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Multiplex Financial Networks: Revealing
the Level of Interconnectedness
in the Banking System
Alejandro de la Concha1(B
), Serafin Martinez-Jaramillo2,
and Christian Carmona3
1Universit´ed’
´
Evry val-d’Essone, Universit´e Paris Saclay, ´
Evry, France
alejandro.delaconchaduarte@ens.univ-evry.fr
2Financial System Risk Analysis Directorate, Banco de M´exico, Mexico
3Department of Statistics, University of Oxford, Oxford, UK
Abstract. Complex networks models have been useful for the study of
systemic risk; however, most of the studies have ignored the true level
of interconnectedness in the financial system; in this work we show the
missing part on the study of interconnectedness. Furthermore, complex-
ity in modern financial systems has been also an important subject of
study. However, we still lack the appropriate metrics to describe such
complexity and interconnectedness; moreover, the data available in order
to describe and study them is still scarce. In addition, most of the focus
on the subject of interconnectedness has been on a single type of net-
work: interbank (exposures) networks. We use a comprehensive set of
market interactions that include transactions in the securities markets,
payment system flows, interbank loans, cross holding of securities, foreign
exchange exposures and derivatives exposures. This the first attempt, to
the best of our knowledge, to describe so comprehensively the complexity
and interconnectedness in a banking system. We are able to identify the
most important institutions in the whole structure in term of their con-
nectedness, the most relevant layer (in structural terms) of the multiplex
and the community structure of the Mexican banking system which can
be seen as a generalization of the well-known Core-Periphery model.
1 Introduction
There has been a lot of recent research on financial networks for the purposes
of studying systemic risk, performing stress testing or determining the relevance
of financial institutions. A commonly shared view is that the financial system
is highly interconnected. Moreover, the term interconnectedness its now com-
monly mentioned by the financial authorities and in research papers. However,
most of the previous works (with some exceptions) focus in very specific types of
The views expressed here are those of the authors and do not reflect the views of
the Mexican central bank.
c
Springer International Publishing AG 2018
C. Cherifi et al. (eds.), Complex Networks & Their Applications VI,
Studies in Computational Intelligence 689, https://doi.org/10.1007/978-3-319-72150-7_92
1136 A. de la Concha et al.
networks: interbank direct exposures networks, CDS networks or payment sys-
tems networks as the only source to measure interconnectedness in the banking
system.
Financial institutions interact in different markets, which can be thought
of as different networks within a meta-structure which can be interpreted as
a multilayer network or a multiplex network [13]. A multiplex or a multilayer
system is one in which the nodes have different types of interactions, each level of
interaction is modeled by a single layer of the multiplex. Depending on the set of
nodes and its heterogeneity, one could refer to multiplex or multilayer systems.
In this work, we adopt the convention that multiplex systems are composed by
the same set of homogeneous nodes and a multilayer system is such that the set
of nodes could be different on each layer and there is heterogeneity on the nodes
attributes.
Multiplex and multilayer systems have been comprehensively studied in the
past from the theoretical point of view and applied in many different contexts
[13]. Examples of applications of the multiplex paradigm range from transporta-
tion networks, infrastructure networks, social networks and more recently finan-
cial networks. In particular, we are interested in modeling only the banking
system, we refer to the Mexican banking system model as a multiplex system or
multiplex network indistinctly. This approach gives rise to a rich set of complex
interactions among these layers. Each layer possesses different topological prop-
erties and the roles played by the institutions might be different depending on
the strategy followed by each of them.
There is already an emerging literature on multiplex financial networks. For
example, in [5] the authors decompose the payment system flows multiplex into
three different layers and study the individual structural properties of each layer.
In [16] an agent-based multilayer network model is used to study interbank con-
tagion. The authors find that there is an important underestimation of contagion
risk if the fact that banks interact in various markets is neglected. In [18], the
authors estimate the systemic risk in the Mexican banking system by model-
ing it as a multiplex network of exposures; the contribution of each layer (type
of exposure) to the systemic risk is also computed. In [2,15] the authors study
multiple layers of exposures networks, their overlapping and the link persistence
in Italy and Mexico; whereas, [4] describes the US financial system as a three
layers system and study how the risk is transmitted from one layer the others
in ways which have not being explored before. [1] study an exposures multiplex
network and propose a novel approach to model solvency and liquidity contagion
by separating the layers by maturity and instrument type. Finally, [17] study
difference dependence structures in financial time series by modeling correlations
among financial stocks as multiplex networks.
