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Characterizing Polarization in Social Networks using the Signed Relational
Latent Distance Model
Nikolaos Nakis Abdulkadir C¸ elikkanat Louis Boucherie Christian Djurhuus
Technical University
of Denmark
Technical University
of Denmark
Technical University
of Denmark
Technical University
of Denmark
Felix Burmester Daniel Mathias Holmelund Monika Frolcov´
a Morten Mørup
Technical University
of Denmark
Technical University
of Denmark
Technical University
of Denmark
Technical University
of Denmark
Abstract
Graph representation learning has become a
prominent tool for the characterization and un-
derstanding of the structure of networks in gen-
eral and social networks in particular. Typically,
these representation learning approaches embed
the networks into a low-dimensional space in
which the role of each individual can be char-
acterized in terms of their latent position. A
major current concern in social networks is the
emergence of polarization and filter bubbles pro-
moting a mindset of ”us-versus-them” that may
be defined by extreme positions believed to ul-
timately lead to political violence and the ero-
sion of democracy. Such polarized networks are
typically characterized in terms of signed links
reflecting likes and dislikes. We propose the
Signed Latent Distance Model (SLDM) utiliz-
ing for the first time the Skellam distribution as
a likelihood function for signed networks. We
further extend the modeling to the characteriza-
tion of distinct extreme positions by constrain-
ing the embedding space to polytopes, form-
ing the Signed Latent relational dIstance Model
(SL IM). On four real social signed networks of
polarization, we demonstrate that the models ex-
tract low-dimensional characterizations that well
predict friendships and animosity while SLIM
provides interpretable visualizations defined by
extreme positions when restricting the embed-
ding space to polytopes.
Proceedings of the 26th International Conference on Artificial
Intelligence and Statistics (AISTATS) 2023, Valencia, Spain.
PMLR: Volume 206. Copyright 2023 by the author(s).
1 INTRODUCTION
For several decades, the origin and influence of political
polarization have been issues receiving considerable at-
tention both within scholarly research and the public me-
dia (Hetherington, 2009). Several studies have demon-
strated an increasing partisan polarization among the po-
litical elites, some of which rely on network science ap-
proaches, for instance, using co-voting similarity networks
and modularity to model and explain the distinct aspects
of the data (Moody and Mucha, 2013). Whereas polariza-
tion has been described in terms of communities and their
boundary properties (Guerra et al., 2013), latent distance
modeling has also been used to extract bipolar structures
(Barber´
a et al., 2015).
Ideological polarization is the distance between policy
preferences, typically of elites taking extreme stands on is-
sues whereas the electoral behavior is denoted affective po-
larization. When these extremes are portrayed as existen-
tial in the media, they typically form an ”us-versus-them”-
mindset (Dagnes, 2019). From a social network perspec-
tive, the process of polarization has been described to occur
when ”homophily and influence become self-reinforcing
when the attraction to those who are similar and differenti-
ation from those who are dissimilar entail greater openness
to influence. The result is network autocorrelation—the
tendency for people to resemble their network neighbors”
(DellaPosta et al., 2015).
To better capture ideological polarization, we turn to signed
networks. Signed networks reflect complex social polar-
ization better than unsigned networks because they capture
positive, negative, and neutral relationships between enti-
ties. The study of signed networks goes back to the ’50s
and was motivated by friendly and hostile social relation-
ships (Harary, 1953). Since then they have been used to
study networks of Twitter users (Keuchenius et al., 2021)
arXiv:2301.09507v3 [stat.ML] 3 Mar 2023
Characterizing Polarization in Social Networks using the Signed Relational Latent Distance Model
and US Congress members (Thomas et al., 2006), two ex-
amples of polarized social networks (Garimella and Weber,
2017; Neal, 2020).
In this paper, we focus on polarization as extreme positions
and argue that the multi-polarity of ”us-versus-them” rein-
forced by homophily and influence can be characterized by
a latent position model (i.e., the latent distance model (Hoff
et al., 2002)) of networks confined to a constrained social
space formed by a polytope, what we denote a sociotope.
As such, the corners of the sociotope define distinct aspects
(i.e., poles) formed by polarized networks’ tendencies to
self-reinforce homophily by positive ties driving those who
are similar close as opposed to those that are negatively
tied being repelled. This can be revealed in terms of the
important multiple poles of social network defining cor-
ners of such sociotope. Within these corners, positive in-
teractions between nodes place them in close proximity in
space thereby accounting for homophily while negative in-
teractions ”push” nodes far apart (towards opposing poles)
yielding the ”us-versus-them” effect.
The conceptual idea of polytopes as formed by pure types
can be traced back to Plato’s forms, which characterize the
physical world as a limited projection of the forms also re-
ferred to as ideal categories. Later, Carl Jung introduced
the concept of universal archetypes, described as a collec-
tive unconscious, in which he related to Plato’s forms by
describing the forms as a Jungian version of the Platonian
archetypes (Williamson, 1985). Employing the theoretical
concept of archetypes to political and ideological polariza-
tion, the archetypes could be interpreted as genuine ideolo-
gies, while the ideological advocates can be expressed as a
mixture of distinct ideologies.
Archetypal Analysis (AA) is a prominent framework for
extracting polytopes in tabular data. AA was originally
proposed by Cutler and Breiman (1994) as an unsupervised
learning method that favors distinct aspects, archetypes, of
the data in which observations are characterized by con-
vex combinations (i.e., mixtures) of these archetypes as op-
posed to clustering procedures extracting prototypical ob-
servations (Mørup and Kai Hansen, 2010). AA has pre-
viously been used to model societal conflicts in Europe
(Beugelsdijk et al., 2022). However, given that AA was
proposed for tabular data, the applications are currently re-
stricted to non-relational data. Thus, whereas the charac-
terization of data in terms of distinct aspects and polytopes
has a long history, such representation learning approaches
have not previously been considered in the context of net-
work analysis for the extraction of polarization by several
extremes.
