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Learning of Weighted Multi-layer
Networks via Dynamic Social Spaces,
with Application to Financial
Interbank Transactions
Chris U. Carmona1,3(B
)and Serafin Martinez-Jaramillo2,3
1Department of Statistics, University of Oxford, Oxford, UK
carmona@stats.ox.ac.uk
2Center for Latin American Monetary Studies, Mexico City, Mexico
3Banco de Mexico, Mexico City, Mexico
Abstract. We propose a general network model suited for longitudinal
data of multi-layer networks with directed and weighted edges. Our for-
mulation built upon the latent social space representation of networks.
It consists of a hierarchical formulation: deep levels of the model rep-
resent latent coordinates of agents in the social space, evolving in con-
tinuous time via Gaussian Processes; meanwhile, top levels jointly man-
age incidence and strength of interactions by considering a Zero-Inflated
Gaussian response. Learning of the model is performed through Bayesian
Inference. We develop an efficient MCMC algorithm targeting the pos-
terior distribution of model parameters and missing data (available in
GitHub). The motivation for our model lies in the context of Financial
Networks, specifically the analysis of transactions between commercial
banks. We evaluate the model in synthetic data, as well as our main
case study: the network of inter-bank transactions in the Mexican finan-
cial system. Accurate predictions are obtained in both cases estimating
out-of-sample link incidence and link strength.
1 Introduction
In a number of natural, social and economic systems the dynamic interaction
between agents across time can be recorded as longitudinal network-valued data,
with directed and weighted edges. Examples include: migration of people and/or
animals between regions, value of trading between countries, financial transac-
tions between banks, etc. These interactions often occur in multiple layers of con-
nectivity, thus generating multi-layer networks, which should be jointly modeled
for an adequate understanding of the system under study.
The analysis and understanding of Networks have advanced rapidly during
the last few years. Nevertheless, there is still a recognized underdevelopment of
statistical analysis for Dynamic Networks (Crane 2018). The lack of available
models is even more pronounced for dynamic complex systems with weighted,
c
Springer Nature Switzerland AG 2020
H. Cherifi et al. (Eds.): COMPLEX NETWORKS 2019, SCI 882, pp. 722–735, 2020.
https://doi.org/10.1007/978-3-030-36683-4_58
Learning of Weighted Multi-layer Networks 723
directed and multilayered interactions, as most of the seminal advancements in
dynamic network models are centered in binary undirected interactions.
In particular, there exist an important gap in current models for banking
interactions that simultaneously incorporate features such as time-dependancy
across layers, external predictors, and other desirable characteristics that poten-
tially improve the predictive performance of models. There is little to precedent
of statistical inference of the underlying dynamic structure of weighted multi-
layer networks in the context of financial system. Arguably, this may be conse-
quence of two factors: (1) the novelty of methods that can handle the combined
complexity of such systems, and (2) the unavailability of open-access datasets
due to the sensitivity of such data, which in turn deters potential academics
interested on statistical models for financial networks.
A successful implementation of this techniques will uncover critical hidden
structures and dependencies in the financial system, significantly improving the
performance of models for prediction, stress testing, and ultimately, systemic
risk assessment. An adequate understanding of the underlying characteristics of
the complex financial networks provides important insights that translate into
concrete policy insights and policy applications to financial stability and macro-
prudential regulation (Battiston and Martinez-Jaramillo 2018).
1.1 Related Literature: Latent Space Models for Networks
One of the current frontiers in statistical models for networks is being developed
under the Latent Social Space representation. This framework, originally studied
in the seminal work of Hoff et al. (2002), assumes that each agent is positioned
within a latent social space, and the probability of interaction with other agent
depends on the relative distance between the two. The model has been expanded
in recent years to accommodate for more complex features (see Kim et al. (2018a)
for a review).
In Ward et al. (2013), a comprehensive model for longitudinal data of
the world-trade network is introduced. Their model presents valuable features,
such as incorporating network dependencies jointly for both link incidence and
strength, supporting directed relations by considering sender and receiver latent
spaces; as well as including external information as predictors. Nevertheless, this
model is not suited to jointly model multi-layer networks, and the dynamics are
considered in discrete time, which complicates inference when there is uneven
sampling in time.
