=4$. The
foliation $\mathcal{F}$ has a dicritical singularity at $[0:1:0]\in M$.

We now give two examples where Theorem 1 is false.

######  Example 6.2.

Consider the Hopf surface $X=(\mathbb{C}^{2}-\\{0\\})/\Gamma_{a,b}$ induced by
the infinite cyclic subgroup of $GL(2,\mathbb{C})$ generated by the
transformation $(x,y)\mapsto(ax,by)$ with $|a|,|b|>1$. Levenberg-Yamaguchi
[19] proved that the domain

$D=\\{(x,y)\cdot\Gamma_{a,b}|\,x\in\mathbb{C},\text{Im}\,\,y>0\\}$

in $X$ with $b\in\mathbb{R}$ is Stein. Furthermore it is bounded by a real-
analytic Levi-flat hypersurface

$M=\\{(x,y)\cdot\Gamma_{a,b}|\,x\in\mathbb{C},\text{Im}\,\,y=0\\}.$

It is clear that the levels of the holomorphic function $f(x,y)=y$ on
$\mathbb{C}^{2}-\\{0\\}$ defines a holomorphic foliation $\mathcal{F}$ on $X$
tangent to $M$. Since any line bundle on $X$ is flat [21] we have
$c^{2}_{1}(N_{\mathcal{F}})=0$ and $M$ has no singularities on $X$.

######  Example 6.3.

Let $X=\mathbb{P}^{1}\times\mathbb{P}^{1}$ and let $\mathcal{F}$ be the
foliation given by the vertical fibration on $X$. Now let
$\pi:X\to\mathbb{P}^{1}$ be the projection on the first coordinate and let
$M=\pi^{-1}(\gamma)$, where $\gamma$ is a real-analytic embedded loop in
$\mathbb{P}^{1}$. Take a fiber $F\subset M$. We have $F$ is isomorphic to
$\mathbb{P}^{1}$ with $F^{2}=0$. Clearly $M$ is a Levi-flat hypersurface on
$X$ tangent to $\mathcal{F}$ and it has no dicritical singularities on $F$.

Acknowledgments.– The authors wishes to express his thanks to Alcides Lins
Neto for several helpful comments during the preparation of the paper.

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