In the last years, representation learning of signed graphs
has gathered substantial attention, with applications such
as signed link prediction (Chiang et al., 2011), and com-
munity detection (Tzeng et al., 2020). Initial works ex-
tended the prominent random walks framework (Perozzi
et al., 2014; Grover and Leskovec, 2016) to the analysis of
signed networks. SID E (Kim et al., 2018b) exploits trun-
cated random walks on the signed graph with interaction
signs for each node pair inferred based on balance theory
(Cartwright and Harary, 1956). Balance theory is a socio-
psychological theory admitting four rules: “The friend of
my friend is my friend,” “The friend of my enemy is my
enemy,” “The enemy of my friend is my enemy,” and “The
enemy of my enemy is my friend.” POLE (Huang et al.,
2022), also utilizes balance theory-based signed random
walks to construct an auto-covariance similarity which is
used to obtain the embedding space. Neural networks have
also been adopted for the analysis of signed networks. Both
SINE (Wang et al., 2017) and SI GNE T (Islam et al., 2018)
combine balance theory and multi-layer neural networks
to learn the network embeddings while SIGNet uses tar-
geted node sampling to provide scalable inference. In ad-
dition, graph neural networks have also been studied in
the context of signed graphs. More specifically, SIGAT
(Huang et al., 2019) and SD GNN (Huang et al., 2021)
combine balance and status theory with graph attention to
learn signed network embeddings. The status theory is an-
other important socio-psychological theory for directed re-
lationships where for a source and a target node, a positive
directed connection assumes a higher status of the target,
i.e. {status(target) >status(source)}, while the inequal-
ity is opposite for a negative connection. Lastly, SLF (Xu
et al., 2019) learns multiple latent factors of the signed net-
work, modeling positive, negative, and neutral, as well as
the absence of a relationship between node pairs.
A prominent approach for graph representation learning is
the Latent Distance Model (Hoff et al., 2002) in which the
tendency of nodes to connect is defined in terms of their
proximity in latent space. Notably, the LDM can express
the properties transitivity (”a friend of a friend is a friend”)
and homophily (”akin nodes tend to have links”). Re-
cently, it has been shown that LDMs can account for the
structure of networks in ultra-low dimensions (Nakis et al.,
2022, 2023; C¸ elikkanat et al., 2022). It has further been
demonstrated that an LDM of one dimension can be used
to extract bipolar network properties (Barber´
a et al., 2015).
For the modeling of signed networks for the characteri-
zation of polarization, we first present the Signed Latent
Distance Model (SL DM). The model utilizes a likelihood
function for weighted signed links based on the Skellam
distribution (Skellam, 1946). The Skellam distribution is
the discrete probability distribution of the difference be-
tween two independent Poisson random variables. It was
introduced by John Gordon Skellam to model the dynamics
of populations (Skellam, 1946). Since then it was used in
medicine to model treatment measurements (Karlis and Nt-
zoufras, 2006), sports results (Karlis and Ntzoufras, 2008),
as well as, econometric studies (Barndorff-Nielsen et al.,
Nakis, C¸ elikkanat, Boucherie, Burmester, Djurhuus, Holmelund, Frolcov´
a, Mørup
2010). Furthermore, we introduce the Signed relational La-
tent dIstance Model (SL IM) being able to characterize the
latent social space in terms of extreme positions forming
polytopes inspired by archetypal analysis enabling archety-
pal analysis for relational data, i.e. relational AA (RAA).
We apply SLDM and SLIM on four real signed networks
believed to reflect polarization and demonstrate how SLIM
uncovers prominent distinct positions (poles). To the best
of our knowledge, this is the first work to model signed
weighted networks using the Skellam distribution and the
first time AA has been extended to relational data by lever-
aging latent position modeling approaches for the charac-
terization of polytopes in social networks. The implemen-
tation is available at: github.com/Nicknakis/SLIM RAA.
2 PROPOSED METHODOLOGY
Let G= (V,Y)be a signed graph where V={1, . . . , N }
denotes the set of vertices and Y:V2→X⊆Ris a map
indicating the weight of node pairs, such that there is an
edge (i, j)∈ V2if the weight Y(i, j )is different from 0.
In other words, E:= {(i, j)∈ V2:Y(i, j )6= 0}indicates
the set of edges of the network. Since many real networks
consist of only integer-valued edges, in this paper, we set
Xto Z, and we will call the graph undirected if the pairs
(i, j)and (j, i)represent the same link. (The directed case
is provided in the supplementary materials.) For simplicity,
yij denotes each edge weight.
2.1 The Skellam Latent Distance Model (SLDM)
Our main purpose is to learn latent node representations
{zi}i∈V ∈RKin a low dimensional space for a given
signed network G= (V,Y)(K |V|). Therefore, the
edge weights can take any integer value to represent the
positive or negative tendencies between the correspond-
ing nodes. We model these signed interactions among
the nodes using the Skellam distribution (Skellam, 1946),
which can be formulated as the difference of two indepen-
dent Poisson-distributed random variables (y=N1−N2∈
Z) with respect to the rates λ+and λ−:
P(y|λ+, λ−) = e−(λ++λ−)λ+
λ−y/2
I|y|2√λ+λ−,
where N1∼P ois(λ+)and N2∼P ois(λ−), and I|y|
is the modified Bessel function of the first kind and order
|y|. To the best of our knowledge, the Skellam distribu-
tion has not been adapted before for modeling the network
likelihood. More specifically, we propose a novel latent
space model utilizing the Skellam distribution by adopting
the latent distance model, which was proposed originally
for undirected, and unsigned binary networks as a logistic
regression model (Hoff et al., 2002). It was later extended
to multiple generalized linear models (Hoff, 2005), includ-
ing the Poisson regression model for integer-weighted net-
works. We can formulate the negative log-likelihood of a
latent distance model under the Skellam distribution as:
L(Y) := log p(yij |λ+
ij , λ−
ij )
=X
i<j
(λ+
ij +λ−
ij )−yij
2log λ+
ij
λ−
ij !−log(I∗
ij ),
where I∗
ij := I|yij |2qλ+
ij λ−
ij . As it can be noticed,
the Skellam distribution has two rate parameters, and we
consider them to learn latent node representations {zi}i∈V
by defining them as follows:
λ+
ij = exp (γi+γj− ||zi−zj||2),(1)
λ−
ij = exp (δi+δj+||zi−zj||2),(2)
where the set {γi, δi}i∈V denote the node-specific random
effect terms, and ||·||2is the Euclidean distance function.
More specifically, γi, γjrepresent the ”social” effects/reach
of a node and the tendency to form (as a receiver and
as a sender, respectively) positive interactions, expressing
positive degree heterogeneity (indicated by +as a super-
script of λ). In contrast, δi, δjprovide the ”anti-social” ef-
fect/reach of a node to form negative connections, and thus
models negative degree heterogeneity (indicated by −as a
superscript of λ).
By imposing standard normally distributed priors elemen-
twise on all model parameters θ={γ,δ,Z}, i.e., θi∼
N(0,1), We define a maximum a posteriori (MAP) esti-
mation over the model parameters, via the loss function to
be minimized (ignoring constant terms):
Loss =X
i<j λ+
ij +λ−
ij −yij
2log λ+
ij
λ−
ij !!