Another line of research with which this work is related is to that of com-
munity detection in complex networks. This is an important line for us as we
are departing from the simplex approach, the community structure of multiplex
financial systems has not being studied in the past. Finally, one of the most
relevant results for financial networks is that of the existence of a core-periphery
Multiplex Financial Networks 1137
structure on interbank networks [8]. However, in a more general context, some
of the not previously studied financial networks do not exhibit such structure. It
is then necessary a more general paradigm and this is the reason why we resort
to stochastic block models for multiplex systems in order to identify community
structures in the Mexican multiplex banking system.
This paper has three main goals: first, we argue that we should overcome
such limited view on interconnectedness by using a multiplex approach; sec-
ond, we propose to use stochastic block models (SBM) in order to generalize
the notion of Core-Periphery in a multiplex context; and finally, we apply some
tools from topological data analysis (TDA) to quantify the level of complex-
ity of the structure. Some additional but also related objectives are: (i) Apply
new metrics and methods in order to characterize and understand the multiplex
network of the Mexican banking system; (ii) Identify important players in the
multiplex structure rather than only on single layers; and (iii) Study the level of
interconnectedness and the complexity of the banking system.
For this purpose we will use two well established approaches to study sys-
tems which are composed by more than one network, and additional complexity
measures, borrowed from Stochastic Block Models (SBM) and Topological Data
Analysis (TDA), which can contribute to reveal the level of interconnectedness
which exists in the banking system. We argue that the complexity and inter-
connectedness in the financial system are multi-faced and that for successfully
revealing their real face, one must consider that agents interact in financial sys-
tems in many different ways and markets.
The main contributions of this work are: (i) we propose a comprehensive
structural analysis of a multiplex banking system and we document some of the
structural aspects of the Mexican multiplex banking system; (ii) by resorting
to some well known approaches in multiplex systems we are able to identify
relevant players in the whole system, relevant layers (market interactions) and
the community structure of such system and the role of these communities among
the system. Additionally, we perform the analysis for several different dates in
order to explore the dynamical aspects of the system.
The rest of the paper is organized as follows: Sect. 2provides some impor-
tant definitions for multiplex systems, stochastic block models and topological
homology. Section 3discusses in detail the data used to build the multiplexes
for this study and the methodological aspects of the paper. Section 4presents
the results obtained by applying the multiplex approach in the Mexican banking
system and in particular the use of SBM and TDA. Finally, Sect. 5concludes
and shows possible lines of extension to this work.
2 Definitions
The multiplex network referred in most of this paper is the multiplex banking
system. For the mathematical description of our paper, we will borrow some
notation and metrics from [3].
The multiplex network Mconsists of Nnodes and Mlayers.
1138 A. de la Concha et al.
The whole structure can be described by the set of adjacency matrices
M≡A={A[1], ..., A[M]}
where A[α]={a[α]
ij }, with a[α]
ij =1ifiand jperformed a financial transaction
in market αand a[α]
ij = 1 otherwise.
If the links have weights, as it is the case in many financial networks, then
the system can be described by the set of weighted matrices
W={W[1], ..., W [M]}
Then we have to move from the degree in one layer k[α]
i=N
i=ja[α]
ij to the
multiplex degree ki={k[1]
i, ..., k[M]
i}
A node, i, is said to be active in a layer, α,ifk[α]
i>0. Let b[α]
idenote the
activity of a node in layer α, then b[α]
i= 1 if the node is active in layer αand 0
otherwise. The activity vector is defined as:
bi={b[1]
i, ..., b[M]
i}
The total activity Bi=M
α=1 b[α]
irepresents the number of layers in which
the node iis active.
Two empirical facts about multiplex networks is that not all nodes have
connections in all layers and the node activity is heterogeneously distributed.
One important concept is that of the overlapping degree, computed as:
oi=
M
α=1
k[α]
i
Another important concept is that of the multiplex participation coefficient:
Pi=M
M−1⎡
⎣1−
M
α=1 k[α]
i
oi2⎤
⎦
If Pi= 1 then all the links incidents in node iare equally distributed across
layers whereas Pi= 0 if node iis only active in one layer.
