Analysis and application of opinion model with multiple topic interactions

Abstract

To reveal heterogeneous behaviors of opinion evolution in different scenarios, we propose an opinion model with topic interactions. Individual opinions and topic features are represented by a multidimensional vector. We measure an agent's action towards a specific topic by the product of opinion and topic feature. When pairs of agents interact for a topic, their actions are introduced to opinion updates with bounded confidence. Simulation results show that a transition from a disordered state to a consensus state occurs at a critical point of the tolerance threshold, which depends on the opinion dimension. The critical point increases as the dimension of opinions increases. Multiple topics promote opinion interactions and lead to the formation of macroscopic opinion clusters. In addition, more topics accelerate the evolutionary process and weaken the effect of network topology. We use two sets of large-scale real data to evaluate the model, and the results prove its effectiveness in characterizing a real evolutionary process. Our model achieves high performance in individual action prediction and even outperforms state-of-the-art methods. Meanwhile, our model has much smaller computational complexity. This paper provides a demonstration for possible practical applications of theoretical opinion dynamics.

Full text

Analysis and application of opinion model with multiple topic interactions
Fei Xiong, , Yun Liu, , Liang Wang, and , and Ximeng Wang
Citation: Chaos 27, 083113 (2017); doi: 10.1063/1.4998736
View online: http://dx.doi.org/10.1063/1.4998736
View Table of Contents: http://aip.scitation.org/toc/cha/27/8
Published by the American Institute of Physics

Analysis and application of opinion model with multiple topic interactions
Fei Xiong,
1,2,a)
Yun Liu,
1,2
Liang Wang,
3
and Ximeng Wang
1,2
1
School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing 100044, China
2
Key Laboratory of Communication and Information Systems (Beijing Jiaotong University), Beijing Municipal
Commission of Education, Beijing 100044, China
3
School of Electrical and Control Engineering, Xi’an University of Science and Technology, Xi’an 710054,
China
(Received 14 April 2017; accepted 31 July 2017; published online 17 August 2017)
To reveal heterogeneous behaviors of opinion evolution in different scenarios, we propose an
opinion model with topic interactions. Individual opinions and topic features are represented by a
multidimensional vector. We measure an agent’s action towards a specific topic by the product of
opinion and topic feature. When pairs of agents interact for a topic, their actions are introduced to
opinion updates with bounded confidence. Simulation results show that a transition from a
disordered state to a consensus state occurs at a critical point of the tolerance threshold, which
depends on the opinion dimension. The critical point increases as the dimension of opinions
increases. Multiple topics promote opinion interactions and lead to the formation of macroscopic
opinion clusters. In addition, more topics accelerate the evolutionary process and weaken the effect
of network topology. We use two sets of large-scale real data to evaluate the model, and the results
prove its effectiveness in characterizing a real evolutionary process. Our model achieves high per-
formance in individual action prediction and even outperforms state-of-the-art methods. Meanwhile,
our model has much smaller computational complexity. This paper provides a demonstration for
possible practical applications of theoretical opinion dynamics. Published by AIP Publishing.
[http://dx.doi.org/10.1063/1.4998736]
When agents interact with neighbors, they may have
multidimensional opinions, which are composed of
beliefs and evaluative components. Meanwhile, agents
can take an action depending on internal opinions. For
instance, users in online social networks may publish a
post about a topic in terms of their own opinions.
Neighbors can read the post and discuss about the topic
to exchange their opinions. In the previous studies,
when pairs of agents have the same opinion distance,
they are often supposed to have similar interaction
behaviors. However, opinion interactions in different
scenarios may be different even with the same opinion
distance, and interactions are also affected by topics. In
this paper, we propose an opinion model with the topic
impact. Individual opinions and topic features are char-
acterized by a multidimensional vector. Agents can take
an action towards a given topic according to their opin-
ions and the topic feature. Meanwhile, actions are
incorporated into opinion updates with bounded confi-
dence. Simulation results show that a transition from a
disordered state to a consensus state depends on the
opinion dimension. Multiple topics promote opinion
interactions and decrease the divergence of individual
opinions. In addition, we use real data to evaluate the
model. The results prove that our model achieves high
performance in individual action prediction and has
much smaller computational complexity. This paper
demonstrates that studies of theoretical opinion dynam-
ics can be applied in other research fields, such as social
data mining, behavior pattern analysis, and decision
support systems.
I. INTRODUCTION
Different kinds of opinion dynamic models have been
presented, and they attract wide attention in the area of statis-
tical physics.
1–6
Opinion models are divided into two groups:
discrete models
7–10
or continuous models.
11–14
As a typical
representative of continuous models, the Deffuant-Weisbuch
(DW) model explores pairwise interactions between agents
who have similar opinions.
15
An individual opinion is a
summary evaluation of a psychological object.
16
Opinions
may have multiple dimensions,
17
each of which represents a
belief or evaluative component associating the object.
Reference 18 considered the general situation of a discrete
opinion space with multiple opinions in a naming game and
calculated steady states with different types of zealots. In
Ref. 19, opinions were considered as multidimensional bit
strings, and agents exchanged opinions if the difference
between the opinion strings was below a similarity threshold.
Non-trivial social structures emerged from the interactions
that took place during the simulation. Agents often base their
opinions on pro and con arguments.
20
With a limited number
of arguments that address a given topic, an agent’s opinion is
based only on a subset of relevant arguments. The model
witnesses opinion bi-polarization without negative influence.
In a continuous case of multidimensional opinions, a main
result is the spontaneous emergence of spatial opinion
a)
E-mail: xiongf@bjtu.edu.cn
1054-1500/2017/27(8)/083113/10/$30.00 Published by AIP Publishing.27, 083113-1
CHAOS 27, 083113 (2017)