−X
i<j
log I|yij |2qλ+
ij λ−
ij
+ρ
2||Z||2
F+||γ||2
F+||δ||2
F,
(3)
where ||·||Fdenotes the Frobenius norm. In addition, ρis
the regularization strength with ρ= 1 yielding the adopted
normal prior with zero mean and unit variance. Impor-
tantly, by setting λ+
ij and λ−
ij based on Eq. (11) and (2),
the model effectively makes positive (weighted) links at-
tract and negative (weighted links) deter nodes from being
in proximity of each other.
2.2 Archetypal Analysis
Archetypal Analysis (AA) (Cutler and Breiman, 1994;
Mørup and Kai Hansen, 2010) is an approach developed
for the modeling of observational data in which the data is
expressed in terms of convex combinations of characteris-
tics (i.e. archetypes). The definition of the embedded data
Characterizing Polarization in Social Networks using the Signed Relational Latent Distance Model
points is given as follows:
X≈XCZ s.t. cd∈∆Nand zj∈∆K(4)
where ∆Pdenotes the standard simplex in (P+ 1) dimen-
sions such that q∈∆Prequires qi≥0and kqk1= 1 (i.e.
Piqi= 1). Notably, the archetypes given by the columns
of A=XC define the corners of the extracted polytope as
convex combinations of the observations, whereas Zdefine
how each observation is reconstructed as convex combina-
tions of the extracted archetypes.
Whereas archetypal analysis constrains the representation
to the convex hull of the data, other approaches to model
pure/ideal forms have been Minimal Volume (MV) ap-
proaches defined by
X≈AZ s.t. vol(A) = vand zj∈∆K,(5)
in which vol(A)defines the volume of A. When Ais a
square matrix this can be defined by vol(A) = |det(A)|,
see also Hart et al. (2015); Zhuang et al. (2019) for a review
on such end-member extraction procedures. A strength is
that, as opposed to AA, the approach does not require the
presence of pure observations, however, a drawback is a
need for regularization tuning to define an adequate vol-
ume (Zhuang et al., 2019) whereas the exact computation
of the volume of general polytopes requires the computa-
tion of determinants of the sum of all simplices defining
the polytope (B¨
ueler et al., 2000). Importantly, Archetypal
Analysis and Minimal volume extraction procedures have
been found to identify latent polytopes defining trade-offs
in which vertices of the polytopes represent maximally en-
riched distinct aspects (archetypes), allowing identification
of tasks or prominent roles the vertices of the polytope rep-
resent (Shoval et al., 2012; Hart et al., 2015). Due to the
computational issues of regularizing high-dimensional vol-
umes and the need for careful tuning of such regularization
parameters, we presently focus on polytope extraction as
defined through the AA formulation rather than the MV
formulation.
2.3 A Generative Model of Polarization
Considering a latent space for the modeling of polarization,
we presently extend the Skellam LDM and define polar-
ization as extreme positions (pure forms/archetypes) that
optimally represent the social dynamics observed in terms
of the induced polytope - what we denote a sociotope, in
which each observation is a convex combination of these
extremes. In particular, we characterize polarization in
terms of extreme positions in a latent space defined as a
polytope akin to AA and MV.
In our generative model of polarization, we further suppose
that the bias terms introduced in the definitions of the Pois-
son rates, (λ+
ij , λ−
ij ), are normally distributed. Since latent
representations {zi}i∈V according to AA and MV lie in the
standard simplex set ∆K, we further assume that they fol-
low a Dirichlet distribution. Formally, we can summarize
the generative model as follows:
γi∼ N(µγ, σ 2
γ)∀i∈ V,
δi∼ N(µδ, σ 2
δ)∀i∈ V,
ak∼ N(µA, σ 2
AI)∀k∈ {1, . . . , K},
zi∼Dir(α)∀i∈ V,
λ+
ij = exp (γi+γj− kA(zi−zj)k2),
λ−
ij = exp (δi+δj+kA(zi−zj)k2),
yij ∼Skellam(λ+
ij , λ−
ij )∀(i, j)∈ V2.
According to the above generative process, positive (γ) and
negative (δ) random effects for the nodes are first drawn,
upon which the location of extreme positions A(i.e., cor-
ners of the polytope denoted archetypes) are generated. In
addition, as the dimensionality of the latent space increases
linearly with the number of archetypes, i.e. Ais a square
matrix, with probability zero archetypes will be placed in
the interior of the convex hull of the other archetypes. Sub-
sequently, the node-specific convex combinations Zof the
generated archetypes are drawn, and finally, the weighted
signed link is generated according to the node-specific bi-
ases and distances between dyads within the polytope uti-
lizing the Skellam distribution.
2.4 The Signed Relational Latent Distance Model
For inference, we exploit how polytopes can be efficiently
extracted using archetypal analysis. We, therefore, de-
fine the Signed Latent relational dIstance Model (SLIM)
by defining a relational archetypal analysis approach en-
dowing the generative model a parameterization akin to
archetypal analysis in order to efficiently extract polytopes
from relational data defined by signed weighted networks.
Specifically, we formulate the relational AA in the context
of the family of LDMs, as:
λ+
ij = exp (γi+γj− kA(zi−zj)k2)(6)
= exp (γi+γj− kRZC(zi−zj)k2).(7)
λ−
ij = exp (δi+δj+kA(zi−zj)k2)(8)
= exp (δi+δj+kRZC(zi−zj)k2).(9)
Notably, in the AA formulation X=RZ corresponds to
observations formed by convex combinations Zof posi-
tions given by the columns of RK×K. Furthermore, in or-
der to ensure what is used to define archetypes A=XC =
RZC corresponds to observations using these archetypes
in their reconstruction Z, we define C∈RN×Kas a gated
version of Znormalized to the simplex such that cd∈∆N
by defining
cnd =(Z>◦[σ(G)]>)nd
Pn0(Z>◦[σ(G)]>)n0d
(10)
Nakis, C¸ elikkanat, Boucherie, Burmester, Djurhuus, Holmelund, Frolcov´
a, Mørup
in which ◦denotes the elementwise (Hadamard) product
and σ(G)defines the logistic sigmoid elementwise applied
to the matrix G. As a result, the extracted archetypes are
ensured to correspond to the nodes assigned the archetype,
whereas the location of the archetypes can be flexibly
placed in space as defined by R. By defining zi=
softmax(˜
zi)we further ensure zi∈∆K.
