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Estimating the Dynamics of Individual Opinions in Online Communities


Abstract and Figures

How do opinions change as a result of public interactions and exchange of ideas? How does the proliferation of online media influence these dynamics? While theoretical research provides several hypotheses, empirical analysis of opinion dynamics in online communities is lagging. We develop a unique method for quantifying users’ opinions in a social news website and estimate the decision rules that regulate website visit, story posting, voting, and opinion change. We find evidence for significant and nonlinear opinion change as a result of exposure to near-opinions. We also find evidence of learning as people adjust their activity based on the feedback they receive online and strategic reciprocal voting. Incorporating these decision rules in a simulation model we show the propensity of this online community to converge to the majority opinion, and discuss the underlying mechanisms and implications.
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Estimating the Dynamics of Individual Opinions in Online
Armin Ashouri Rad, Ph.D. Student, Grado Department of Industrial and Systems Engineering,
Virginia Tech, Northern Virginia Center, Falls Church, VA, 22043, USA (
Hazhir Rahmandad, Associate Professor, Grado Department of Industrial and Systems
Engineering, Virginia Tech, Northern Virginia Center, Falls Church, VA, 22043, USA
Mehdi Yahyanejad, Project Leader, USC Information Sciences Institute, 4676 Admiralty Way,
Suite 1001, Marina del Rey, CA 90292 (
How do opinions change as a result of public interactions and exchange of ideas?
How does the proliferation of online media influence these dynamics? While
theoretical research provides several hypotheses, empirical analysis of opinion
dynamics in online communities is lagging. We develop a unique method for
quantifying users’ opinions in a social news website and estimate the decision
rules that regulate website visit, story posting, voting, and opinion change. We
find evidence for significant and nonlinear opinion change as a result of exposure
to near-opinions. We also find evidence of learning as people adjust their activity
based on the feedback they receive online and strategic reciprocal voting.
Incorporating these decision rules in a simulation model we show the propensity
of this online community to converge to the majority opinion, and discuss the
underlying mechanisms and implications.
1. Introduction
Humans form their opinion by interacting with each other and diverse information sources.
Therefore, the evolution of a person’s opinion is the result of highly complex feedback process in
which one can affect others’ opinion and get affected by them. As a result of such dynamics,
different trajectories of opinion could emerge in a community (e.g. polarization, plurality, and
consensus) (Van Alstyne and Brynjolfsson 2005; Rahmandad and Mahdian 2011)
Several theories have been proposed to explain opinion formation and its dynamics. In general,
opinion formation models study the adjustment of individuals’ opinion through time based on
their interaction with opinions of others. The classical model of opinion dynamics proposed by
De Groot (DeGroot 1974) assumes each agent adjust his opinion for the next time step by taking
a weighted average on the opinion of others. This model is able to generate consensus or
fragmentation of opinion depending on a few conditions. The key feedback mechanism here
includes the slow conversion of members to the opinions held by the majority (Figure 1, R1).
Friedkin-Johnen model (Friedkin and Johnsen 1990) on the other hand, assumes an individual
holds on to his initial opinion to a certain degree at any time step while also adjusting towards a
weighted average of the community. In the bounded confidence model (Hegselmann and Krause
2002) agents adjust to those opinions that are closer to their own than a threshold. Another
theoretical model (Rahmandad and Mahdian 2011) follows a similar idea on the opinion
formation and introduces motivation as a separate and relevant component. Individuals’
motivation for participating in a community erodes when they are exposed to disagreeable ideas,
and that influences the sharing of points of view different from the majority perspective, leading
to another reinforcing loop (R2 in Figure 1).
Figure 1. Conversion and motivation feedbacks
Despite these theoretical elaborations, the actual process of opinion change is often very
complex, opinions are hard to track, and the influencing factors so many, that empirical
assessment of the main contributing factors and the resulting dynamics have not been attempted.
Yet, an increasing fraction of social interactions happen online in social networks, social news
websites, forums, and other internet based media through posting, sharing, commenting, and
other forms of digital interaction. In such structures, people express themselves and interact
using socially generated digital objects such as “like”, “vote”, “retweet”, and “Digg”. These
online interactions provide us with a valuable source of data on trails of individual choices, and
thus opinions, in the online media that can be used to better understand how their opinions
change through these interactions.