structures within which all individuals share the same opin-
ion.
21
Multidimensional continuous opinions were applied in
the DW and Hegselmann-Krause model, and results showed
that the number of minority clusters rose with the dimension-
ality and higher dimensionality led to better chances for a
vast majority consensus.
22
Reference 23 assumed that indi-
vidual nodes had different certainties of belief on multiple
truth statements, and agents interacted in terms of their own
and others’ displayed certainties of belief. The interdepen-
dencies among statements were expressed as a matrix of
logic constraints.
While agents have internal opinions, they may take an
external action to express their opinions. An individual action
can be regarded as an outward manifestation in terms of its
opinion and a topic. For instance, users have their beliefs
toward a given product on an online shopping website, and
they may take external actions, i.e., publish a rating within a
certain range on the product. Moreover, users in online social
networks communicate with neighbors to exchange their
opinions, and they may take an external action, i.e., publish a
post about a topic. If neighbors read the post, their opinions
may be affected. A model with continuous opinions and dis-
crete actions (CODA)
24–26
was presented, characterizing the
dependence of actions based on opinions. Agents in the
model may have an action to select one of two objects, and
they also have an opinion about which object is better with a
probability. Individual actions are determined by the proba-
bility, and when agents observe a neighbor’s action, they will
update their opinions. If a neighbor selects an object, the
agent will increase the probability that the object is better.
The model witnesses the appearance of extremists. The con-
crete relationship between individual opinions and actions
has also been investigated. A neutral opinion was introduced
to the CODA model, implying the unclear or unknown atti-
tude toward two objects.
27
Before opinion updates, agents
collect the evidences of neighboring external actions and use
the Dempster-Shafer evidence theory to fuse them into one as
reference. Then, agents conclude their opinions by the fused
reference. In the prediction of user actions,
28,29
an action,
which is a numerical value given by a user towards a topic, is
calculated by the inner product of the user opinion vector and
topic feature vector. These user and topic vectors are trained
according to existing actions. Once the vectors have been
obtained, they can be used to predict unknown user actions
towards different topics. How opinion interactions are influ-
enced by external actions still needs further exploration. In
this paper, we assume that an agent’s action is determined by
the multidimensional vector of its opinion, and the actions of
two agents affect a pairwise opinion update.
When pairs of agents have the same opinion distance,
they are often supposed to have similar interaction behaviors.
However, opinion interactions in different scenarios may be
different, even if the opinion distance between pairs of
agents is the same. In online social networks, user behaviors
are determined both by their opinions and features of infor-
mation. Two users may not interact towards a topic but they
may change their opinions towards another topic.
In this paper, we present an opinion model with topic
interactions. We mimic an agent’s action from its
multidimensional opinion and topic feature. To describe the
feedback of actions, individual actions are deemed to affect
opinion updates while agents interact toward a specific topic.
We study the transition of the system from a disordered state
to a consensus state. Analysis and Monte-Carlo simulations
are conducted to reveal the relationship of opinions and
topics. Results show that multiple topics promote opinion
interactions and decrease the divergence of individual opin-
ions. In addition, the underlying topology has a notable
effect only when there are few topics in the system. We use
different real social data to evaluate the opinion model, and
the results prove its effectiveness in characterizing online
opinion evolution. This paper demonstrates the practical use
of opinion models.
The rest of this paper is structured as follows. Section II
presents an opinion model with topic interactions. Section III
contains simulation results and discussions about the model.
Section IV presents the evaluation of the model for two sets of
real social data. Section Vis devoted to concluding remarks.
II. OPINION MODEL WITH MULTIPLE TOPIC
INTERACTIONS
In this section, we present our opinion model that
includes interactions between agents and topics. There are m
agents in the system. An agent i’s opinion at time tis defined
as ~
uiðtÞthat is a k-dimensional vector, representing its beliefs
and evaluative components on a small number of latent fac-
tors. Agent i’s opinion vector reveals how each factor applies
to the agent. At time t,i’s opinion ~
uiðtÞis the same when it
faces any topic. The initial value of each dimension in ~
uiðtÞ
is uniformly distributed in the continuous interval ½0;1.
Agents select a topic to interact with each other, and their
opinions evolve with time. Assume that there are ntopics in
the system. A topic lis denoted by ~
vl, representing the fea-
ture vector of the topic. Topic feature vectors are indepen-
dent of each other. ~
vlis also a k-dimensional vector, and it is
different for different topics. ~
vlcharacterizes the inherent
contents of the topic about latent factors. During the dynam-
ics, when agents change their opinions, topic features always
remain constant.
If a topic lis selected for an interaction agent i’s action
rl
iðtÞtowards lis determined by its opinion and the topic.
rl
iðtÞcan be approximated by how agent i’s opinion is coordi-
nated with topic l’s feature about latent factors. For instance,
when agents discuss about a movie, latent factors might
include actor popularity, movie duration, and movie genre,
etc. The topic feature is treated as the contents of the movie
about latent factors, and an agent’s action is treated as a rat-
ing given by the agent towards the movie. We calculate rl
iðtÞ
as the inner product
rl
iðtÞ¼~
uiðtÞ~
vT
l;(1)
where ~
vT
lmeans the transpose of ~
vl.The system is governed
by the following discrete-time step rules:
(i) In an update, an agent iand one neighbor jare
selected at random. They randomly select a topic lto
interact.
083113-2 Xiong et al. Chaos 27, 083113 (2017)