Durante and Dunson (2014b,2014a) introduced continuous-time dynamics
by considering Gaussian Processes for the evolution of latent coordinates of
agents across time, later expanded in to incorporate locally adaptive dynamics
Durante and Dunson (2016). Durante et al. (2017) introduced a model for multi-
layer networks, the formulation considers one shared latent space to capture
the global structure between agents, and Klayer-specific latent spaces, which
characterises the idiosyncratic structure of each layer. These models, however,
are suitable only for undirected (symmetric) unweighted (binary) edges.
724 C. U. Carmona and S. Martinez-Jaramillo
Sewell and Chen (2015,2016,2017) proposed an alternative formulation for
dynamic networks with directed and weighted edges. Instead of considering two
latent spaces (sender/receiver) as in Ward et al. (2013), they consider one latent
space, together with a set of node-specific parameters r1:nto reflect the social
reach of agents, and two global parameters, βIN and βOUT , to express the impor-
tance of popularity and sociability, respectively. multi-layer networks and con-
tinuous time are not considered in their formulation.
Recently, Linardi et al. (2017) developed a model for inter-bank data based
on Sewell and Chen (2015) model. Their aim is to characterize only the presence
or absence of links between banks in discrete time, in a single layer.
1.2 Our Contribution
We fill a gap in the current literature of statistical network models by combining
features from preceding research into a comprehensive model. Compared with
previous network models, we expanded the inferential tools to allow the inference
and prediction of a more general class of dynamic networks, accommodating for
links with direction, weight, or both. Moreover, the simultaneous incorporation
of additive effects and external covariates had not been discussed before, such
factors reveal important insight about the underlying structure of the network
and its evolution.
The correct implementation of this features together is not trivial. We devel-
oped the DynMultiNet package1based on R and C++. The application of the
model and software developed are not restricted to financial networks, there is
plethora of fields in which our model could be beneficial for the understanding
and prediction of such complex networks.
In the context of financial stability, our work is pioneer. No previous work
has performed statistical inference on the joint network of transactions between
banks. Our probabilistic framework opens the door for future work in anomaly
detection, stress testing and risk management which considers agents interactions
adequately.
This work is motivated by the study of transactions between commercial
banks within the Mexican Banking System. In Sect. 5we apply the proposed
methodology to real transactions observed in the Mexican Banking System
between 2010 and 2018. Our model achieves accurate in- out-of-sample esti-
mation of probabilities of connection and transaction value.
The model is defined and discussed in Sect. 2, followed by a description of
the MCMC method developed for estimation in Sect. 3.
2 The Model
Consider a dynamic system of Vinteractive agents across time, with activity
recorded as a multi-layer network with Klevels. Let Y(k)(t)={y(k)
ij (t)}∈RV×V
1Available at https://github.com/christianu7/DynMultiNet.
Learning of Weighted Multi-layer Networks 725
be the weighted adjacency matrix for the network in layer kat time t, with
y(k)
ij (t) measuring the interaction from agent ito agent j,fori, j =1,...,V,
t=t1,...,t
Tand k=1,...,K.
The network model consists of a hierarchical model. The observational model
is given by a mixture between a Gaussian distribution and a probability mass
at zero. The first component accounts for the strength of positive interactions
between agents, while the second comprise the pairs with no activity. This is
y(k)
ij (t)∼λ(k)
ij (t)∗N(μ(k)
ij (t),σ
2
(k))+[1−λ(k)
ij (t)] ∗δy(k)
ij (t)(0).(1)
independently for i, j =1,...,V,t=t1,...,t
Tand k=1,...,K. Here, N(μ, σ2)
denotes the Gaussian distribution with mean μand variance σ2and δy(0) the
Dirac measure concentrated at zero.
λ(k)
ij (t) is interpreted as the probability of incidence for an interaction from
agent ito agent jin layer kat time t,andμ(k)
ij (t)istheexpected strength of
such the interaction -if it existed-. σ2
(k)is the variance of all interactions within
the corresponding layer k. We incorporate the latent space representation of
networks through μ(k)
ij (t)andλ(k)
ij (t).