In this paper, we contribute to this literature by providing a blueprint for empirically modeling
the dynamics of online communities. We first propose a novel estimation method to empirically
map individuals into an opinion space so that their opinions could be estimated over time. We
apply this method to data from a social news website and identify the empirical influence
patterns that shape individual opinions over time. We also empirically estimate the decision rules
of users in their visiting, posting and voting, the other key decisions that contribute the dynamics
of this online community. Finally combining the estimated agent rules with the dynamics of
online interactions, we develop a simulation model that projects the opinion dynamics in the
study community and uses that to tease out the most likely opinion trajectories possible in this
community. By combining simulation modeling and empirical estimation, we bridge a common
gap, in which empirical studies of opinion formation only focus on a single aspect of the
problem, and the systems level simulation studies are not empirically grounded.
2. Methods and Data
While qualitative coding for assessing opinion is sometimes applied, there is little literature on
empirically measuring opinions on a large scale. Opinion space (Faridani, Bitton et al. 2010) and
Fraction of
expressions consistent
with majority view
Fraction of
population holding
on majority view
Motivation of
EU Profiler (Trechsel and Mair 2011) are two tools developed to measure and map opinion on a
2-dimensional space based on answers provided by users to some predefined questions. Both use
principal component analysis (PCA) based and reduce the dimension of data collected from users
(on a continuous or discontinuous Likert scale) to two dimensions. These methods require
original surveys to be administered and therefore time series data collection, which requires
multiple measures of the same person over time, is challenging. Here we propose a method for
measuring opinions based on matrix factorization and collaborative filtering techniques that
could be used on any form of unary, integer or continuous rating data commonly available from
online interaction traces of individuals.
In recommendation system literatures, collaborative filtering is defined as “a method of making
automatic predictions about the interests of a user by collecting taste information from many
users” (Wang 2006). Collaborative filtering starts with a common data structure, in which users
and items (e.g. products, movies, stories) are separately identified. It then moves forward based
on a simple assumption: users with similar rating on a set of items have close tastes/opinions to
each other compared to other users who do not show such similarity in their ratings. In order to
find the closeness of opinion, these methods could use matrix factorization methods to form taste
vectors for users and items. By estimating the elements of a k-dimensional taste vector assigned
to each user and item, matrix factorization collaborative filtering techniques find the
taste/opinion values that minimize the difference between the observed and expected ratings. In
essence, this procedure estimate both the user, and the item taste vectors so that similarly rated
items and corresponding raters (users) have small distance in their taste vectors. As a result,
users (and corresponding rated items) with small similarity in rating have larger distance in their
taste vector than those with more similarity. Considering that, optimized taste vector of users
(acquired from such factorization) could be a representative of users’ position with respect to
each other in an opinion space.
We extend the existing matrix factorization methods to the case were ratings are zero-one based
(e.g. “like” button in Facebook) and observations are sparse (not every user has seen every
story). The details of this method are discussed below, but in short, we use a novel method for
identifying the items a user has likely observed (Ashouri-Rad and Rahmandad 2013) and then
define a maximum likelihood estimation procedure for estimating user and item opinion vectors.
Applying this method on time stamped data (using time based rating matrices as inputs) we are
able to map changes in peoples’ opinion through time and study the effect of online interactions
on opinion changes. Using our method, we are also able to capture the effect of opinion distance
on people’s decision rules in their interaction. Equipped with empirically estimated decision
rules we are able to predict the dynamics of opinion change in a community using a well-
grounded simulation model.
Next we describe our empirical setting, a social news website (, and identify the
key empirical decision rules used by its users to interact with each other. Then we map users’
opinion on a 2-dimensional space through time and estimate the identified rules based on opinion
information and other data we collected from the website. Finally, we discuss the results of
simulations that benefit from the statistical estimation work we report in the rest of the paper.