(ii) In terms of the opinion and topic feature vectors,
agent iand jtake an action rl
iðtÞ¼~
uiðtÞ~
vT
land
rl
jðtÞ¼~
ujðtÞ~
vT
l, respectively.
(iii) According to their actions, agent iand jchange their
opinions. Considering the bounded confidence, the
difference between their actions on the topic lis cal-
culated by rl
ijðtÞ¼krl
iðtÞrl
jðtÞk. If the action differ-
ence is smaller than a tolerance threshold h, i.e.,
rl
ijðtÞ<h(0 <hk), their opinions are updated by
~
uiðtþ1Þ¼~
uiðtÞþlðrl
ijðtÞÞð~
ujðtÞ~
uiðtÞÞ;
~
ujðtþ1Þ¼~
ujðtÞþlðrl
ijðtÞÞð~
uiðtÞ~
ujðtÞÞ;(2)
where lðrl
ijðtÞÞ denotes the trust function of agents in
terms of the action difference rl
ijðtÞ. It holds true that
0lðrl
ijðtÞÞ  0:5, and lðrl
ijðtÞÞ is a non-increasing
function of rl
ijðtÞ.Ifrl
ijðtÞh, nothing will take place.
For instance, we assume that the dimensionality is k¼2,
and agent i’s opinion and j’s opinion are ð0:5;0:2Þand
ð0:5;0:5Þ, respectively. There are two topics in the system,
and their feature vectors are ð0:5;0:2Þand ð0:2;0:5Þ, respec-
tively. The tolerance threshold is set at h¼0:1. In an update,
if the topic ð0:5;0:2Þis selected for discussion, the difference
between actions of iand jis 0.06. Therefore, the two agents
will change their opinions. However, if the topic ð0:2;0:5Þis
selected in an update, the difference between actions is 0.15.
In this situation, individual opinions remain unchanged.
If we have the dimensionality of vectors k¼1, n¼1,
and lis treated as a constant, then the model reduces to the
DW model. In other cases, topic features play a significant
role in opinion interactions.
Now, we make an analysis of the model. We consider a
special case that only one topic exists, i.e., n¼1. Therefore,
a topic l’s feature vector is simply written by ~
v, and rl
ijðtÞis
simplified as rijðtÞ. To make a more general expression of the
tolerance condition, we use a truncated trust function
lhðrijðtÞÞ as
lhðrijðtÞÞ ¼ lðrijðtÞÞ;rij ðtÞ<h
0;rijðtÞh:
((3)
The squared distance between the opinion of agent iand
jis denoted by dijðtÞ¼k
~
uiðtÞ~
ujðtÞk2. After the update
event, dijðtþ1Þis obtained as
dijðtþ1Þ¼k
~
uiðtþ1Þ~
ujðtþ1Þk2
¼ð12lhðrijðtÞÞÞ2k~
uiðtÞ~
ujðtÞk2
¼dijðtÞ4lhðrij ðtÞÞð1lhðrijðtÞÞÞdij ðtÞ:(4)
After the update event, the average opinion of the popu-
lation Eð~
uðtÞÞ remains constant. EðÞ means the expectation
operation, and Eð~
uðtÞÞ¼ð0:5;0:5;…;0:5Þ. We define the
average squared distance of individual opinions DðtÞat time
tas
DðtÞ¼X
ik~
uiðtÞEð~
uðtÞÞk2=m;(5)
where mis the number of agents in the system. If agents i
and jinteract at time t, the average opinion distance Di;jðtþ
1Þafter the interaction of iand jcan be obtained as
Di;jðtþ1Þ¼Di;jðtÞ2lhðrijðtÞÞð1lhðrij ðtÞÞÞdijðtÞ=m:
(6)
We calculate the expectation of Eq. (6) over all possible
pairs (i,j)
EðDi;jðtþ1ÞDi;jðtÞÞ
¼X
ði;jÞ
pi;jðDi;jðtþ1ÞDi;jðtÞÞ
¼2X
ði;jÞ
pi;jðlhðrijðtÞÞð1lhðrij ðtÞÞÞdijðtÞÞ=m;(7)
where pi;jmeans the probability that agent iinteracts with j.
pi;jis determined by the network topology, and it is indepen-
dent of opinion interactions.
In an asynchronous update event, two agents are
selected at random to interact with each other. For an m-
sized system, a time step contains mupdate events.
Therefore, for asynchronous evolution, we obtain the transi-
tion of DðtÞby
Dðtþ1=mÞDðtÞ
¼2X
ði;jÞ
pi;jðlhðrijðtÞÞð1lhðrij ðtÞÞÞdijðtÞÞ=m:(8)
In synchronous evolution, all agents select a neighbor at
random, and these mpairs of agents update their opinions
simultaneously each time. For synchronous evolution, we
have
Dðtþ1ÞDðtÞ
¼m2X
ði;jÞ
pi;jðlhðrijðtÞÞð1lhðrijðtÞÞÞdij ðtÞÞ=m:(9)
From Eqs. (8) and (9), we obtain the derivative of DðtÞ
that is the same for synchronous and asynchronous updates
@DðtÞ=@t¼2X
ði;jÞ
pi;jðlhðrijðtÞÞð1lhðrijðtÞÞÞdij ðtÞÞ:(10)
In homogeneous networks, Eq. (10) can be simplified as
@DðtÞ=@t¼2EðlhðrijðtÞÞð1lhðrij ðtÞÞÞdijðtÞÞ:(11)
When mis large enough, we have EðdijðtÞÞ2DðtÞ.
From Ref. 30, when the system in homogeneous networks
converges, the initial condition h>Eðrijð0ÞÞ holds true. The
condition is necessary but not sufficient one. Since each
dimension of ~
uið0Þand ~
vlis randomly distributed from ½0;1,
the expectation of an action rið0Þis k=4, where kis the
dimensionality of the vector. It can be inferred that the
expectation of r2
ijð0Þis ÐÐÐð~
uið0Þ~
vT~
ujð0Þ~
vTÞ2d~
uid~
uj
d~
v¼k=18. Therefore, with large k,Eðrij ð0ÞÞ increases, and
less pairs of agents will update their opinions, leading to
larger divergence of opinions.
083113-3 Xiong et al. Chaos 27, 083113 (2017)