Importantly, the loss function of Eq. (13) is adopted for
the relational AA formulation forming the SL IM, with the
prior regularization applied to the corners of the extracted
polytope A=RZC instead of the latent embeddings Z
imposing a standard elementwise normal distribution as
prior ak,k0∼ N(0,1). Furthermore, we impose a uniform
Dirichlet prior on the columns of Z, i.e. (zi∼Dir(1K),
this only contributes constant terms to the joint distribu-
tion, and therefore the maximum a posteriori (MAP) opti-
mization only constant terms. As a result, the loss function
optimized is given by Eq. (13) replacing kZk2
Fwith kAk2
F.
Complexity analysis. With SLDM/SLIM being distance
models, they scale prohibitively as O(N2)since the node
pairwise distance matrix needs to be computed. This does
not allow the analysis of large-scale networks. For that, we
adopt an unbiased estimation of the log-likelihood through
random sampling. More specifically, gradient steps are
based on the log-likelihood of the block formed by a sam-
pled (per iteration and with replacement) set Sof network
nodes. This makes inference scalable defining an O(S2)
space and time complexity. More options for scalable infer-
ence of distance models have also been proposed in Nakis
et al. (2022); Raftery et al. (2012).
3 RESULTS AND DISCUSSION
We extensively evaluate the performance of our proposed
methods by comparing them to the prominent GRL ap-
proaches designed for signed networks. All experiments
regarding SLDM/SLI M have been conducted on an 8GB
NVIDIA RTX 2070 Super GPU. In addition, we adopted
the Adam optimizer Kingma and Ba (2017) with learning
rate lr = 0.05 and for 5000 iterations. The sample size
for the node set was chosen as approximately 3000 nodes
for all networks. The initialization of the SLDM/SL IM
frameworks is deterministic and based on the spectral de-
composition of the normalized Laplacian (more details are
provided in the supplementary).
Artificial networks. We first, introduce experiments on
artificial networks, as generated by the generative process
described in Section 2.3. We create two networks express-
ing different levels of polarization. Results are presented
in Fig. 1. More specifically, sub-Figs 1a and 1e show the
ground truth latent spaces generating the networks with ad-
jacency matrices as shown by sub-Figs 1b and 1f, respec-
tively. The inferred latent spaces of the two networks are
provided in sub-Figs 1c and 1g where it is clear that the
Table 1: Network statistics; |V|: # Nodes, |Y+|: # Positive
links, |Y−|: # Negative links.
|V| |Y+| |Y−|Density
Reddit 35,776 128,182 9,639 0.0001
Twitter 10,885 238,612 12,794 0.0021
wiki-Elec 7,117 81,277 21,909 0.0020
wiki-RfA 11,332 117,982 66,839 0.0014
model successfully distinguishes the difference in the level
of polarization of the two networks. We also verify the gen-
erated networks based on the inferred parameters given by
sub-Figs 1d and 1h. We observe that the model success-
fully generates sparse networks accounting for the positive
and negative link imbalance.
Real networks. We employed four networks of varying
sizes and structures. (i)Reddit is constructed based on hy-
perlinks representing the directed connections between two
communities in a social platform (Kumar et al., 2018). (ii)
wikiRfA and (iii)wikiElec are the election networks cov-
ering the different time intervals in which nodes indicate
the users and the directed links show supporting, neutral,
and opposing votes to be selected as an administrator on
the Wikipedia platform (West et al., 2014; Leskovec et al.,
2010). Finally, (iv)Twitter is an undirected social network
built on the corpus of tweets concerning the highly polar-
ized debate about the reform of the Italian Constitution (Or-
dozgoiti et al., 2020).
In our experiments, we consider the greatest connected
component of the networks, and if the original network
is temporal, we construct the static network by summing
the weights of the links through time. For the experiments
performed on undirected graphs, we similarly combine di-
rected links to obtain the undirected version of the net-
works.
Baselines. We benchmark the performance of our pro-
posed frameworks against five prominent graph representa-
tion learning methods, designed for the analysis of signed
networks: (i) PO LE (Huang et al., 2022) which learns
the network embeddings by decomposing the signed ran-
dom walks auto-covariance similarity matrix, (ii) SLF (Xu
et al., 2019) learns embeddings that are the concatenation
of two latent factors targeting positive and negative rela-
tions, (iii) SIGAT (Huang et al., 2019) is a graph neural
network approach using graph attention to learn node em-
beddings, (iv) SI DE (Kim et al., 2018b) is another random
walk based method for signed networks, (v) SI GNE T (Is-
lam et al., 2018) is a multi-layer neural network approach
constructing a Hadamard product similarity to accommo-
date for signed proximity on the network pairwise relations.
Characterizing Polarization in Social Networks using the Signed Relational Latent Distance Model
(a) Ground Truth (b) (.017,77,23) (c) Inferred space (d) (.018,73,27)
(e) Ground Truth (f) (.012,63,37) (g) Inferred space (h) (.014,59,41)
Figure 1: Two artificially generated networks with different levels of polarization {zi∼Dir(1)(top row), and
zi∼Dir(0.1·1)(bottom row)}. Both size N= 5000 nodes and K= 3 archetypes. The first column shows the
first two principal components of the original latent space ˜
Z=AZ, the second column the original adjacency matrix,
while the parenthesis shows the network statistics as: (density,% of positive (blue) links,% of negative (red) links). The
third column displays the first two principal components of the inferred latent space, and the fourth column is the SL IM
generated network based on inferred parameters. All network adjacency matrices are ordered based on zi, in terms of
maximum archetype membership and internally according to the magnitude of the corresponding archetype most used for
their reconstruction.
Table 2: Area Under Curve (AUC-ROC) scores for representation size of K= 8.
WikiElec WikiRfa Twitter Reddit
Task p@n p@z n@z p@n p@z n@z p@n p@z n@z p@n p@z n@z
POLE .809 .896 .853 .904 .921 .767 .965 .902 .922 x x x
SLF .888 .954 .952 .971 .963 .961 .914 .877 .968 .729 .955 .968
SIGAT .874 .775 .754 .944 .766 .792 .998 .875 .963 .707 .682 .712
SIDE .728 .866 .895 .869 .861 .908 .799 .843 .910 .653 .830 .892
SIG NET .841 .774 .635 .920 .736 .717 .968 .719 .891 .646 .547 .623
SLIM (OU RS ) .862 .965 .935 .956 .980 .960 .988 .963 .972 .667 .955 .978
SLDM (OU RS ) .876 .969 .936 .963 .982 .963 .986 .962 .973 .648 .951 .975
Table 3: Area Under Curve (AUC-PR) scores for representation size of K= 8.
