Carmona, Christian,

Martinez-Jaramillo, Serafin

English

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Submitted on 1 Aug 2019
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Learning of Weighted Dynamic Multi-layer Networks via
Latent Gaussian Processes
Christian Carmona, Serafin Martinez-Jaramillo
To cite this version:
Christian Carmona, Serafin Martinez-Jaramillo. Learning of Weighted Dynamic Multi-layer Networks
via Latent Gaussian Processes. LatinX in AI Research at ICML 2019, Jun 2019, Long Beach, CA,
United States. ￿hal-02213097￿
Learning of Weighted Dynamic Multi-layer Networks
via Latent Gaussian Processes.
Christian U. Carmona (Univ. of Oxford), Serafin Martinez-Jaramillo (CEMLA, Banxico)
carmona@stats.ox.ac.uk
Motivation
In a number of natural, social and eco-
nomic systems the dynamic interaction
between agents across time can be recor-
ded as longitudinal network-valued data,
with directed and weighted edges (eg. fin-
ancial transactions, trading between coun-
tries, migration between regions, etc).
These interactions often occur in mul-
tiple types of connectivity, thus generat-
ing Multilayer networks, which should
be jointly modelled for an adequate un-
derstanding of the system under study.
The motivation for our model lies in the context of Financial Networks, specifically the
analysis of transactions between commercial banks and its implications for systemic
risk. We aim to characterise the underlying probabilistic structure that dictates
interactions between banks and across various types of transactions.
There have been a recent surge of statistical models for dynamic networks using latent
factors (see [3]), but is focused on undirected, binary and single-layer Networks.
Dynamic Network Mixture Model via Latent GP Factors
Consider a dynamic system with V agents. Their activity generates relational data
recorded as a Multiplex Network with K levels. We denote as Y (k)ij (t) the directed inter-
action from agent i to agent j (i, j = 1, . . . , V ), for the layer k = 1, . . . , K, recorded at
time t = t1, . . . , tT .
The data of interest has a semi-continuos nature, Y (k)ij (t) ≥ 0, with a probability mass at
zero for pairs with no interaction, and a continuous distribution over the value of positive
interactions between agents.
Observational equations
The observational model consist of a mixture between a Gaussian distribution and a
probability mass at zero.
y
(k)
ij (t) ∼ λ(k)ij (t)N
(
µ
(k)
ij (t), σ
2
(k)
)
+
(
1− λ(k)ij (t)
)
δ
y
(k)
ij (t)
(0). (1)
Here λ(k)ij (t) is the probability of incidence of an interaction from agent i to agent j in
layer k at time t. The Gaussian mean µ(k)ij (t) is the expected strength of such interaction
-if it existed-. Finally, σ2(k) accounts for the variance of interactions, which is shared by
all pairs within layer k.
Evolving Latent Space
The complexity of the system and the dynamics of the relation between agents is captured
by latent effects for both the incidence probability λ(k)ij (t), and the expected strength
µ
(k)
ij (t).
The latent structure for the expected strength is given by
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)
(2)
The baseline processes, θ(k)(t) ∈ R, captures the average intensity of interactions
between all agents in layer k.
The multiplicative effects, ui(t)′vj(t) and u
(k)
i (t)
′v(k)j (t), are interpreted as the distance
between agents within the latent social space, where ui(t) = {ui,1(t), . . . , ui,H(t)}′ ∈
RH is the vector of latent coordinates of node i at time t within the global sender space;
similarly, vj(t) is the location of agent j in the global receiver space. u
(k)
i (t) and v
(k)
j (t)
denote the corresponding coordinates within the layer-specific spaces.
The additive effects, sµ,i(t) and pµ,j(t), represent agent i sociability (out-degree) and
agent j popularity (in-degree), respectively.
The systemic structure of dependency that governs all layers will be captured by the