III. SIMULATION RESULTS
We conduct Monte-Carlo simulations to investigate how
feature dimensionality affects opinion evolution and whether
the final system state is determined by the topic set. We
implement the opinion model asynchronously. At the begin-
ning, each dimension of all individual opinions and topic
features is assigned randomly from ½0;1. In an update event,
an agent and one of its neighbor are selected at random, and
they take actions following the rules of our bounded-
confidence model. After msuch update events, time is
increased by 1. The dynamics continue until no opinion
update takes place. In the simulations of this section, we fix
the system size at m¼1000. Both regular networks and
Newman-Watts small-world networks
31
are used as interac-
tion topology. A small-world network is constructed as fol-
lows: initially, each node connects 4 neighboring nodes, and
these nodes form a closed loop; every time, two nodes are
selected at random, and an edge is added to connect them;
the process of edge growth is repeated until the average
degree of all nodes reaches a given value. Since the value of
lðrl
ijðtÞÞ only determines the relaxation rate of the system
and does not influence the convergence of individual opin-
ions, we fix the trust function at lðrl
ijðtÞÞ¼0:4 for rl
ijðtÞ<h.
Figure 1shows the distribution of individual opinions in
the three-dimensional space after the evolution. Here, we
only depict 200 agents. In this figure, we set k¼3, and
therefore, the opinion of each agent is described by a three-
dimensional vector. A circle in the figure represents an
agent’s opinion, and each axis represents a dimension of
opinions. During the dynamics, agents interact for the given
topic and reduce the difference among their opinions. In the
left plot, with a small tolerance threshold, many opinion
clusters exist, and individual opinions are divergent in the
final system. Increasing the threshold h, a main opinion
cluster at ~
uðtÞ¼ð0:5;0:5;0:5Þoccurs, together with several
small clusters. Agents in these small clusters have a large
opinion distance from those in the main cluster, preventing
the convergence of the whole system. If h0:5, all agents
update their opinions to ð0:5;0:5;0:5Þ, and the average opin-
ion does not change.
We investigate the final average squared distance of
individual opinions Dð1Þ with different thresholds h,as
shown in Fig. 2. From Fig. 2, a clear transition of Dð1Þ can
be observed, especially for small dimensionality k. When
k¼1 like the traditional DW model, Dð1Þ drops quickly
for h<0:3 with the increase in h. At the critical point
h0:3, opinions of all agents stick to the average opinion
0.5. For k¼5, the critical point is h0:7. For k¼10,
although the average squared opinion distance decreases dra-
matically and the divergence is reduced, the system cannot
reach consensus under the condition h1. Thus, the dimen-
sionality of individual opinions greatly affects the conver-
gence of the system, and a large kleads to more disordering.
This phenomenon is in accordance with the theoretical anal-
ysis. Comparing plots (a) and (b) in Fig. 2, the average opin-
ion distance is reduced in a small-world network, as a result
of shortcuts among nodes. Furthermore, we investigate the
critical point of habove which a consensus state is achieved.
For n¼1, agents interact for the same topic, and then, we
can calculate the average initial difference among individual
actions Eðrijð0ÞÞ. We change the dimensionality kto adjust
the initial difference Eðrijð0ÞÞ and obtain the relationship
between the critical point and Eðrijð0ÞÞ, as shown in Fig. 3.
The critical point of hincreases monotonically with the ini-
tial difference of actions. When the system converges, we
have h>Eðrijð0ÞÞ, coinciding with the theoretical analysis.
Topics play a significant role in opinion interactions,
and the final system state correlates with the number of
FIG. 1. Scatter diagrams of final indi-
vidual opinions after 500 time steps,
k¼3 and n¼1. For all three plots,
the underlying network is a regular
network, and each node in the network
connects 10 other nodes. (a) h¼0:1;
(b) h¼0:3; and (c) h¼0:5.
083113-4 Xiong et al. Chaos 27, 083113 (2017)