WikiElec WikiRfa Twitter Reddit
Task p@n p@z n@z p@n p@z n@z p@n p@z n@z p@n p@z n@z
POLE .929 .922 .544 .927 .937 .779 .998 .932 .668 x x x
SLF .964 .926 .787 .983 .922 .881 .994 .870 .740 .966 .956 .850
SIGAT .960 .724 .439 .969 .646 .497 .999 .861 .582 .965 .692 .232
SIDE .907 .779 .608 .920 .806 .739 .974 .831 .469 .957 .820 .614
SIG NET .944 .670 .298 .950 .572 .417 .998 .647 .248 .956 .510 .083
SLIM (OU RS ) .953 .956 .785 .973 .969 .907 .999 .962 .813 .958 .960 .850
SLDM (OU RS ) .960 .963 .787 .977 .971 .912 .999 .963 .809 .954 .955 .846
Nakis, C¸ elikkanat, Boucherie, Burmester, Djurhuus, Holmelund, Frolcov´
a, Mørup
3.1 Link prediction
We evaluate performance considering the link prediction
task considering the ability of our model to predict links of
disconnected network pairs which should be connected, as
well as, infer the sign of these links (positive or negative).
For this, we remove/hide 20% of the total network links
while preserving connectivity on the residual network. For
the testing set, the removed edges are paired with a sample
of the same number of node pairs that are not the edges of
the original network to create zero instances. To learn the
node embeddings, we make use of the residual network.
Predictions and evaluation metrics. For our methods
we fit a logistic regression classifier on the concatena-
tion of the corresponding Skellam rates and log-rates, as
χij = [λ+
ij , λ−
ij ,log λ+
ij ,log λ−
ij ]. Since our Skellam like-
lihood formulation relies both on the ratio and products
of the rates, a concatenation can take advantage of a lin-
ear function of the rates, as well as, their ratio or prod-
uct as allowed from the log transformation. For the base-
lines, we use five binary operators {average, weighted L1,
weighted L2, concatenate, Hadamard product}to construct
feature vectors. For each of these feature vectors, we fit a
logistic regression model (except for the Hadamard prod-
uct which is used directly for predictions). Since different
operators provide different performances, for the baselines
we choose the operator that returns the maximum perfor-
mance per individual task. As a consequence of the class
imbalances and the sparsity present in signed networks, we
adopt robust evaluation metrics, such as area-under-curve
of the receiver operating characteristic (AUC-ROC) and
precision-recall (AUC-PR) curves. Lastly, we denote with
”x” the performance of a baseline if it was unable to run
due to high memory/runtime complexity.
Link sign prediction. In this setting, we utilize the link
test set containing the negative/positive cases of removed
connections. We then ask the models to predict the sign
of the removed links. We denote the task of the link sign
prediction task as p@n. In Table 2 we provide the AUC-
ROC scores while in Table 3 the AUC-PR scores for the
undirected case. Here we observe that our proposed mod-
els outperform the baselines in most networks while be-
ing competitive in the Reddit network against SLF. This
specific baseline is the most competitive across networks
showing high and consistent performance similar to SLIM
and SLDM. Comparing now SLIM with SLDM we get
mostly on-par results, verifying that constraining the model
to a polytope still provides enough expressive capability as
the unconstrained model while allowing for accurate ex-
traction of ”extreme” positions.
Signed link prediction. The second and more challeng-
ing task is to predict removed links against disconnected
pairs of the network, as well as, infer the sign of each
link correctly. For that, the test set is split into two sub-
(a) (b)
Figure 2: wikiElec: Performance of SL IM across di-
mensions for different tasks, (a) Area-Under-Curve Re-
ceiver Operating Characteristic scores, (b) Area-Under-
Curve Precision-Recall scores. Both AUC-ROC and AUC-
PR scores are almost constant across different dimensions
sets positive/disconnected and negative/disconnected. We
then evaluate the performance of each model on those sub-
sets. The tasks of signed link prediction between positive
and zero samples are denoted as p@zwhile the negative
against zero is n@z. We summarize our results by present-
ing AUC-ROC and AUC-PR scores in Table 2 and Table 3
respectively. Once more our models outperform the base-
lines in most networks and for both versions of signed link
prediction. The SLF baseline is again the most competitive
baseline yielding on-par results for Reddit.
Directed networks. Directed network results are provided
in the supplementary. Since SLF has higher modeling
capacity it outperforms the simple model formulation of
SLDM and SLIM. For that, we explore and discuss for-
mulations allowing for more capacity in the SLDM/SLI M
model for the directed case (see supplementary).
Effect of dimensionality. In Figure 2, we provide the per-
formance across dimensions for the different downstream
task and for the wikiElec dataset. We observe that both
AUC-ROC and AUC-PR scores are almost constant across
different dimensions (note that as RK×Kdimensions for
the SL IM is given by the number of archetypes), showcas-
ing that increasing the models’ capacity (in terms of dimen-
sions) does not have a significant effect on the performance
of these downstream tasks (similar results were observed
for all networks and most of the baselines).
Visualizations. The RAA formulation facilitates the infer-
ence of a polytope describing the distinct aspects of net-
works. Here, we visualize the latent space across K= 8
dimensions for all of the corresponding networks. To facil-
itate visualizations we use Principal Component Analysis
(PCA), and project the space based on the first two princi-
pal components of the final embedding matrix ˜
Z=AZ.
In addition, we provide circular plots where each archetype
of the polytope is mapped to a circle every radk=2π
Kra-
dians, with Kbeing the number of archetypes. Figure 3
contains three columns with the first denoting the PCA-
Characterizing Polarization in Social Networks using the Signed Relational Latent Distance Model
(a) WikiElec (b) WikiElec (c) WikiElec
(d) WikiRfa (e) WikiRfa (f) WikiRfa
(g) Reddit (h) Reddit (i) Reddit
(j) Twitter (k) Twitter (l) Twitter
Figure 3: Inferred polytope visualizations for various networks. The first column showcases the K= 8 dimensional
sociotope projected on the first two principal components (PCA) — second and third columns provide circular plots of the
sociotope enriched with the negative (red) and positive (blue) links, respectively.
Nakis, C¸ elikkanat, Boucherie, Burmester, Djurhuus, Holmelund, Frolcov´
a, Mørup
induced space while the second and third columns corre-
spond to the circular plots enriched by the negative (red)
and positive (blue) links, respectively. We observe how the
polytope successfully uncovers extreme positional nodes.