Piand oiare useful to classify the nodes in multiplex hubs (high Piand oi);
focused hubs (high oiand low Pi); multiplex leaves (low oiand high Pi)and
focused leaves (low oiand low Pi).
2.1 Stochastic Block Models in the Context of Multiplex Networks
Stochastic Block Models were first introduced by Nowiki and Snijders [2001] and
are an useful tool to uncover the latent structure in complex networks. The main
hypothesis is that the attributes of agents affect the way they interact with each
other. For example, it is expected that a small bank will have less activity with
Multiplex Financial Networks 1139
other institutions compared to a big bank. Agents with similar attributes are
classified into the same class and the relation between agents is conditioned to
the cluster they belong. The formal definition of SBM is:
As before, let Vbe the set of vertex and Ethe set of nodes. Consider Q=
1,2, ..q classes on nodes such that Qdefines a partition on E. We define the
membership matrix Zas Zi,q =1ifi∈q,fori∈Eand q∈Q. In a Stochastic
Block Model the distribution of Xi,j , the link between i and j, is conditioned to
the membership of iin q-th class and node jin the l-th class:
Xi,j |ZiqZjl =1∼Fi,j
q,l
It is possible to use external information in order to infer the parameters and
classes. This is done by adding covariates related to each vertex (i, j), but in this
paper we will just use the network topology as input.
SBM are mixture models which approximates the distribution of Xi,j.In
order to estimate the parameters involved in a SBM it is necessary to solve three
problems: the inference of the membership of nodes to clusters, the estimation
of the parameters and the selection of the number of clusters.
Our objective is to find the mixture model which best cluster each node,
one way to do it is to look for a model MQwith Qblocks, which maximizes the
complete data likelihood and penalizes models with a high number of parameters.
SBM models can describe uniplex as well as multiplex networks. We are
interested in describing the role each bank play in the Mexican Financial System
and in understanding how their activity in a given market affects their activity
in others. For this reason, a multiplex network will offer a better framework to
model this phenomena. In most of the research done in financial networks, the
interrelations among markets are ignored leading to a poor description of the
level of interconnectedness of a financial system. We believe that the approach
proposed in this work would help us to understand how banks interact.
Let Ethe set of banks in the Financial System which participate in pdifferent
markets and Xi,j represents the relations among bank iand bank jwith i, jE.
Xi,j takes values over {0,1}p.IfXr
i,j = 1 that means that iand jare connected
in the rmarket. The definition of being connected can be different according to
the layer. It is clear that Xi,j follow a multivariate Bernoulli distribution. We
suppose that the parameters of this distributions depends on the membership of
bank iand bank jto group qand group lrespectively. That is:
∀x∈{0,1}pP(Xi,j =x|Ziq Zil)=πi,j (x)
A Core-Periphery structure can be thought as a Stochastic Block Model
in a uniplex network where the number of blocks is 2, designed by core and
periphery, and the probability of two banks of being connected is one if both
banks are members of the core and 0 if both belong to the periphery. This
definition can intuitively been extended to the multiplex context by asking that
the joint probability of two banks of being connected is one or 0 depending
whether they belong to the core or to the periphery. Nevertheless, as we will see,
when we try to adjust this model to the observed networks the error committed
is to high.
1140 A. de la Concha et al.
2.2 Topological Data Analysis and Multiplex Networks
There exists some empirical studies on financial networks which reveal some of
the important structural characteristics of such systems: [5,14,15,19].
Multiplex networks is a well established field with many important devel-
opments and important applications. Nevertheless this paradigm has only been
used in the context of financial networks recently.
Topological Data Analysis, TDA, is a powerful framework for extracting
insight from high-dimensional, complex data sets. TDA represents a fundamen-
tal advance in machine learning, it is a new and growing field of the applied
mathematics.
These new techniques have provided important advances in the study of
data in an increasingly diverse set of applications, such as contagion spread on
networks [20], collective behavior in biology [6], viral evolution [12], among many
others.
The basic idea behind TDA is to study the shape of a dataset through the
use of different techniques developed in Algebraic Topology. Some of the most
important advances in this field have being done by Carlson [7] and Edelsbrunner
[9,10].
Topology is the sub-field of mathematics concerned with the study of shape.
