IEICE Communications Express, Vol.11, No.12, 741–747

Special Cluster in Conjunction with IEICE General Conference 2022

A multi-dimensional opinion
formation model for onlinesocial networks
Naoki Hirakura1, a) and Masaki Aida1
1 Graduate School of Systems Design, Tokyo Metropolitan University

6-6, Asahigaoka, Hino, Tokyo 191-0065 Japana) hirakura-naoki@ed.tmu.ac.jp

Abstract: In this paper, we propose a multi-dimensional opinion formation
model for online social networks. Previous studies have not taken into accountthe “curse of dimensionality” that comes from the higher-dimensional statespace used in multi-dimensional opinion formation; the dimensionality is highas opinions with various types of preferences must be handled simultaneously.
Conventionally, the interactions between users are limited to static models inwhich the topics that users discuss are predetermined and do not change.
Our proposed method can avoid the above problems with previous studies
by introducing low-dimensional subspaces for interacting user pairs by using
word distributed representation. We propose a dynamic model that allows
users to discuss various topics at each interaction. Simulations of the proposedmodel show that discussion focused on a particular topic encourages opinionformation.
Keywords: multi-dimensional opinion formation, polarization, online social networks
Classification: Multimedia Systems for Communication
References©

IEICE 2022

[1] C.R. Sunstein, # Republic, Princeton University Press, Princeton, New Jersey,
2018.
[2] F. Baumann, P. Lorenz-Spreen, I.M. Sokolov, and M. Starnini, “Emergenceof polarized ideological opinions in multidimensional topic spaces,” PhysicalReview X, vol. 11, no. 1, p. 011012, 2021. DOI: 10.1103/physrevx.11.011012[3] R. Axelrod, J.J. Daymude, and S. Forrest, “Preventing extreme polarization ofpolitical attitudes,” Proceedings of the National Academy of Sciences, vol. 118,
no. 50, e2102139118, 2021.
[4] T. Mikolov, I. Sutskever, K. Chen, G.S. Corrado, J. Dean, “Distributed representations of words and phrases and their compositionality,” Advances in NeuralInformation Processing Systems, vol. 26, 2013.
[5] N. Hirakura, M. Aida, and K. Kawashima, “Modeling polarization caused by empathetic and repulsive reaction in online social network,” IEICE Trans. Commun.,vol. E105-B, no. 8, 2022, in press. DOI: 10.1587/transcom.2021ebp3150[6] P. Erdős and A. Rényi, “On Random Graphs. I,” Publicationes Mathematicae,
vol. 6, pp. 290–297, 1959.

DOI: 10.1587/comex.2022COL0002
Received May 18, 2022
Accepted June 21, 2022
Publicized June 28, 2022
Copyedited December 1, 2022

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1 Introduction
Recently, polarization, in which users on online social networks (OSNs) divided
themselves into groups with opposing opinions, has become a social problem [1].
Opinion formation models have been proposed to elucidate the mechanism of polarization. Opinion formation models introduce the opinion vector to quantitatively
represent users’ opinions, and construct rules for opinion change due to interactions
among users. Opinion formation models have been actively developed, and thepresence or absence and strength of interaction are usually determined based on thedistance between opinion vectors [2, 3]. However, these studies did not take accountof the “curse of dimensionality” that occurs in the higher-dimensional space necessary for multi-dimensional opinion formation, which deals with opinions on multiple
topics. In a higher-dimensional Euclidean space, the distances between almost allvectors tend to be similar. This is because most of the volume in a hypersphere in ahigher-dimensional space is concentrated near the surface. For example, the volumeof the part of n-dimensional hypersphere S n of radius 1 that is more than 0.9 awayfrom the center is approximately 19% of the total volume for S 1 (circle), 27% for S 2(sphere), and 99.997% for S 100 ; this is called the “curse of dimensionality”. This cancause problems when measuring the distance between opinion vectors. Also, mostuser interaction models deal with only static topics. However, since the topics ofconversation on OSNs change frequently, a dynamic model is needed. We propose
a model in which users interact in a lower-dimensional subspace corresponding totheir communication on a particular topic. This is a dynamic model that interacts
with the topic at each point in time while avoiding the “curse of dimensionality”.