We consider multiple social spaces to deal with the directed and multi-layer
nature of our networks. To accommodate for directed interactions, we follow
Ward et al. (2013) and duplicate the social spaces into the sender and receiver
spaces. Additionally, following Durante et al. (2017), we will use K+ 1 spaces
for a K-layered network: one global space capturing the systemic interaction
between agents, and Klayer-specific spaces that speak for idiosyncratic behavior
for activity of type k.
μ(k)
ij (t)=θ(k)(t)+
ui(t)vj(t)+u(k)
i(t)v(k)
j(t)+
sμ,i(t)+pμ,j (t)+
βμ(t)x(k)
ij (t),
γ(k)
ij (t)=η(k)(t)+
ai(t)bj(t)+a(k)
i(t)b(k)
j(t)+
sλ,i(t)+pλ,j (t)+
βλ(t)x(k)
ij (t).
(2)
Let us start by describing the expected strength. Here ui(t)∈RHis a
dynamic vector of latent coordinates which represents the location of node i
at time twithin the global sender space; similarly, vj(t) is the position of agent j
in the global receiver space.u(k)
i(t)andv(k)
j(t) denote the corresponding coordi-
nates within the layer-specific spaces. The baseline processes θ(k)(t)∈Rcaptures
the average intensity of interactions between all agents in layer k.
Now, sμ,i(t)andpμ,j (t) are additive effects, induced by the sender agent i
and receiver agent j, respectively. Additive effects represent agent i“sociabil-
ity” (out-degree) and agent j“popularity” (in-degree). They capture significant
heterogeneity in activity levels across nodes (Ward et al. 2013;Hoff2018;Kim
et al. 2018b).
726 C. U. Carmona and S. Martinez-Jaramillo
External covariates are also a desirable feature that incorporate exogenous
variation to our system (Ward et al. 2013; Durante and Dunson 2014a;Kim
et al. 2018b). We introduce them using a vector of Pedge-specific covariates
x(k)
ij (t)∈Rp. We define βμ(t)=(βμ,1(t),...,β
μ,P (t)) ∈Rpand βλ(t)∈Rp
as the corresponding -dynamic- coefficients for the strength and incidence of
interactions.
The dynamic of the network is captured by assuming changes in the parame-
ters described before. We adopt smooth trajectories in our formulation introduc-
ing Gaussian Processes (GPs) with squared exponential correlation. C(t, t)=
exp ( t−t
δ)2, as the priors for the coordinates,
θ(k)(·)∼GP (¯
θ(k),C
μ),
ui,h(·)∼GP (¯ui,h,C
μ),v
i,h(·)∼GP (¯vi,h ,C
μ),
u(k)
i,h (·)∼GP (¯u(k)
i,h ,C
μ),v
(k)
i,h (·)∼GP (¯v(k)
i,h ,C
μ),
(3)
independently for k=1,...,K,i=1,...,V and h=1,...,H, with Cμ(t, t)=
exp ( t−t
δμ)2. The parameter and δμcontrol the smoothness of changes across time
for these latent coordinates. The current implementation of our model uses a sin-
gle smoothness parameter for all latent coordinates to ease the MCMC imple-
mentation2. We use a constant (non-zero) mean function for the GPs centering
parameter.
More elaborate dynamics can be considered for the latent locations. One
option is incorporating non-stationarity through the GPs covariance Ras-
mussen and Williams (2005). Another possibility are Nested Gaussian Processes
(Durante and Dunson 2016) which induce Locally Adaptive trajectories for the
latent factors.
The dimension of the social spaces His fixed. We suggest choosing Hby
optimizing a metric for predictive performance, such as the WAIC Watanabe
(2009) or the Pareto-smoothed approximation Vehtari et al. (2017) to leave-one-
out cross-validation.
The probability of incidence λ(k)
ij (t) is treated in a similar way to the strength
of interaction. However, we need to map the latent similarity measure among
units into the probability space. We use a logistic link for this mapping, obtaining
λ(k)
ij (t)= 1
1 + exp(−γ(k)
ij (t)),
γ(k)
ij (t)=η(k)(t)+ai(t)bj(t)+a(k)
i(t)b(k)
j(t).