More specifically, all networks have at least one archetype
which acts as a ”dislike” hub and at least one as a ”like”
hub. Meaning that these archetypes contain high values of
negative/positive interactions. For the wiki-RfA and Twit-
ter networks we observe archetypes of very low degree,
this is explained due to some only ”disliked” nodes being
pushed away from the main node population. These can be
regarded as ”outliers” of the sociotope. Nevertheless, such
outliers are discovered since they provide high expressive
power for the model.
Discussion. The Signed Relational Latent Distance Model
has been presented for the undirected case setting, and
we employed the Euclidean distance for both Skellam
rates λ+
ij , λ−
ij . The capacity of the current formulation
works well for undirected networks. Nevertheless, there
are alternative model formulations, and keeping the dis-
tance identical for the positive and negative rates constrains
the models’ expressive capability, especially for the di-
rected/bipartite signed network case. We therefore explore
additional model formulations such as setting the Skel-
lam rates as, λ+
ij = exp(βi+βj− ||zi−wj||2)and
λ−
ij = exp(γi+γj− ||ui−wj||2)in the supplementary
material. Under this assumption, a positive directed rela-
tionship (i→j)shows that node i”likes” node jand ”dis-
likes” node jif it is negative. The latent embedding wjis
then the receiver position for the ”likes” and ”dislikes” with
embeddings ziand uibeing the sender positions for pos-
itive and negative relationships, respectively. In this case,
we introduce three latent embeddings instead of the con-
ventional two for the undirected case. The disparity of lo-
cation ziand uihere can point out how polarity is formed
between the two regions of the latent space (Please see the
supplementary material for further discussion and results).
Another important design characteristic for the
SLDM/SLI M frameworks is the choice of the
prior/regularization of the different parameters. So
far, we did not tune any regularization strength of the
priors and simply adopted a normal distribution on the
model parameters and non-informative uniform Dirichlet
prior on Zin the case of SLIM. Potential tuning of
the priors with cross-validation is expected to boost the
performance and results.
A prominent characteristic of signed networks is the spar-
sity or, in other words, the excess of ”zero” weights
among node pairs. An intriguing direction to account for
it might be the zero-inflated version of the Skellam dis-
tribution (Karlis and Ntzoufras, 2008). Here essentially,
we can define a mixture model responsible for the im-
balance between cases (sign-weighted links) and controls
(neutral zero links) in the network. Such zero-inflated
SLDM/SLI M models can thereby define a generative pro-
cess that can straightforwardly address different levels of
network sparsity.
Whereas we consider the generalization of SL DM and
SLIM to directed networks in the supplementary, a pos-
sible future direction should consider generalizations to bi-
partite networks in which we expect the directed general-
izations to be applicable (Kim et al., 2018a; Nakis et al.,
2022). Furthermore, networks of polarization typically
evolve over time. Future work should thus investigate
how the proposed modeling framework can be extended to
characterize dynamic networks leveraging existing works
by exploring dynamic extensions of latent space model-
ing approaches, including the diffusion model of (Sarkar
and Moore, 2005) and approaches reviewed in Kim et al.
(2018a).
4 CONCLUSION AND LIMITATIONS
The proposed Skellam Latent Distance Model (SL DM)
and Signed Latent Relational Distance model (SL IM) pro-
vide easily interpretable network visualization with favor-
able performance in the link prediction tasks for weighted
signed networks. In particular, endowing the model with
a space constrained to polytopes (forming the SLIM) en-
abled us to characterize distinct aspects in terms of ex-
treme positions in the social networks akin to conventional
archetypal analysis but for graph-structured data. The
Skellam distribution is considerably beneficial in modeling
signed networks, whereas the relational extension of AA
can be applied for other likelihood specifications, such as
LDMs in general. This work thereby provides a founda-
tion for using likelihoods accommodating weighted signed
networks and representations akin to AA in general for an-
alyzing networks.
The optimization for the SL DM/S LIM frameworks is a
highly non-convex problem and thus relies on the quality
of initialization in terms of convergence speed. In this re-
gard, we use a deterministic initialization based on the nor-
malized Laplacian. In addition, we observed that a max-
imum likelihood estimation of the model parameters be-
came unstable when the network contained some nodes
having only negative interactions. This is a direct conse-
quence of the presence of the distance term (exp(+||·||2))
for negative interactions, which can lead to overflow during
inference. Nevertheless, the adopted MAP estimation was
found to be stable across all networks. For real networks,
the generative model created an ”excess” of negative links
increasing the overall network sparsity. For that, a modified
SLIM excluding the regularization over the model param-
eters was introduced which achieved correct network spar-
sity (as shown in supplementary). Assuming priors over the
model parameters created a bias over the generated network
when compared to the ground truth network statistics.
Characterizing Polarization in Social Networks using the Signed Relational Latent Distance Model
Acknowledgements
We would like to express sincere appreciation and thank the
reviewers for their constructive feedback and their insight-
ful comments. We gratefully acknowledge the Independent
Research Fund Denmark for supporting this work [grant
number: 0136-00315B].
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Characterizing Polarization in Social Networks using the Signed Relational Latent Distance Model
A Directed Case Model Formulations
In this section, we describe how our proposed frameworks can be extended to the study of directed networks, and we
further explore additional model formulations allowing for more capacity and expressive power.
A.1 The Skellam Latent Distance Model for the Directed Case (LDM)
Our main purpose here is to learn two latent node representations {zi}i∈V ∈RKand {wi}i∈V ∈RKin a low dimensional
space for a given directed signed network G= (V,Y)(K |V|). The two sets of the latent embeddings correspond to
modeling directed relationships i→jof nodes, with zithe source node and wjthe target node, and vice-versa for an
oppositely directed relationship i←j. Similar to the main paper, we can formulate the negative log-likelihood of a latent
distance model under the Skellam distribution as:
L(Y) := log p(yij |λ+
ij , λ−
ij )
=X
i,j
(λ+
ij +λ−
ij )−yij
2log λ+
ij
λ−
ij !−log I|yij |2qλ+
ij λ−
ij ,
For the directed case, the Skellam distribution has two rate parameters as well, and we consider them to learn latent node
representations {zi}i∈V and {wj}j∈V ∈RKby defining them as follows:
λ+
ij = exp (βi+γj− ||zi−wj||2),(11)
λ−
ij = exp (δi+j+||zi−wj||2),(12)
where the set {βi, γi, δi, i}i∈V denote the node-specific random effect terms, and ||·||2is the Euclidean distance function.