latent elements that affect all layers, i.e. the global social spaces and the agent-specific
additive effects.
The evolution of the network through time will be produced by dynamic latent compon-
ents. We adopt smooth trajectories in our formulation by using Gaussian Processes with
squared exponential correlation, C(t, t′) = exp (t−t
′
δ )
2, as the priors for the coordinates,
θ(k)(·) ∼ GP (θ¯(k), Cµ),
ui,h(·) ∼ GP (u¯i,h, Cµ) , vi,h(·) ∼ GP (v¯i,h, Cµ),
u
(k)
i,h (·) ∼ GP (u¯(k)i,h , Cµ) , v(k)i,h (·) ∼ GP (v¯(k)i,h , Cµ),
sµ,i(·) ∼ GP (s¯µ,i, Cµ) , pµ,i(·) ∼ GP (p¯µ,i, Cµ),
(3)
The latent structure of the probability of incidence λ(k)ij (t) is analogous. We use a
separate set of latent components and map the measure of similarity into the unit interval
using a logistic link.
logit(λ
(k)
ij (t)) =η
(k)(t)+
ai(t)
′bj(t) + a
(k)
i (t)
′b(k)j (t)+
sλ,i(t) + pλ,j(t)
(4)
The dynamics of the latent components in λ(k)ij (t) also follow Gaussian Processes priors.
Posterior Computation
We perform MCMC using a Gibbs sampler. The method is based on [1, 2], but expan-
ded to consider the additional latent elements in our model. The implementation is
available in the R package DynMultiNet.
Interbank transactions in the Mexican Financial System
We applied our model to financial transactions between V=46 banks in Mexico, aggreg-
ated monthly between Jan-2014 and Dec-2017. The layers correspond to three types
of transactions: 1) Unsecured lending, 2) Repo agreements, and 3) Securities trading
(CVT).
To asses the performance of the model we consider criteria for both classification (AUC)
and regression (RMSE), due to the semi-continuos nature of the data. The plots below
illustrate an exercise training the model with 36 months (2014-2016) and testing on the
next 12 months (2017). We calculate the criteria for in- and out-of-sample periods.
We evaluate the prediction of edge incid-
ence by computing the AUC (Area Under
the ROC curve). The performance is out-
standing, reaching almost 1 for in-sample
data, and values above 0.85 for out-of-
sample periods in the three layers.
To assess the accuracy of the predicted
weight, we calculate two versions of the
RMSE: one using predictions for all ob-
servations, and one only using connected
pairs (Y > 0).
As a reference, the average weights for
each layer (CVT,Repo,Unsecured) are 13.4,
15.2 and 13.8, respectively, with standard
deviations of 2.1, 1.8, and 2.3.
The probabilistic structure captured by the
model provides information that can be used for
insightful visualization of the system under study.
For example, combining posterior expected
weights with force-directed network plots, it is pos-
sible to visualize the core-periphery structure of the
financial system and identify important actors.
It is possible to visualise changes in the relationship between any pair of agents across
time. The plots below show the evolution of the probability of connection and expected
weight for two banks (blue points indicate observed values).
Conclussions
This work provides a flexible model that efficiently captures the underlying systemic
dependency in weighted multilayer networks, showing an exceptional prediction per-
formance and good interpretability.
References
[1] DURANTE, D., AND DUNSON, D. B. Nonparametric Bayes dynamic modelling of relational data.
Biometrika 101, 4 (12 2014), 883–898.
[2] DURANTE, D., MUKHERJEE, N., AND STEORTS, R. C. Bayesian Learning of Dynamic Multilayer
Networks. Journal of Machine Learning Research 18 (2017), 1–29.
[3] KIM, B., LEE, K. H., XUE, L., AND NIU, X. A review of dynamic network models with latent variables.
Statistics Surveys 12 (2018), 105–135.


Learning of Weighted Dynamic Multi-layer Networks via Latent Gaussian Processes

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Learning of Weighted Dynamic Multi-layer Networks via Latent Gaussian Processes

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