(4)
We introduced a separate baseline processes η(k)∈RHand new set of latent
coordinates ai,h,b
i,h,a
(k)
i,h ,b
(k)
i,h ∈RH. These terms follow similar dynamics as we
2It is possible to optimize for the parameter δfor each agent using Variational Infer-
ence (Tran et al. 2015), which will be explored in future works.
Learning of Weighted Multi-layer Networks 727
defined previously for θ,uand v,
η(k)(·)∼GP (¯η(k),C
μ),
ai,h(·)∼GP (¯ai,h,C
μ),b
i,h(·)∼GP (¯
bi,h,C
μ),
a(k)
i,h (·)∼GP (¯a(k)
i,h ,C
μ),b
(k)
i,h (·)∼GP (¯
b(k)
i,h ,C
μ),
(5)
independently for k=1,...,K,i=1,...,V and h=1,...,H, with Cλ(t, t)=
exp ( t−t
δλ)2.
We use separate sets of latent coordinates for the incidence and for the
strength of interactions. In case studies datasets, we observed that the empirical
distribution of the log-strength log(y(k)
ij (t) + 1) for existing links (i.e. y≥0)
is concentrated far from zero, even for pairs with low activity. Using a zero-
truncated gaussian distribution to account for both incidence and strength (as
in Sewell and Chen (2016)) would notably reduce the performance of the model.
We preserve GPs for the dynamics of additive effects and coefficients associ-
ated with external covariates. We use different parameters for the GPs for each
external covariates. In financial networks, agents may react fast to changes in
stock markets levels and slow to changes in macroeconomic variables such as
employment or GDP growth. We have
sμ,i(·)∼GP (¯sμ,i ,C
μ),s
λ,i(·)∼GP (¯sλ,i ,C
λ),
pμ,i(·)∼GP (¯pμ,i,C
μ),p
λ,i(·)∼GP (¯pλ,i,C
λ),
βμ,l(·)∼GP (¯
βμ,l,C
βl),β
λ,l(·)∼GP (¯
βλ,l,C
βl),
(6)
independently for each agent i=1,...,V and each covariate l=1,...,P.
The formulation of the model allows to straightforwardly handle missing
values, infer missing links and perform forward and backward predictions.
Sampling from missing link is an optional step. Inference of all the latent
parameters does not rely on the imputed values, as they can be integrated out
for the computation of conditional posteriors.
3 Estimation
We adopt a Bayesian approach to learn the parameters in the model. The pos-
terior distribution is computed via a Gibbs sampler. We follow Durante and
Dunson (2014b) and employ a P`olya-Gamma data augmentation strategy Pol-
son et al. (2013) to sample the dynamic terms related to the link incidence
(i.e. η, a, b, a(k),b
(k),s
λ,p
λ,β
λ). Under this approach, every dynamic latent in
the model has a multidimensional Gaussian distribution as conditional poste-
rior. The main complication arise in correctly setting the updating sequence
to maintain conditional independence, and in arranging the associated design
and response matrices. We use Metropolis-Hastings (MH) steps to sample the
variance of the weights σ(k).
728 C. U. Carmona and S. Martinez-Jaramillo
We provide an implementation of the sampler using the R and C++ program-
ming languages3, as well as detailed derivation of the posterior in the author’s
website.
4 Simulation Study
We conduct a simulation study to evaluate the performance of our model in
a synthetic dataset, constructed to mimic a possible generating process in a
financial application4.
Consider a system with V= 15 agents, whose interactions are recorded in a
Weighted multilayer network with K= 2 layers. We assume a total of T=30
observational times, which are randomly (uniform) distributed in the interval
(0,30).
Probabilities of positive interactions are constructed considering two compo-
nents. First, core-periphery static probabilities that differentiate between highly-
connected and dissociated nodes, as shown in heat maps on the left side of
Fig. 1(top layer 1, bottom layer 2). Second, dynamic probabilities that emu-
late a scenario of financial crisis happening at time t= 10, displayed on Fig. 1.
Two of these trajectories affect the probability λ(k)
ij (t), one for each layer: a
sharp increase in connectivity before the crisis, followed by a prolonged decrease
(top-left); a systemic decrease of trust that start few periods before the crisis
(top-right).