Algebraic topology offers different methods for gauging the global properties of
a particular topological space by associating with it a collection of algebraic
objects. One of this methods is a set of invariants known as the Homology.
Homology is a mathematical formalism for talking in a quantitative and
unambiguous manner about how a space is connected.
Homology groups of dimension k,Hk(X), provide information about
properties of chains formed from simple oriented units known as simplices.
The elements of homology groups are cycles (chains with vanishing boundary).
Homology groups can be computed using the methods of linear algebra. It should
be remarked that these computations can be quite time-consuming in spite of
recent advances in computational techniques.
3 Data and Methods
3.1 Data
The data used for this work consists on different types financial transaction
between bank. The time series used for this study are of a formidable length
and frequency for any previous study on financial networks in the past. With
data from SPEI and regulatory reports available at the Mexican central banks
Datawarehouse we are able to construct daily matrices from January 2005 to
December 2015. This represents a 11 years time window of daily networks.
As we want to characterize the complexity in the financial system we wanted
to use different market activities. For example, In the case of the SPEI network,
there will be a link between a pair of banks, Cand D, if money has been sent
from bank Cto bank D. A link between Eand Fin the Total network represents
Multiplex Financial Networks 1141
that bank Eis exposed to bank F. For the CVT network two banks Gand H
have a relationship if banks Gbuys assets from bank H.
In this work we study 6 different layers or types of interactions among banks:
1. Transactions on the securities market (CVT layer)
2. Payment system flows (SPEI layer)
3. Exposures arising from interbank deposits and loans (D&L layer)
4. Exposures arising from cross holding of securities (Securities layer)
5. Exposures arising from derivatives transactions (Derivatives layer)
6. Exposures arising from foreign exchange transactions (FX layer)
It is a possibility to simple aggregate all the layers in order to study the
multiplex system of interest; nevertheless, insightful information regarding inter-
actions is lost when the multiplex structure is neglected. Given that the last 4
layers are built under the common concept of exposure at default, we will add
up such layers and we will call this new layer as the Total layer. The exposures
multiplex system has been studied in [18] from the systemic risk point of view,
not from the structural side. In all the following computations and results we
will use the Total layer instead of all the 4 exposures layers for computational
costs saving purposes.
3.2 Methods
There is now a good wealth of common metrics used to describe financial net-
works. In order to precise the type of metrics used to characterize complexity
in a financial system some basic definitions are needed. The topological metrics
used in this section are described in [11].
Our study analyses the complexity of different financial networks by con-
sidering its representation as a simplicial complex. A simplicial complex is a
topological space constructed by the union of points, line segments, triangles,
and their n-dimensional counterparts. A formal definition can be found in [10].
Let Vt=(vt,1, ..., vt,n ) denote the collection of vertices and Et=(et,1, ..., et,r)
the collection of edges for a given network Δtat time t. A simplicial complex
can be constructed from Vtand Et. There will be an edge between two vertices
if there is a financial relationship between this two market participants.
We can analize the complexity of the topological structure given by a network
by calculating the Betti numbers of the associated simplicial complex.
The Betti numbers are used to distinguish topological spaces based on the
connectivity of n-dimensional simplicial complexes. The nth Betti number rep-
resents the rank of the nth homology group, denoted Hn, which tells us the
maximum amount of cuts that must be made before separating a surface into
two pieces or 0-cycles, 1-cycles, etc. We can think of the kth Betti number as
the number of k-dimensional holes on a topological surface.
We will focus in the study of the 0 and 1 dimensional betti numbers, denoted
by b0and b1. A common interpretation of this numbers is to understand b0as
the number of connected components and b1as the number of one-dimensional
1142 A. de la Concha et al.
or “circular” holes. In the context of financial market infrastructure, an inter-
pretation could be to consider b0as the number of trading blocs, and b1as the
number of netting opportunities.
We computed the betti numbers for the networks described in previous sec-
tions, and the results provided us with a new perspective of the complexity of
each network. In the case of the payments network, the results shown in the
right plot of Fig. 1show us an increase in the magnitude of b1, this could be
interpreted as a constant increase in the complexity of this network.
2
4
6
(a)
210
240
270
2008 2010 2012 2014 2016 2008 2010 2012 2014 2016
(b)
Fig. 1. Betti numbers for the 0 (a) and 1 (b) homologies for the payments network.