topics n, as illustrated in Fig. 4. Multiple topics promote
opinion interactions. Agents may not interact towards a topic
but they may update their opinions towards another topic.
With more topics, two agents iand jcan more easily find a
topic ~
vltowards which they take similar actions, i.e., rl
ijðtÞ
¼k
~
uiðtÞ~
vT
l~
ujðtÞ~
vT
lkis small. Then, they can update
their opinions. If n<10, Dð1Þ has a severe drop as n
increases; if n>10, Dð1Þ gradually reaches a plateau. A
large topic number causes more ordering of the system, but
injecting more topics cannot make the system converge.
From the data, we obtain the relation Dð1Þ / nc, where
c1 for all situations. In the power-law fitting, the sum of
squares for error is below 105. When n¼1, Dð1Þ for
h¼0:2 in a small-world network is 35.5% smaller than in a
regular network, but when n¼20, Dð1Þ only decreases by
9.54% in a small-world network. It is concluded that the
effect of shortcuts is restricted when there are a lot of topics
that agents can choose for interactions.
We investigate the evolution of opinion clusters. If the
opinion distance between two agents is smaller than
0:01 maxði;jÞk~
uiðtÞ~
ujðtÞk ¼ 0:01 ffiffiffi
k
p, we consider that
these agents belong to the same cluster. Here, opinion clus-
ters are considered to depend on k, accounting for the effect
of increasing distance with k. This specific measurement of
opinion clusters will not affect the macroscopic characteris-
tics of opinion cluster evolution. We also use other measure-
ments of opinion clusters, and simulation results are
analogous. Figure 5shows the number of opinion clusters as
a function of time in a small-world network. The number of
opinion clusters drops rapidly within 50 time steps, and then
they tend to level off. Large network connectivity leads to
less small clusters during the dynamics, but it has nothing to
do with the relaxation time. From the right plot, more topics
accelerate the evolutionary process, and the relaxation time
is reduced. In addition, for a larger n, the network connectiv-
ity takes a less obvious effect, and the result is similar to the
role of shortcuts. Therefore, the effect of the network struc-
ture depends on the topic distribution. Similar results are
found in a regular network, as shown in Fig. 6, but more
small opinion clusters exist in the system.
We calculate the sizes of the largest opinion cluster and
the second largest cluster in a small-world network, as illus-
trated in Fig. 7. For few topics, the sizes of the largest and
second largest clusters increase at first as tolerance threshold
hincreases. When the second largest cluster is above a cer-
tain level, the size of the largest cluster decreases, and thus
FIG. 2. Final average squared distance
of individual opinions Dð1Þ versus
the tolerance threshold h,n¼1. The
results are averaged over 50 realiza-
tions. (a) The underlying network is a
regular network, and the degree of
each node is 10. (b) A small-world net-
work with the average degree 10 medi-
ates opinion interactions.
FIG. 3. Critical point of hversus the average initial difference of actions
Eðrijð0ÞÞ in a regular network and small-world network, n¼1. The average
degrees of underlying networks are 10. The results are averaged over 50
realizations.
083113-5 Xiong et al. Chaos 27, 083113 (2017)

an inflection point of hcan be observed clearly. When
0:05 <h<0:3, two main opinion clusters coexist in the sys-
tem. When his large enough, the system reaches a consensus
state. For the case in which many topics exist, the second
main cluster emerges with a smaller h. The difference for the
largest cluster is that its size continues to decrease until the
size of the second largest cluster reaches the peak. Large n
impels the occurrence of macroscopic-sized clusters.
Especially, for n¼10 and h¼0:01, the final proportion of
the largest cluster approaches 70%. Multiple topics and
shortcuts in the underlying network increase the chance of
opinion interactions. Agents may more easily find a neighbor
with a similar action towards a randomly selected topic and
then update their opinions. However, for n¼10, when we
decrease the threshold hbelow 0.004, the proportion of the
largest cluster drops greatly to a low value. If a regular net-
work is used as interaction topology, the results are similar,
as shown in Fig. 8. The largest opinion cluster has a much
smaller size in regular networks than that in small-world net-
works, when the tolerance threshold his small.
In addition, we also conduct simulations in Watts-
Strogatz small-world networks
32
and Barabasi-Albert scale-
FIG. 4. Final average squared distance
of individual opinions Dð1Þ versus
the number of topics n,k¼5. Each
plot is an average of 50 simulations.
(a) The underlying network is a regular
network, and the degree of each node
is 10. (b) A small-world network with
the average degree 10 mediates opin-
ion interactions.
FIG. 5. Time evolution of the number
of opinion clusters, h¼0:05 and
k¼5. The underlying network is a
small-world network. Each plot is an
average of 50 simulations. (a) n¼1
and (b) n¼10.
083113-6 Xiong et al. Chaos 27, 083113 (2017)