2 Proposed method
We describe our method for constructing the subspace of user interaction and theinteraction rule.

©

IEICE 2022

DOI: 10.1587/comex.2022COL0002
Received May 18, 2022
Accepted June 21, 2022
Publicized June 28, 2022
Copyedited December 1, 2022

2.1 Subspace construction method
Humans have various values, and trying to express them perfectly creates a highdimension vector to encompass the huge variety of values possible. When a user
posts some articles to SNS, it is natural to think that the user’s values are reflected
in it. However, the post should not be taken to reflect the entire personality of theuser, only some of the values associated with the limited topics being discussed.
The characteristics of users’ posts can be categorized by using the word distributed representation [4]. This representation handles words as vectors with dimensionality of several hundred, where the word vector orientation corresponds tothe meaning of the word.
We adopt this approach and assume that the characteristics (or opinions) of users
can be described by vectors in the (higher-dimensional) space spanned by word
vectors of the word distributed representation. Also, we assume the words used in apost reflect the topic at that time, and construct a subspace with the corresponding
word vector as its basis. Opinion vectors of a user in the higher-dimensional space
represent the whole personality of the user, but a user’s limited opinions on the742

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topic are projected onto the constructed subspace; interaction between users occurs
between the corresponding projected vectors.
Let word vectors be M-dimensional vectors, and W be the set of words. The∑M i
w j ei , where ei is the standard
word vector of word j ∈ W is denoted by w j = i=1basis and M is the dimension of the space in which the word vector is embedded. Letopinion vectors also be M-dimensional vectors in the same space as word vectors,∑M iand U = {1, 2, . . . , N } be the set of users. The opinion vector xk = i=1
xk ei ofuser k ∈ U exists in the space where word vectors are embedded.
Suppose that only D words are involved in a particular SNS communication. TheD words are selected the smallest number of word vectors to be linearly independent.
If linearly dependent vectors exist, they are excluded. Consider a D-dimensionalej ( j = 1, . . . , D) be a reindexed
subspace spanned by the selected D vectors. Let w
word vector corresponding to D words in the subspace. To assess interactions in
the D-dimensional subspace, opinion vectors are projected to this subspace. The
e1, . . . , weD } isprojection matrix for the subspace span{ w() −1
P = A A⊤ A A⊤,(1)eD ] and the projection vector of user k is pk = Pxk . Projection
where A = [ew1, . . . , w
vector pk is represented using the basis matrix A of the subspace:
pk = P xk = A ok =D∑

jej ,ok w(2)

j=1()⊤
where ok = ok1 , . . . , okD is a column vector of coeﬃcients and is corresponding
to opinion value for each word vector. Using the pseudo-inverse matrix A+ of thebasis matrix A, defined as()
A+ = A⊤ A−1A⊤,

it is possible to obtain ok as
ok = A+ pk = A+ Pxk ,and we can represent the projection vector as a linear combination of word vectors.
It is a known problem that word vectors are similar between antonyms. In otherwords, there is no word vector of antonyms in the opposite direction to a given wordvector. However, in considering user interaction, it is necessary to deal with positive
jand negative opinions on a topic. If o k in equation (2) increases, we consider
ej ,that user k has more positive opinions about the corresponding word vector w
jand if o k decreases, we consider that user k has more negative opinions about thecorresponding word vector.

©

IEICE 2022

DOI: 10.1587/comex.2022COL0002
Received May 18, 2022
Accepted June 21, 2022
Publicized June 28, 2022
Copyedited December 1, 2022

2.2 Interaction rules
We extend the interaction rules of our previously proposed one-dimensional opinion
formation model [5] to multiple dimensions. In this one-dimensional model, periodic
boundary conditions are imposed at both ends of the interval where opinion values
can be taken, and two types of reactions, empathy and repulsion, are defined by
the same form exponential functions. To extend this model to multiple dimensions,
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we impose periodic boundary conditions on the subspace constructed in section
2.1 and consider the subspace as a multi-dimensional torus. We determine thetype and strength of the interaction by comparing the projection vectors on a multidimensional torus.
We describe the interaction rules step by step. First, for each dimension corresponding to each word vector, empathy or repulsion is selected with a certain
probability. Projected vectors of user k and user l, pk and pl , can be represented
by a linear combinations of word vectors. Along the( word) vector of word j, wej
jintroduce the scaling of opinion value as eo k = a tanh o k so that the user’s opinion values take the range (−a, a) (a ∈ R). The projection vector is also written as∑
jjje
ej . Let ∆ j = |epk = Aeok = Dokw
ol − eo k | be the distance in opinion values along
j=1 e
the word vector of word j between user k and user l as shown in Fig. 1(a). Anotherdistance ∆′j = 2a − ∆ j can be defined by considering periodic boundary conditions.
The significant characteristic of this model is that the reaction of empathy and repulsion are described by the same framework by using the distance of vectors withperiodic boundary conditions [5]. Given that smaller diﬀerences in opinion makeit easier to establish empathy and that larger diﬀerences in opinion make repulsion
more likely, the probabilities of empathy pe, j and repulsion pr, j in dimension j aredefined as follows:()
exp −∆ j pe, j =(),()′exp−∆+exp−∆jj()(3)′exp −∆ j().
pr, j =
()exp −∆ j + exp −∆′j
Next, the projection vector is changed so that it approaches the interaction partner
on the multi-dimensional torus corresponding to the determined combination ofreactions. As shown in Fig. 1(b), e
pk is changed toward epl . Assuming that the closer
the distance is to the interaction partner, the stronger is the influence and the larger
is the change in opinion value, and also assuming that there is no absolute standardof opinion, the projection vector changes according to the following rule:
epk (t) = epk (t − 1) + Xcλ exp (−λ∆) u.
Here, u is the unit vector
u=(4)(epl − epk ),|epl − epk |where ∆ = | epl − epk |. Also, c and λ ∈ (0, 1/c] are parameters, and X denotes thedistance that projection vector e
pk can be changed. X can be obtained as X = τ∆using the minimum τ at the intersection of vector
epk + τ(epl − epk )©