4 Results
In this section we report some of the results of the characterization of the
Mexican Financial System as a multiplex network. First some general struc-
tural metrics are shown and interpreted. Second, applications of the SBM are
given and the results are interpreted.
4.1 General Structural Results
A first result from the application of the multiplex approach to the Mexican
banking system is to determine the distribution of the overlapping degree and
of the activity, metrics which were defined in Sect. 2. In Fig. 1it is possible to
observe two characteristics which has been observed on other multiplex systems:
an heterogeneous distribution of node activity and a power law distribution of
the overlapping degree distribution. From this figure one can deduct that most
of the nodes are active in many layers and that the nodes which present low
Multiplex Financial Networks 1143
Frequency
(a) Total Overlappin
g
De
g
ree
0 50 100 150 0 1 2 3 4 5 6 7
04812
0510
(b) Total Activit
y
Fig. 2. Overlapping degree and total activity
activity are the minority. Regarding the distribution of the overlapping degree,
a few nodes have many connections in the whole multiplex structure and the
majority has only a few connections.
In addition to investigate into the community structure of the Mexican inte-
bank multiplex system and to provide also information of the most relevant
nodes, it is desirable to establish some criteria to determine the relevance of
each of the layers that compose the multiplex system. This question has been
answered in [21] by using the correlations among simplex networks. In [21], the
authors first define the inter-simplex correlations for each node as:
Cisr =jas
ij ar
ij
jas
ij +jar
ij −jas
ij ar
ij
where as
ij is the interaction of banks iand jin layer s. This coefficient takes
valus in [0,1]. From there, the authors define the correlation among layers as:
Csr =1
n
i
Cisr
where nis the number of nodes in the multiplex system, Csr takes values in
[0,1].
Finally, by computing such correlations between all layers, the authors in [21]
define the importance of a layer as:
Is=rCrs
srCrs
In the above table we can see the results of the computation of layer impor-
tance. It is noticeable that in all the evaluated periods, the total exposures layer
is the one with the highest importance, followed by the payment systems net-
work, then the CVT network follows, with the repo network located at the end
of the ranking.
1144 A. de la Concha et al.
Network June 2007 October 2008 October 2010 June 2015
Repo 0.359 0.360 0.312 0.360
SPEI 0.648 0.572 0.555 0.530
Tota l 0.780 0.645 0.678 0.640
This result, provides important information on how the layers are correlated.
In the next section we will find results that considers the Total exposures network
as an important layer. However, this does not mean that the exposures segment of
interaction is more important for the banking system than the payment system.
The only implication that we can extract is that if two banks interact on the
total exposures network, such interaction is more informative than interactions
in other markets. Nevertheless, the congruence of both approaches pointing to
the same layer is encouraging about the application of this type of analysis.
4.2 Results on SBMs
Understanding interconnectedness of a system helps market regulators to iden-
tify different groups with respect to their relationships, a common approach to
address this problem is to think that markets have a Core-Periphery structure.
The idea behind this model is that there are markets in which some agents are
forced to limit the number of agents they interact with. When this happens a
group of agents intermediate between agents that cannot interact among them.
The group of agents that play as intermediates are part of the core, while those
that do no interact among them belong to the periphery. If the market has a
perfect Core-Periphery structure then each agent from the Core will be con-
nected with each other and connected agents from the Periphery. On the other
hand, agents belonging to the Periphery will be completely disconnected among
them and they will interact with agents from the Core. Nevertheless, the lack
of flexibility of this model can lead us to a poor fit and as far as we know, this
model has not been generalized to the multiplex case.
In order to relax the definition of the Core, we will redefine a Core as a
group of agents that are interconnected with a probability near to 1, whereas
a Periphery will be a group of agents that interact with each other with a low
probability near to 0. Additionally, this definition can be easily extend to the
multiplex case, where instead of analyzing the marginal probabilities of each
market we will check the joint probability distribution of the links in the multi-
plex market. In a fist stage, we analyze the relationships among banks in three
different layers: SPEI, CVT and Total, without taking into account the strength
of this connections.