The expected weights of the links are created through an inverse logis-
tic transformation of the static probabilities, then adding a fixed value for
each layer, and finally including two additional trajectories. The trajecto-
ries here assume that strength of connection increases/decreases between cen-
tral/periphery agents after the crisis (bottom-left/bottom-right).
Finally, both expected weights and probabilities were perturbed considering
agent-specific trajectories during the period.
For inference, we choose a value H= 3 for the dimension of the latent spaces
and a value δ= 20 for the smoothing parameter. We ran 2500 Gibbs iterations,
and discarded the first 500 as warm-up period. We verified good mixing of the
parameters in the observational equations.
In Fig. 2we compare the generative vs. estimated probabilities of connection
for three relevant times: before, at, and after the crisis (t=0.3,9.1,24.6 respec-
tively). The structure of the network is learned adequately. Figure 3shows an
overall comparison of the estimated probabilities and weights across all agents,
layers and times. The panels compare horizontally three types of estimation: in-
sample, imputation, and projective forecasting. Top panels correspond to proba-
bilities, bottom panels to weights. The model effectively captures the underlying
structure of the network in all periods, but it perform much better when more
information is available, as expected.
3Available at https://github.com/christianu7/DynMultiNet.
4Code to replicates this experiment is available at the author’s website.
Learning of Weighted Multi-layer Networks 729
Fig. 1. Base probabilities and systemic trends in synthetic data. Left panel shows base
probabilities for the V= 15 agents and two K= 2 layers. Right panel show the
systemic trends that modify μ(k)
ij (t)andμ(k)
ij (t) through time for all agents. The points
corresponding values at observational times. Shaded areas display periods with missing
data.
Fig. 2. Comparison of real vs. estimated probabilities. We show the probability of con-
nection λ(k)
ij (t) for three times (t=0.3,9.1,24.6) and the K= 2 layers. The left panel
shows the original generative probabilities, while the right panel show the estimation
given by the posterior mean.
Lastly, Fig. 4displays how the model captures the dynamics of the network.
For the pair of agents (i=1,j = 7), we show the evolution of probability of
connection and link strength. The smooth dynamic of the latent elements in the
model effectively captures changes in the adjacency matrices.
730 C. U. Carmona and S. Martinez-Jaramillo
Fig. 3. Overall comparison of real vs. estimated probabilities and weights. We dis-
play the prediction accuracy for probability λ(k)
ij (t) (top) and mean strength μ(k)
ij (t)
(bottom). We compare the prediction under three conditions: Period with observations
(left), period with missing data within sample 15 <t<18 (center), and forecasted
values for the future t>23 (right).
Fig. 4. Dynamics of probability of connection λ(k)
ij (t) (left) and expected weight μ(k)
ij (t)
(right) for the pair of agents (i=1,j =7)inlayerk= 1. Red lines show the posterior
mean for both quantities, together with their 95% and 75% credibility intervals. Blue
lines are the true generative quantities. Blue circles show the network data: solid for
observed, empty for missing data.
5 Case Study
5.1 Interbank Activity in the Mexican Financial System
The data used for this case study is derived from a database on exposures built
and operated by the Central Bank of Mexico (Banxico) with the specific pur-
pose of studying contagion and systemic risk. Banxico gathers information using
daily, weekly, and monthly regulatory reports. These reports contain every single
funding transaction between most financial institutions in the Mexican Financial
Learning of Weighted Multi-layer Networks 731
System, on a daily basis, in local and foreign currency. For empirical network
studies using this data see Poledna et al. (2015); Molina-Borboa et al. (2015);
de la Concha et al. (2018).
We aim to characterise the underlying probabilistic structure that dictates
interactions between banks across markets. The results of the inferential pro-
cess will provide enlightening information about the underlying connectivity of
the banking system. We focus our analysis on interbank transaction within the
Mexican financial system in three layers:
OCIMN + OCIME: Unsecured lending. This network is built using the infor-
mation from the regulatory report which contain the unsecured loans in
domestic currency (OCIMN) and in foreign currency (OCIME). A link ij
is the sum of all the transacted loans granted by ito jin any currency during
the time period of aggregation (daily or monthly).