Balatarin is a social news website where users can post stories (i.e. links to different news,
websites, videos and other medias; these stories are the “items” we discuss in the methods
section), read other users’ stories, and vote or comment on the posted stories. Balatarin as the
largest social news website for the Persian-speaking community includes over 30000 registered
users, a million stories, thirty million votes and several million comments (Bandari, Rahmandad
et al. 2013). Popular stories in Balatarin (those with more than a specific threshold) get promoted
to its first page and when users visiting the website have the choice of reading first page stories
or recently posted stories. We have data only on stories (story ID, posting User’s ID, Time of
posting, user identified story category (political, economics, sports, social, etc.)) and votes (story
ID, time of vote, voting user’s ID). Using an algorithm we report elsewhere (Ashouri-Rad and
Rahmandad 2013) we rebuild the history of Balatarin in a simulation environment, calculate
status of stories and reconstruct users’ online behavior over time. Doing that, we are able to
collect different forms of data on users and stories, such as the location of each story at any
moment, pages users most likely have visited and stories they have read (with some probability)
at each point in time, online and offline time of each user, and almost any other key feature of
user behavior online. Using this detailed profile for every user from the beginning of this
community, we are able to identify and calibrate the empirical decision rules user use in visiting,
voting and posting.
3. Analysis and Results
3.1. Mapping users’ opinion
We use the collaborative filtering idea to measure and map users’ opinion vector on an opinion
space through time. We form daily time windows and estimate users and stories opinion vectors
on user-story vote data of the day. Doing so, we have enough information to capture the effect of
reading stories with similar or diverse opinions on users’ opinion change. We assign a 2
dimensional taste vector to each user and story, along with a fixed effect parameter for each user
and each story to capture the attractiveness of each published story and the amount of activity
each user has on the website regardless of their taste/opinion location. We use the following cost
function to factorize the user-story vote matrix to opinion vectors:
Argmin,(,) = 1 × ∑ ∑ ,× log ,×...+1,×
log 1,+,×1... (1)
Where × is the user-story vote input matrix (,= 1 if user voted for story and ,= 0
otherwise), and represent the number of users and stories respectively. × is the
exposure probability matrix which is estimated to represent the probability that each user has
seen each story. This weight matrix is calculated based on the algorithm we proposed on
(Ashouri-Rad and Rahmandad 2013) and details of the algorithm are available in Appendix A
and ()= 1/(1 + ).
In (1) . and . are the opinion vectors (for user and story respectively) we need to estimate.
The cost function defined above has two desirable features. First, it is the log-likelihood function
for the observed voting pattern, providing maximum likelihood estimates for and vectors.
Moreover, the optimization problem is convex, therefore despite the high dimensionality of the
problem (with and users and stories, we have ×× parameters to estimate),
convergence to the optimum solution is possible with simple gradient descent methods.
However, due to the nonlinearity of the problem, with the same input data we may get different
optimum values for and with the same payoff, all results of rotations and scaling of and
matrices. In order to track the changes in users’ opinions through multiple time windows, we
need to have the same scaling and rotation across different time windows. We therefore include a
fraction of stories from the previous time window in the next input matrix and fix their taste
values to those estimated in the previous window. As a result, the estimated opinion vectors in
the current window will follow the rotation and scaling of the previous one, and thus the
estimates remain comparable across time.
Using limited memory BFGS optimization algorithm (Liu and Nocedal 1989; Kalhor,
Akbarshahi et al. 2013), we calculate opinion vectors of users on a daily basis for a week (longer
time horizons are feasible and desirable, but core insights are similar). Figure 2.a shows the
opinion values of users on one time window, Figure 2.b represents the change in opinion values
for 3 selected users over a one week period (only 100 users mapped on this plot).
a. Opinion values of users on one time window
b. Tracking opinion values for 3 selected users over time
Figure 2. Opinion space
Using optimized opinion vectors, we have the ingredients to estimate how individual opinions
interact with story opinions to influence user activity and retention. We also can estimate how
opinions change as a result of different factors. In the next part we discuss the statistical models
we use to extract these relationships.
3.2. Statistical Estimation
In this section we discuss the key decision rules that shape the basic dynamics of communication
and interaction in online communities in general, and Balatarin in particular. These functions
identify how users change their opinion due to exposure to different stories, how they decide to
show up online, what influences the rate by which they post stories, and how they decide to vote
for a story.