free networks
33
and obtain similar results with Newman-
Watts networks.
IV. REAL DATA EVALUATION
We use real data to evaluate the effectiveness of our
opinion model for describing social interactions. We col-
lected data from product review websites. In the websites,
users rate products, publish their opinions, and discuss the
products with their friends. Users can vote for products with
five discrete ratings 1–5. Products in the websites can be
regarded as topics in our model, and ratings of users repre-
sent individual actions towards a given product. If a user is
not satisfied with a product, the rating on the product is 1;
otherwise, when the user likes the product very much, the
rating is 5. Meanwhile, the websites also serve as online
social networks in which users exchange opinions with
others. User-product ratings are applied to check whether
our model can predict individual actions through opinion
evolution and the model adequately characterizes the essence
of real interactions.
We used Epinion (epinions.com) and Ciao (ciao.co.uk),
which are well-known consumer review websites. The data can
be freely collected from the websites by a directed robot.
Alternatively, one can conveniently download the datasets from
a scholar’s website (www.cse.msu.edu/tangjili/trust.html),
and the datasets have widely been used for algorithm evaluation.
We obtained 7411 users from Epinion and 7267 users from
Ciao. We extracted relationships among users to construct a net-
work for these two websites separately. The statistics of the
FIG. 6. Time evolution of the number
of opinion clusters, h¼0:05 and
k¼5. The underlying network is a
regular network. Each plot is an aver-
age of 50 simulations. (a) n¼1 and
(b) n¼10.
FIG. 7. Final proportion of the largest and second largest opinion clusters
versus h,k¼5. The underlying network is a small-world network with the
average degree of 10. Each plot is an average of 50 simulations.
FIG. 8. Final proportion of the largest and second largest opinion clusters
versus h,k¼5. The underlying network is a regular network, and the degree
of each node is 10. Each plot is an average of 50 simulations.
083113-7 Xiong et al. Chaos 27, 083113 (2017)

datasets are shown in Table I. The density of actions can be cal-
culated by
the number of actions
the number of users the number of topics :
Therefore, the density in the Epinion dataset is 0.427%,
while that in the Ciao dataset is 0.183%.
Algorithms of recommender systems can predict ratings
of users on products. In the algorithms, each user’s opinion
is expressed as a multidimensional vector, and the feature of
a product is given by a vector with the same dimensionality.
A rating is calculated by the product of the user opinion vec-
tor and topic feature vector. We use two popular metrics:
Mean Absolute Error (MAE) and Root Mean Square Error
(RMSE), to measure the prediction quality of our proposed
model (Appendix B). We randomly select 50% and 75%
data as training data to obtain opinion and topic vectors, and
the remaining data are used as test data. We apply the inter-
action rules of our opinion model and evaluate the accuracy
of action prediction through the remaining data. The random
data selection is carried out 5 times independently, and we
take the average results. Detailed approaches can be found in
Appendix A. To demonstrate performance improvement, we
compare our model with representative algorithms on the
same data: matrix factorization
28
and matrix factorization
with social regularization.
34
We simply write the algorithms
of matrix factorization, and matrix factorization with social
regularization as MF and SoReg, respectively.
Table II shows the performance of these algorithms on
the Epinion and Ciao dataset. Here, the dimensionality of
individual opinions and topic features is set at k¼20, and
75% data are used for training. The trust function is empiri-
cally set as
lhðrl
ijðtÞÞ ¼ 0:03;rl
ijðtÞ<2
0;rl
ijðtÞ2:
((12)
The iteration time is 100 for all three methods. From
Table II, our method has notably lower MAE and RMSE
than traditional methods. Low MAE and RMSE mean
greater accuracy in user action prediction. Our method
reduces MAE by 5.44% compared with MF and 3.29% com-
pared with SoReg for the Epinion data, while the proportion
for the Ciao data is 8.68% compared with MF and 2.35%
compared with SoReg. Thus, our model outperforms MF and
SoReg in predicting user actions, and our opinion model
adequately characterizes the real evolutionary process.
Furthermore, our model has comparable computation as MF
and has much less computational complexity than SoReg
(Appendix C), implying that our model can be integrated in
real social applications. In addition, we also use 50% data as
training data, as shown in Table III and obtain similar
results.
V. CONCLUSIONS
We investigated heterogeneous interaction behaviors in
opinion dynamics under different scenarios. We proposed an
opinion model with external actions under the impact of mul-
tiple topics. Individual opinions were characterized by a
multidimensional vector, of which each element represents a
belief on a certain factor of topics. Topic features were also
denoted by a vector with the same dimensionality. Agents’
actions towards a given topic were measured according to
their opinions and the topic feature. Individual actions were
incorporated into opinion updates with bounded confidence.
We conducted mean-field analysis and numerical simulations
and used real social data to evaluate the model.
From the model, we observe a clear transition from a
disordered state to a consensus state at the critical point of
the tolerance threshold. Increasing the dimensionality of
opinions leads to more disordering, and the critical point for
the state transition also increases. With multiple topics, the
average opinion distance among agents decreases. Small
opinion clusters become less in the final system, and macro-
scopic clusters occur. However, increasing the number of
topics cannot lead to a consensus state. The properties of net-
work topology like the average degree or shortcuts have a
notable effect only when few topics exist. Under heteroge-
neous topic distribution, the size of the largest cluster has
quite different evolution with tolerance.
We applied our model in recommender systems to predict
user actions. The results show that our model achieves better
performance than traditional methods and has lower MAE and
RMSE. The computation complexity of our model is
TABLE II. Performance comparison on two real data sets with 75% training
data (a smaller MAE or RMSE value means a better performance).
Epinion MAE RMSE
MF 0.8415 1.0899
SoReg 0.8228 1.0662
Ours 0.7957 1.0345
Ciao MAE RMSE
MF 0.8085 1.0593
SoReg 0.7561 0.9991
Ours 0.7383 0.9821
TABLE III. Performance comparison on two real data sets with 50% train-
ing data.
Epinion MAE RMSE
MF 0.8640 1.1163
SoReg 0.8421 1.0903
Ours 0.8106 1.0529
Ciao MAE RMSE
MF 0.8419 1.1060
SoReg 0.7816 1.0362
Ours 0.7626 1.0186
TABLE I. Statistics of users and actions of real data.
Statistics User Topic Action Link
Epinion 7411 8728 2 76 116 52 982
Ciao 7267 11 211 1 49 147 1 10 755
083113-8 Xiong et al. Chaos 27, 083113 (2017)