IEICE 2022

DOI: 10.1587/comex.2022COL0002
Received May 18, 2022Accepted June 21, 2022
Publicized June 28, 2022
Copyedited December 1, 2022(b ∈ [0, 1]) ,and the boundaries ± a ( j = 1, . . . , D) of the multi-dimensional torus. An exceptionpl − epk ) does not intersect any boundaries in the range 0 ≤ τ ≤ 1
is when e
pk + τ (e(when empathy is selected in all dimensions), in that case X = ∆.

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Fig. 1. Illustrations of opinion distances in each dimension
and opinion change rule
Finally, the opinion vector is changed as the projection vector changes in the()subspace. That is, the inverse transformation of eo j (t) = a tanh o j (t) ,)(1a+eo j (t)o (t) = log,
2a−eo j (t)j∑jej as well.
is applied, and the projection vector is changed to pk (t) = D
j=1 o k (t) w
Using these, opinion vector xk (t) after the interaction is xk (t) = xk (t − 1) +( pk (t) − pk (t − 1)).

3©

IEICE 2022

DOI: 10.1587/comex.2022COL0002
Received May 18, 2022
Accepted June 21, 2022
Publicized June 28, 2022
Copyedited December 1, 2022

Experimental evaluation

We conduct simulations and evaluate the impact of the dimension of the interacting
subspace on opinion formation.

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Fig. 2. Opinion changings in the case of interaction in diﬀerent
dimensional subspaces
We consider opinion formation through interaction between networked users.
A randomly selected user on the network performs some action such as posting onan OSN, and neighboring users are influenced and their opinion vector changes;
this process is repeated 5000 times. We conduct this simulation in two diﬀerent
dimensional subspaces and compare the time evolution of opinion values for aparticular word vector.
The experimental conditions are given below. The network structure is given
by Erdős-Rényi model [6] with 100 nodes with link connection probability of 0.3.
For each node k (k = 1, . . . , 100), each element of the opinion vector xk is given
randomly in the range [−1, 1]. We use ten randomly generated vectors as the word
e1, . . . , w
e10 and these vectors are normalized. We prepare span{ we1 } andvectors w
e1, . . . , we10 } as the subspaces in which the interaction occurs. Interaction in aspan{ wone-dimensional space corresponds to a situation in which discussion is concentrated
on a specific topic, while interaction in ten-dimensional space corresponds to asituation in which multiple topics are mixed and the topics of conversation aree1 } due tounfocused. We compare the time evolution in opinion value o1 of { we1 } and span{ we1, . . . , we10 }. The parameters in the interaction
interactions in span{ wrule (4) are set to c = 0.05 and λ = 14.
e1 } is shown in Fig. 2(a). In this case,
The change of o1 when interacting in span{ wtwo groups with diﬀerent opinions are formed. On the other hand, Fig. 2(b) showse1, . . . , we10 }. In this case, the opinion value
the change of o1 for interaction in span{ wof each user changes little. This comparison implies that overheated discussions on
specific topics contribute to opinion formation.

4 Conclusion©

IEICE 2022

DOI: 10.1587/comex.2022COL0002Received May 18, 2022
Accepted June 21, 2022
Publicized June 28, 2022
Copyedited December 1, 2022

We proposed a multi-dimensional opinion formation model that can avoid the problems caused by higher-dimensional state spaces. In addition, we evaluated the eﬀectof the dimension of the interacting subspace by simulation. The results imply that
smaller subspace dimensionality, i.e., concentration of discussion on a particular746

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topic, contributes to opinion formation.
In the simulations, randomly generated vectors were used as word vectors. The
word vectors are approximately orthogonal to each other. Future work will include
evaluations of the eﬀect of the angle of intersection of word vectors and verification
using actual word vectors and text data collected from OSNs.

Acknowledgments
This research was supported by Grant-in-Aid for Scientific Research (B) No.
20H04179 (2020–2022), and No. 21H03432 (2021–2023), and Grant-in-Aid for
Challenging Research (Exploratory) No. 21K19775 (2021–2023) from the Japan
Society for the Promotion of Science (JSPS), and TMU local 5G research support.

©

IEICE 2022

DOI: 10.1587/comex.2022COL0002
Received May 18, 2022
Accepted June 21, 2022
Publicized June 28, 2022
Copyedited December 1, 2022

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