More specifically, the sender βiand the receiver γjrandom effects represent the ”social” reach of a node and the tendency
to form positive interactions, expressing positive degree heterogeneity (indicated by +as a superscript of λ). In contrast, δi
and jprovide the ”anti-social” sender and receiver effect of a node to form negative connections, and thus model negative
degree heterogeneity (indicated by −as a superscript of λ).
By imposing (as in the undirected case) standard normally distributed priors elementwise on all model parameters θ=
{β,γ,δ,,Z,W}, i.e., θi∼ N(0,1), We define a maximum a posteriori (MAP) estimation over the model parameters,
via the loss function to be minimized (ignoring constant terms):
Loss =X
i,j λ+
ij +λ−
ij −yij
2log λ+
ij
λ−
ij !!−X
i,j
log I|yij |2qλ+
ij λ−
ij
+ρ
2||Z||2
F+||W||2
F+||γ||2
F+||β||2
F+||δ||2
F+||||2
F,
(13)
where ||·||Fdenotes the Frobenius norm. In addition, ρis the regularization strength with ρ= 1 yielding the adopted
normal prior with zero mean and unit variance.
A.2 The Signed Relational Latent Distance Model for Directed Networks
We formulate the relational AA in the context of the family of LDMs and for directed networks, as:
λ+
ij = exp (βi+γj− kA(zi−wj)k2)(14)
= exp (βi+γj− kR[Z;W]C(zi−wj)k2).(15)
λ−
ij = exp (δi+j+kA(zi−wj)k2)(16)
= exp (δi+j+kR[Z;W]C(zi−wj)k2).(17)
Notably, in the AA formulation X=R[Z;W]corresponds to observations formed by the concatenations of the convex
combinations Zand Wof positions given by the columns of RK×K. Furthermore, in order to ensure what is used to define
archetypes A=XC =R[Z;W]Ccorresponds to observations using these archetypes in their reconstruction [Z;W], we
define C∈R2N×Kas a gated version of [Z;W]normalized to the simplex such that cd∈∆2Nby defining
cnd =([Z;W]>◦[σ(G)]>)nd
Pn0([Z;W]>◦[σ(G)]>)n0d
(18)
Nakis, C¸ elikkanat, Boucherie, Burmester, Djurhuus, Holmelund, Frolcov´
a, Mørup
in which ◦denotes the elementwise (Hadamard) product and σ(G)defines the logistic sigmoid elementwise applied to the
matrix G. As a result, the extracted archetypes are ensured to correspond to the nodes assigned the archetype, whereas
the location of the archetypes can be flexibly placed in space as defined by R. By defining zi= softmax(˜
zi)and
wi= softmax( ˜
wi)we further ensure zi,wi∈∆K.
As in the undirected case, the loss function of Eq. (13) is adopted for the relational AA formulation forming the SLI M,
with the prior regularization applied to the corners of the extracted polytope A=R[Z;W]Cinstead of the latent embed-
dings Z,Wimposing a standard elementwise normal distribution as prior ak,k0∼ N(0,1). Furthermore, we impose a
uniform Dirichlet prior on the columns of Z,W, i.e. (zi,wi∼Dir(1K), this only contributes constant terms to the joint
distribution. As a result, the loss function is given by Eq. (13) replacing kZk2
Fand kWk2
Fwith kAk2
Ffor the maximum a
posteriori (MAP) optimization.
A.3 Model Extensions for Additional Capacity
In the main paper, we briefly introduced an additional formulation for the rates of the Skellam distribution as adopted by
our models. In this case (and for directed networks), the rates are:
λ+
ij = exp(βi+γj− ||zi−wj||2)and λ−
ij = exp(δi+j− ||ui−wj||2)(19)
In this proposition, we have adopted three latent embeddings instead of the two previously described for the directed case.
The disparity of location ziand uihere can point out how polarity is formed between the two regions of the latent space.
This model specification introduces an additional regularization for the third embedding matrix Uin the loss function
of Equation (13). For the RAA case, we thereby define X=R[Z;U;W], i.e., as the concatenation of all three latent
positions and with C∈R3N×K.
A.4 Directed case — Results and performance
In Table 4 and Table 5, we provide the results for the directed networks against various prominent baselines. Note that
POLE is not defined for the directed case while SID E failed to create embeddings for one-degree nodes. For the frame-
works, we use two additional variations for SL DM and SLIM. The first ones are the SLDM R EG=0.01 and SLIM
RE G= 0.01, where we have used a regularization power ρ= 0.01 in Equation (13). This shows how performance is af-
fected by less regularized parameters. In addition, we also provide results for SL DM-EXP R and SL IM-EXP R which denote
the more expressive model as described in Subsection A.3. The results showcase our models’ capability to outperform the
baselines or provide competitive performance. Comparing now the SLDM and SLIM different variations we observe that
performance is boosted by just using the vanilla methods. It seems that the most important trait is the regularization power
of the model rather than the expressive capabilities that extra parameters provide to the model. Lastly, Figure 4 provides
the same visualizations as in the main paper but for the directed networks.
Table 4: Area Under Curve (AUC-ROC) scores for varying representation sizes (Directed). The symbol ’-’ denotes that
the corresponding model is not able to run on directed networks while ’x’ that the model returned errors.
WikiElec WikiRfa Reddit
Task p@n p@z n@z p@n p@z n@z p@n p@z n@z
POLE - - - - - - - - -
SLF .938 .971 .980 .991 .980 .985 .823 .974 .984
SIGAT .921 .750 .871 .988 .772 .927 .982 .713 .980
SIDE x x x x x x x x x
SIG NET .929 .907 .835 .991 .921 .873 .881 .757 .719
SLIM (OU RS ) .910 .981 .963 .984 .989 .981 .713 .973 .982
SLDM (OU RS ) .914 .977 .966 .983 .987 .978 .657 .937 .964
SLIM RE G=0.01 (O URS ) .927 .989 .980 .992 .994 .990 .827 .982 .989
SLDM RE G=0.01 (O URS ) .940 .989 .980 .984 .987 .976 .774 .982 .986
SLIM-EX PR (O URS ) .922 .984 .977 .987 .988 .982 .706 .930 .949
SLDM-EX PR (O URS ) .915 .987 .985 .989 .994 .992 .657 .965 .965
Characterizing Polarization in Social Networks using the Signed Relational Latent Distance Model
Table 5: Area Under Curve (AUC-PR) scores for varying representation sizes (Directed). The symbol ’-’ denotes that the
corresponding model is not able to run on directed networks while ’x’ that the model returned errors.