comparable with matrix factorization and is much less than
social regularization. These results prove the effectiveness of
our model in characterizing online opinion interactions.
ACKNOWLEDGMENTS
This work has been supported by the National Natural
Science Foundation of China under Grant No. 61401015, the
Fundamental Research Funds for the Central Universities
under Grant No. 2017JBM013, and the Talent Fund of
Beijing Jiaotong University under Grant No. 2015RC013.
APPENDIX A: APPROACH OF INDIVIDUAL ACTION
PREDICTION
In the product rating data, the number of users is given by
m, and the number of topics (i.e., products) is n. The actions
(i.e., ratings) of users on topics are denoted by a matrix
R2Rmn. Matrix factorization is a classical method of recom-
mender systems and has widely been applied in recent stud-
ies.
35,36
It has been proven that matrix factorization has high
accuracy in predicting user actions. Matrix factorization
decomposes the mnrating matrix into two low-rank matri-
ces U2Rmkand V2Rnk, obviously k<minðm;nÞ.An
element rl
iin Rmeans the rating user igave to topic lfrom
existing data. Each row vector ~
uiin Urepresents user i’s opin-
ion, while each row vector ~
vlin Vrepresents topic l’s feature.
The rating matrix is expressed by R¼UVT, meaning that the
target matrix Rcan be approximated by the product of two
low-rank matrices. Therefore, user i’sactiontowardtopic lis
estimated by ~
ui~
vT
l. To make a prediction of user actions, we
should minimize the following objective function over the
observed ratings:
‘¼1
2X
m
i¼1X
n
l¼1
Ii;l
ðÞ
rl
i~
ui~
vT
l

2
þk1
2kUk2
Fþk2
2kVk2
F;
(A1)
where k1and k2are regularization parameters to avoid over
fitting, and kUkFmeans the Frobenius norm of U.Iði;lÞ
equals 1 if user ihad an action on topic lin existing data,
and it equals 0 otherwise. A local minimum of the objective
function given by Eq. (A1) can be found by performing gra-
dient descent on Uand V
@‘
@~
ui¼X
n
l¼1
Ii;l
ðÞ
~
ui~
vT
lrl
i

~
vlþk1~
ui;
@‘
@~
vl¼X
m
i¼1
Ii;l
ðÞ
~
ui~
vT
lrl
i

~
uiþk2~
vl:(A2)
We integrate our opinion model with matrix factoriza-
tion. In each iteration of gradient decent in matrix factoriza-
tion, we introduce mup opinion update events following the
interaction rules. In an update event, two users iand jare
selected at random. User irandomly selects a topic lon
which user jtook an action rl
jin existing data. Then, user i
takes an action on topic lby ~
uiðtÞ~
vT
l. After observing user
j’s action, user ichanges its opinion ~
uiin terms of our model.
If rl
ijðtÞ¼k
~
uiðtÞ~
vT
lrl
jk<h, user i’s opinion is updated as
~
uiðtþ1Þ¼~
uiðtÞþlðrl
ijðtÞÞð~
ujðtÞ~
uiðtÞÞ:(A3)
For all baseline methods, we use cross validation to
determine their parameters. In all methods, we set k1¼k2
¼0:1. For the social regularization method (SoReg), the
social regularization parameter is set to 5, and Pearson
Correlation Coefficient is used to measure user similarity. In
our opinion model, we empirically set the number of update
events in each iteration at mup ¼0:8m, where mis the num-
ber of users. For each realization, we randomly select 50%
and 75% data as training data
37
to obtain user opinions and
topic features and use the remaining data to test the perfor-
mance of these methods.
APPENDIX B: EVALUATION METRICS
MAE is defined as
MAE ¼1
jRtestjX
rl
i2Rtest
jrl
i~
ui~
vT
lj;(B1)
where jRtestjmeans the number of actions in the test data set
Rtest, and rl
irefers to an action in Rtest.
RMSE is defined as
RMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
jRtestjX
rl
i2Rtest
rl
i~
ui~
vT
l