A Bernoulli Multilayer SBM was enough to uncover the underlying structure
of the system. In order to have an idea of the time component, we analyze the
system in four different periods: June 2007, October 2008, June 2010 and June
2015. In all cases, banks were categorized into three groups (Fig.3). It is worth
noting that the topological measures Overlapping degree and the Multiplex
Multiplex Financial Networks 1145
Participation were enough to distinguish the groups. As it can be seen, banks
in group 1 can be clearly identified as focused leaves or multiplex leaves, those
in group 2 can be either focused hubs or multiplex hubs, depending of the time,
while group 3 contains in all cases multiplex hubs. Figure 3also includes infor-
mation about the number of assets of each bank represented by the size of the
point. Surprisingly, in some cases big banks are not categorized in group 3 as
expected.
Benoulli Multilayer SBM cluster banks according to their behavior not only
in one market but in the whole structure. Banks that belong to group 1 are
those less active in all markets, group 2 is better connected in the CVT layer
than the other groups while it is more diversified in the other layers and group
3 seems to be formed by banks more active in the Total and SPEI layer and less
interconnected in the CVT layer. Once the parameters of a multiplex SBM have
been estimated, it is possible to have a clearer picture of the activity within and
between groups.
0
50
100
150
Participation
OverlappingDegree
Cluster
1
2
3
Assets
1e+06
2e+06
3e+06
June 2007
0
50
100
150
Participation
OverlappingDegree
Assets
0e+00
1e+06
2e+06
3e+06
4e+06
Cluster
1
2
3
October 2008
0
50
100
150
Participation
OverlappingDegree
Assets
1e+06
2e+06
3e+06
4e+06
Cluster
1
2
3
June 2010
0
50
100
150
0.4 0.6 0.8 1.0 0.00 0.25 0.50 0.75 1.00
0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00
Participation
OverlappingDegree
Cluster
1
2
3
Assets
2e+06
4e+06
6e+06
October 2015
Fig. 3. Clusters found using Bernoulli Multiplex SBM
1146 A. de la Concha et al.
Another advantage of these model is that it provides a total description of
the jointly distribution of the links. This information enables us to compute
conditional probabilities, which help us to understand the relationship between
markets. By using the marginal and joint distributions, directed networks can
be built, Fig. 4shows the probability of banks to be connected according to the
group they belong to, the size of the links depends on the magnitude of this
probability with no link if it is less than 0.1, this figure sheds more light on the
connection profile of each cluster. The networks showed in Fig.4represent the
probability of two banks to be connected in two markets at the same time and
how it changes when there is a connection in a third market. Being connected in
a layer have a positive effect in the probability of being connected in other layers,
the magnitude of this effect depends, as it can be seen, on the membership of
the bank.
For example, we found that if a bank from cluster 2 and a bank from cluster
3 make transactions in the securities market, they are more likely to be exposed
to each other and to interact more in the SPEI layer, while connections between
banks from the same group increases marginally their connections in other lay-
ers. The SPEI layer has a similar behavior that the CVT layer, it strengthens
the connections in the other two layers, specially between banks from group 2
and 3, but in a more subtle way. Things become completely different when we
analyse the CVT layer. Having more transactions on securities to a bank seems
to increase the activity in other markets, in general, no matter the membership
of the banks. It is really interesting that banks in group 1, which is the least
(a) June 2007 (b) October 2008
(c) June 2010 (d) October 2015
Fig. 4. Joint Distributions of SPEI and CVT networks
Multiplex Financial Networks 1147
active group, increases dramatically their chances to interact among its mem-
bers and other groups. During June 2007, October 2008 and June 2015, the links
that become stronger once there is a connection in the CVT, are those between
cluster 1 and cluster 3 in both directions, while in June 2010 are those between
1 and 2. In all cases except in June 2007, keeping transactions of banks from
cluster 1, help banks in cluster 1 to interact with each other.
5 Conclusion
This is the first attempt to characterize complexity and interconnectedness in
a comprehensive way for a financial system. The Topological and Multiplex
approaches provided the right tools to perform such analysis. However, there
is still a lot to do in order to really characterize financial systems and more
importantly to translate such characterization into concrete policy guidance. The
main takeaways from this approach for studying multiplex banking networks are:
(i) We studied the multiplex of the Mexican financial system; (ii) It is possible
to include more layers related to different market activities than in previous
exercises; and (iii) The multiplex approach deliver important information not
available on a individual layer view.