REPO: Repurchase agreement transactions. This network is built using the
information from the regulatory report on Repo transactions. A link ij is
the sum of all the Repos transacted during the time period of aggregation,
regardless of the collateral, currency and maturity.
CVT: Purchase and sale of securities, bilateral transactions. This network is
built with the information from the regulatory report on the bond market
activity by banks. A link ij represents the total amount of all the securi-
ties sold by ito jregardless of the type of securities and the terms of such
purchases.
We discuss here the results from two main experiments. We analyze financial
data from V= 46 banks in the layers described above. We used aggregated
monthly data during two periods of analysis to train the model:
– 2008-01-01 to 2009-12-31 (two years observed)
– 2008-01-01 to 2010-12-31 (three years observed)
The predictive horizon was set as 1 year forward in both cases. For inference,
we set δ= 24, H=5.
There is a clear structure captured by the model, which allows to easily
identify the centrality and relevance of each actor in the system. The estimated
propensity of connection between agents in the system for a given month is
illustrated in Fig. 5. Each bank is represented by a circle in this graph, whereas
the distance between them and the width of line linking them represents the
probability of being connected.
The dynamic of the interaction between banks is also learned by the model.
In Fig. 6, the evolution of the relation between two banks is depicted. The left
panel shows the probability of connection between them as a function of time.
We see that there is a clear gradual increase in unsecured transactions from
bank “11” to bank “1” during 2019 (time 13 to 24). The right panel shows the
expected volume for the transaction (conditional on its occurrence). The blue
points in both graphs correspond to the observed outcome (0/1 for link existence
in the left and a positive weight for the transaction volume in the right.).
732 C. U. Carmona and S. Martinez-Jaramillo
Fig. 5. Force–directed visualization of unsecured market for a (randomly) selected day,
estimated by the model. The width of the edges are proportional to the probability
of connection between two agents on that day. The size of the nodes is given by their
aggregated in-strength.
Fig. 6. For a (random) pair of banks (i=11,j = 1), we show the dynamic of the
strength of unsecured transactions from bank ito bank jacross time (monthly activ-
ity, time = 0 corresponds to Jan-2008). On the left, the posterior distribution for the
probability of connection, Mean ˆpin red, and the observed activity represented by the
blue dots (0/1). On the right, The expected amount for the transaction (conditional
on being connected), and the actual amount also represented by the blue dots. All
quantities are estimated simultaneously within a comprehensive Bayesian model.
We evaluate predictive accuracy of edge formation for each layer as measured
by the ROC curve and the associated area under the ROC Curve, AUC. Out-
of-sample performance was obtained by predicting 12 months following after the
training period and comparing to the observed (left-out) values. In Fig. 7we
show the Area-Under-the-Curve for predictions made for both in-sample and
out-of-sample periods. The performance is outstanding, reaching almost 1 for
in-sample edges; slowly decaying for out-of-sample data, but still with values
above 0.85 for all layers.
Regarding the prediction of weights, in Fig.8we compare observed vs pre-
dicted weights for the three layers. The left plot corresponds to in-sample obser-
vations, reaching a strong correlation of 0.89; whereas the right plot corresponds
to predicting the first six months after the training data, showing a correlation
of 0.77.
Learning of Weighted Multi-layer Networks 733
Fig. 7. Assessing model accuracy. Area Under the ROC curve.
Fig. 8. Assessing model accuracy. Comparison of observed vs predicted link weights.
6 Conclusions
This work provides a comprehensive statistical model that can be useful to cap-
ture in a very broad sense complex aspects of interconnectedness in weighted
multilayer networks. There very limited number of statistical models which com-
prise weighted and directed multiplex structures, something which is a crucial
element in financial networks applications, in particular those related to systemic
risk analysis. The proposed model proved to be effective in capturing important
aspects of the complexity of financial networks.
The possible implications in the field of stress testing and systemic risk mea-
surement are many and important. Until now, most of the studies on financial
networks rely on limited information to build financial networks (one layer) or
single shots (limiting the analysis of the dynamics of such structures).
The model and the software developed usefulness is not restricted to financial
networks, there is plethora of fields in which it could be useful and benefit from
all these novel techniques which were applied in this work.
734 C. U. Carmona and S. Martinez-Jaramillo
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