WikiElec WikiRfa Reddit
Task p@n p@z n@z p@n p@z n@z p@n p@z n@z
POLE - - - - - - - - -
SLF .981 .949 .890 .995 .954 .951 .978 .972 .919
SIGAT .977 .689 .562 .993 .685 .714 .998 .727 .659
SIDE x x x x x x x x x
SIG NET .979 .831 .577 .995 .840 .671 .988 .675 .233
SLIM (OU RS ) .971 .974 .852 .989 .981 .951 .962 .971 .874
SLDM (OU RS ) .972 .967 .862 .988 .978 .939 .952 .948 .861
SLIM RE G=0.01 (O URS ) .976 .983 .910 .995 .988 .973 .980 .982 .918
SLDM RE G=0.01 (O URS ) .981 .983 .912 .991 .976 .930 .972 .981 .911
SLIM-EX PR (O URS ) .976 .978 .914 .992 .980 .953 .958 .938 .823
SLDM-EX PR (O URS ) .973 .981 .936 .993 .987 .979 .949 .966 .871
B Initialization
For the SLDM model, we used the Eigen-decomposition of the normalized Laplacian for singed networks (Atay and
Tunc¸el G ¨
olpek, 2014). Solving the eigenproblem for a few eigenvalues can be done efficiently through the Lanczos method
(Golub and Van Loan, 1996), due to the high sparsity of real large-scale networks.
For SLIM, we would like to initialize matrix Abased on the convex hull of the spectral decomposition of the normalized
Laplacian. This is very costly since finding the convex hull has an exponential increase in complexity in terms of the
dimensionality of the space. For that purpose, we use the furthest sum algorithm (Mørup and Kai Hansen, 2010) to
discover guaranteed distinct aspects of the spectral space. Lastly, since we are unable to directly initialize A, we use the
furthest sum discovered points to initialize Rwhile also tuning Gfor picking up the correct points in the latent space.
C Bessel Function Approximation
We need to compute the modified Bessel function of the first kind and of order yfor the implementation of our proposed
approach, which is defined by
Iy(x)=(x
2)y
∞
X
k=0
(x2
4)k
k! Γ(y+k+ 1)
We approximate the actual value by only considering the first 50 terms of the infinite sum. Since we have small orders of
yand small values of x, the series components converge to zero quickly. We observed that taking the first 50 components
does not affect the performance/accuracy of the model.
D Generating based on real networks
Here, we test how the model generates based on real networks. We use the wikiElec to train an K= 8 dimensional SLIM
model and we then generate a network based on the inferred parameters. Results are shown in Fig. 5 where we observe
that the generated network learns successfully the main structure of the network but generates more non-zero elements and
more negative links thereby decreasing the sparsity and increasing the percentage of negative links when compared to the
ground truth. Modifying SLIM to exclude the regularization over the model parameters achieves correct network sparsity
as shown in Fig 6 with only a 2% increase in the inferred percentage of negative links when compared to the ground
truth. Adding priors to the model creates a bias over the network generation. Lastly, the un-regularized SL IM boosted
performance in the link prediction tasks ranging from 1% to 5% for the wikiElec network. Nevertheless, priors over the
model parameters stabilize the inference when ”extreme” negative nodes exist in the network (nodes with only negative
links) that can also be considered outliers.
Nakis, C¸ elikkanat, Boucherie, Burmester, Djurhuus, Holmelund, Frolcov´
a, Mørup
(a) WikiElec (b) WikiElec (c) WikiElec
(d) WikiRfa (e) WikiRfa (f) WikiRfa
(g) Reddit (h) Reddit (i) Reddit
Figure 4: Inferred polytope visualizations for various directed networks. The first column showcases the K= 8 di-
mensional sociotope projected on the first two principal components (PCA) of the combined embeddings (source, target)
[Z;W]— second and third columns provide circular plots of the sociotope enriched with the negative (red) and positive
(blue) links, respectively.
E Effect of Sampling Size
In the main paper, the sample size was set to the maximum number (∼3000 nodes) that our 8GB GPU could fit in memory.
Here, we provide a study on how different sample sizes affect the performance of the SLIM model. In Fig 7, we provide the
performance across different tasks for the wikiElec dataset, considering sampling size of {10%,20%,30%,40%,50%}.
We observe with small differences almost constant performance across different sampling sizes. As we decrease the
sampling size to 10% and 20% we observe some more significant decreases in the p@ntask performance. This is because
we keep the total number of training iterations (it=5000) for all cases. Overall, smaller sampling sizes require additional
iterations to converge to the performance of the model with larger sampling sizes.
F Effect of Learning Rate
The learning rate for SL DM and S LIM was set to lr = 0.05. In Fig 8, we provide the performance across different tasks
for the wikiElec dataset, considering three different learning rates lr ∈ {0.01,0.05,0.1}. We observe that the performance
can be considered constant for the different learning rates, showing small sensitivity to the choice of this hyperparameter.
Characterizing Polarization in Social Networks using the Signed Relational Latent Distance Model
(a) Ground Truth: (.003,78%,22%) (b) Generated: (.006,63%,37%)
Figure 5: wikiElec ground truth (left) adjacency matrix and generated (right) adjacency matrix based on inferred param-
eters. The parenthesis shows the network statistics as: (density,% of positive (blue) links,% of negative (red) links). All
network adjacency matrices are ordered based on zi, in terms of maximum archetype membership and internally according
to the magnitude of the corresponding archetype most used for their reconstruction.
(a) Ground Truth: (.003,78%,22%) (b) Generated: (.003,76%,24%)
Figure 6: wikiElec ground truth (left) adjacency matrix and generated (right) adjacency matrix based on inferred param-
eters with a SL IM without regularization priors over the parameters. The parenthesis shows the network statistics as:
(density,% of positive (blue) links,% of negative (red) links). All network adjacency matrices are ordered based on zi, in
terms of maximum archetype membership and internally according to the magnitude of the corresponding archetype most
used for their reconstruction.
Nakis, C¸ elikkanat, Boucherie, Burmester, Djurhuus, Holmelund, Frolcov´
a, Mørup
(a) (b)
Figure 7: wikiElec : Performance of SLI M across sample sizes for different tasks, (a) Area-Under-Curve Receiver Operat-
ing Characteristic scores, (b) Area-Under-Curve Precision-Recall scores. Both AUC-ROC and AUC-PR scores are almost
constant across different dimensions
(a) (b)
Figure 8: wikiElec: Performance of S LIM across learning rates for different tasks, (a) Area-Under-Curve Receiver Op-
erating Characteristic scores, (b) Area-Under-Curve Precision-Recall scores. Both AUC-ROC and AUC-PR scores are
constant across different dimensions