Additionally, we provide some initial but useful metrics to start studying such
structures and propose a concrete procedure to detect the community structure
of the multiplex banking system. We also determine the most relevant layer of
the multiplex in terms of node correlation and detect systemically important
banks in such structure. Nevertheless, there are still many things to be done as
future work like extending the multiplex analysis to more financial intermedi-
aries, explore the dynamical aspects of the Mexican multiplex banking system
and apply all these results for weighted multiplex networks.
References
1. Aldasoro, I., Alves, I.: Multiplex interbank networks and systemic importance: an
application to European data. J. Financial Stab. (2017). (In press)
2. Bargigli, L., di Iasio, G., Infante, L., Lillo, F., Pierobon, F.: The multiplex structure
of interbank networks. Quantitative Finance 15(4), 673–691 (2015)
3. Battiston, F., Nicosia, V., Latora, V.: Structural measures for multiplex networks.
Phys. Rev. E 89, 032804 (2014)
4. Bookstaber, R., Kennet, D.: Looking deeper, seeing more: a multilayer map of the
financial system. Brief series, Office of Financial Research (2016)
5. Bravo-Benitez, B., Alexandrova-Kabadjova, B., Martinez-Jaramillo, S.: Centrality
measurement of the Mexican large value payments system from the perspective of
multiplex networks. Comput. Econ. 1–29 (2016)
6. Topaz, C.M., Ziegelmeier, L., Halverson, T.: Topological data analysis of biological
aggregation models. ArXiv e-prints (2014)
7. Carlsson, G.: Topology and data. Bull. Am. Math. Soc. (2009)
8. Craig, B., von Peter, G.: Interbank tiering and money center banks. J. Finan.
Intermediation 23(3), 322–347 (2014)
1148 A. de la Concha et al.
9. Edelsbrunner, H., D., M.: Persistent homology: theory and practice. Proc. Eur.
Congress Math. (2012)
10. Edelsbrunner, H., Harer, J.: American Mathematical Society (2010)
11. Flood, M.D., Simon, J., Timm, M.: Measures of financial network complexity a
topological approach. In: Network Models, Stress Testing and Other Tools for
Financial Stability Monitoring and Macroprudential Policy Design and Implemen-
tation (2015)
12. Joseph Minhow Chan, G.C., Rabadan, R.: Topology of viral evolution. Proc. Nat.
Acad. Sci. (2013)
13. Kivela, M., Arenas, A., Barthelemy, M., Gleeson, J.P., Moreno, Y., Porter, M.A.:
Multilayer networks. J. Complex Netw. 2(3), 203 (2014)
14. Martinez-Jaramillo, S., Alexandrova-Kabadjova, B., Bravo-Benitez, B., Solorzano-
Margain, J.P.: An empirical study of the Mexican banking system’s network and
its implications for systemic risk. J. Econ. Dynamics Control 40, 242–265 (2014)
15. Molina-Borboa, J.L., Martınez-Jaramillo, S., L´opez-Gallo, F., van der Leij, M.:
A multiplex network analysis of the Mexican banking system: link persistence,
overlap and waiting times. J. Netw. Theor. Finance 1(1), 99–138 (2015)
16. Montagna, M., Kok, C.: Multi-layered interbank model for assessing systemic risk.
Technical report, University of Kiel (2013)
17. Musmeci, N., Nicosia, V., Aste, T., Matteo, T.D., Latora, V.: The multiplex depen-
dency structure of financial markets. Mimeo (2016)
18. Poledna, S., Molina-Borboa, J.L., Mart´ınez-Jaramillo, S., van der Leij, M.,
Thurner, S.: The multi-layer network nature of systemic risk and its implications
for the costs of financial crises. J. Financial Stab. (2015)
19. Solorzano-Margain, J.P., Martinez-Jaramillo, S., Lopez-Gallo, F.: Financial con-
tagion: extending the exposures network of the Mexican financial system. CMS
10(2–3), 125–155 (2013)
20. Taylor, D., Klimm, F., Harrington, H.A., Kramar, M., Mischaikow, K., Porter,
M.A., Mucha, P.J.: Topological data analysis of contagion maps for examining
spreading processes on networks. Nature Communications (2015)
21. Zhu, G., Li, K.: A Unified Model for Community Detection of Multiplex Networks,
pp. 31–46. Springer International Publishing, Cham (2014)