2
v
u
u
t:(B2)
APPENDIX C: COMPLEXITY ANALYSIS
Let tmax denote the number of iteration times, kdenote
the dimensionality of user opinions, and jRjdenote the num-
ber of existing actions. The computational complexity of
matrix factorization is Oðtmax jRjkÞ. The average number
of existing actions for each user is jRj=m, and therefore, the
computational complexity for calculating user similarity is
Oðm2jRj=mÞ¼OðmjRjÞ, where mis the number of users.
Implementing matrix factorization with social regularization
takes computational complexity at Oðtmax kðjRjþmfþ
þmfÞþmjRjÞ, where fþmeans the average number of
outgoing links for each user and fmeans the average num-
ber of ingoing links. Since mis often larger than tmax k,
more computation time is needed for user similarity calcula-
tion than for matrix factorization. For the Ciao data, fþand
fequal to 15.24, while for the Epinion data, they are 7.149.
In our method, opinion interactions result in computational
complexity for each iteration at OðmkÞ. Therefore, the
computational complexity of our method is Oðtmax kðjRj
þmÞÞ. Since the number of users mis much smaller than jRj,
our method does not lead to larger complexity. Our method
reduces the computation by Oðtmax kð2fþ1Þmþm
jRjÞ compared with matrix factorization with social
regularization.
1
N. Ananikian, R. Artuso, and I. Chakhmakhchyan, Commun. Nonlinear
Sci. 19, 3671 (2014).
2
H. U. Stark, C. J. Tessone, and F. Schweitzer, Phys. Rev. Lett. 101,
018701 (2008).
3
A. Chmiel and K. Sznajd-Weron, Phys. Rev. E 92, 052812 (2015).
083113-9 Xiong et al. Chaos 27, 083113 (2017)

4
E. J. P. Biral, P. F. C. Tilles, and J. F. Fontanari, J. Stat. Mech. 2015,
P04006.
5
S. Tanabe and N. Masuda, Chaos 23, 043136 (2013).
6
Y. L. Shang, Commun. Nonlinear Sci. 19, 3766 (2014).
7
V. Sood and S. Redner, Phys. Rev. Lett. 94, 178701 (2005).
8
S. Galam, Physica A 285, 66 (2000).
9
K. Sznajd-Weron, J. Szwabinski, R. Weron, and T. Weron, J. Stat. Mech.
2014, P03007.
10
N. Crokidakis and C. Anteneodo, Phys. Rev. E 86, 061127 (2012).
11
R. Hegselmann and U. Krause, JASSS - J. Artif. Soc. S. 5, 1 (2002);
available at http://jasss.soc.surrey.ac.uk/5/3/2/2.pdf.
12
A. R. Vieira, C. Anteneodo, and N. Crokidakis, J. Stat. Mech. 2016,
023204.
13
M. Pineda, R. Toral, and E. Hernandez-Garcia, Eur. Phys. J. B 86, 490
(2013).
14
X. Chen, X. Xiong, M. H. Zhang, and W. Li, Chaos 26, 083105 (2016).
15
G. Deffuant, D. Neau, F. Amblard, and G. Weisbuch, Adv. Complex Syst.
3, 87 (2000).
16
D. Urbig and R. Malitz, “Dynamics of structured attitudes and opinions,”
in Proceedings of the 3rd Conference of the European Social Simulation
Association (2005), p. 5.
17
D. Baldassarri and P. Bearman, Am. Soc. Rev. 72, 784 (2007).
18
A. Waagen, G. Verma, K. Chan, A. Swami, and R. D’Souza, Phys. Rev. E
91, 022811 (2015).
19
S. Banisch, T. Araujo, and J. Louca, Adv. Complex Syst. 13, 95 (2010).
20
M. Mas and A. Flache, PLos One 8, e74516 (2013).
21
W. Zhang, C. C. Lim, G. Korniss, and B. K. Szymanski, Sci. Rep. 4, 5568
(2014).
22
J. Lorenz, “Continuous opinion dynamics of multidimensional allocation
problems under bounded confidence: More dimensions lead to better chan-
ces for consensus,” European Journal of Economic and Social Systems 19,
213 (2006).
23
N. E. Friedkin, A. V. Proskurnikov, R. Tempo, and S. E. Parsegov,
Science 354, 321 (2016).
24
A. C. R. Martins, Int. J. Mod. Phys. C 19, 617 (2008).
25
A. C. R. Martins, Phys. Rev. E 78, 036104 (2008).
26
A. C. R. Martins and S. Galam, Phys. Rev. E 87, 042807 (2013).
27
X. Lu, H. M. Mo, and Y. Deng, Chaos Soliton Fract. 73, 98 (2015).
28
Y. Koren, R. Bell, and C. Volinsky, Computer 42, 30 (2009).
29
R. Kannan, M. Ishteva, and H. Park, Knowl. Inf. Syst. 39, 491 (2014).
30
L. Li, A. Scaglione, A. Swami, and Q. Zhao, IEEE J. Sel. Area Commun.
31, 1072 (2013).
31
M. E. Newman and D. J. Watts, Phys. Lett. A 263, 341 (1999).
32
D. J. Watts and S. H. Strogatz, Nature 393, 440 (1998).
33
A. L. Barabasi and R. Albert, Science 286, 509 (1999).
34
H. Ma, D. Zhou, C. Liu, M. Lyu, and I. King, “Recommender systems
with social regularization,” in Proceedings of the 4th ACM International
Conference on Web Search and Data Mining (2011), p. 287.
35
A. F. da Costa and M. G. Manzato, Inf. Syst. 56, 120 (2016).
36
L. Yu, C. Liu, and Z. K. Zhang, Knowl.-Based Syst. 85, 307 (2015).
37
Y. Zhang, G. Zhang, H. Chen, A. L. Porter, D. Zhu, and J. Lu, Technol.
Forecast. Soc. 105, 179 (2016).
083113-10 Xiong et al. Chaos 27, 083113 (2017)