3.2.1. How do opinions change?
Using extracted opinion vectors over time, we extract the impact of reading stories (with
different embedded opinions) on users’ opinion using an adaptation of the bounded confidence
model (Hegselmann and Krause 2002). Bounded confidence model proposes that a change in the
opinion of an agent may happen due to interactions with other agents “whose opinions differ
from his own not more than a certain confidence level”. Such model is able to generate different
trajectories of opinion in a community (i.e. polarization, plurality, and consensus) using only two
parameters: a confidence interval and an opinion dependent bias factor. Using a one dimensional
opinion space, bounded confidence model assumes all others in the confidence bound of the
agent are equality likely to influence the focal person’s opinion. Also, it does not consider any
differences among agents in how likely they are to change their opinions (e.g. the heterogeneity
of stubbornness).
We model the impact of interacting with other opinions using the same idea of bounded
confidence, by studying the change in user’s opinion through time caused by reading stories with
opinions close to her/his. We estimate this impact by tracking the change in each user’s opinion
through time (in daily time windows) and predicting that change using the opinions embedded in
the stories the user read in the time window. Specifically, consider capturing the impact of
reading stories on users’ opinions in a 2 dimensional space, where () indicates 
dimension of opinion for user at the time . (1) is the distance between user and
story opinion value for each dimension . , is the exposure weight we introduced above and
is the confidence threshold in the  dimension. We define a fixed effect, ,, which
captures the changes in each user’s opinion during the week of study which is due to factors
external to the interactions in the online community. By including this fixed effect we control for
the time trends in opinions and stories which are independent of the causal relationship between
the two. Finally represents the impact of reading stories on user’s opinion.
×,×(1)(1)<+,= 1,2
 (2)
The regression above can be estimated using any values of the confidence bound size , which
determines the stories that influence the focal actor’s opinion. We select this parameter by
running multiple regressions with different values of confidence bound ( and ) and finding
the combination that provides the best fit based on R-squares. The values 0.25 and 0.5 were
chosen as the best fit for and respectively, and these values were used in reporting the
results. Table 1 provides regression coefficients for the impact of reading on user’s opinion in
each dimension (R-squared: 0.8176, number of data points: 11507 on 1st dimension and R-
squared: 0.8706, number of data points: 12359 on 2nd dimension).
Estimated Value
(Standard Deviation)
0.09271 (0.02237) 3.45E-05
0.03342 (0.01002) 8.59E-04
Table 1. Regression coefficients, standard errors and p-values for variables in model (2)
Both of these estimates are statically significant and the directions of effects are consistent across
the two regressions. The magnitudes are different, which is expected because each dimension on
the opinion space captures a very different characteristic. Interpreting the coefficients, we can
say that when user read story (,= 1) if the taste value of story on dimension () falls
in user’s confidence bound ( <), story changes user’s opinion (on dimension ) as
times its distance from user’s opinion value on that dimension (× ). These
effects are rather strong but not unrealistic: they require dozens of slightly different (but not far-
fetched) stories to change the opinion of an individual in each dimension.
3.2.2. How often users visit the site?
Users’ activity in any online community is directly related to how frequently they go online.
Number of times each user visits Balatarin is thus another decision rules we are interested in
estimating. In analyzing this measure we build on theories of learning in repeated choice to
specify the factors that may influence user visiting pattern and estimate these effects statistically.
In summary, we develop a Poisson regression model to capture the impact of different
parameters on daily rate of visiting the site. Setting the number of times each user visits the site
in each day as the dependent variable, we include following independent variables in our
regression: time spent online in the previous day (to capture both habituation and saturation
effects), proximity of opinion between user and stories s/he read (effect of opinion feedback on
motivation to visit) in the previous day, average number of votes each of his posted stories
receive (effect of reinforcement from peers), and a dummy variable for days of week (to capture
weekly cyclical patterns). Details of these factors follow.
A fundamental finding of learning literature is the law of effects: actions that are reinforced by
some reward are more likely to be repeated. One such reward in an online community is reading
stories that are consistent with one’s opinions. Proximity of opinion between user and stories
(calculated by equation (4) for each story-user pair) combines the exposure weight (whether the
user has read the story) with how favorable the user’s opinion is towards the story. Larger (closer
to one) values for this function indicate a user who has a strong opinion on some topic and reads
stories that support that strong opinion. Values close to zero suggests the stories the user has read
are inconsistent with her opinion. Summing over all the stories in the period, this measure
provides an estimate of the positive feedback users received because of reading stories that are
consistent with their opinions. This measure is in fact close to the total number of stories that the
user positively votes for but more precisely measures the impact of reading attractive stories
even if they are not voted for. We hypothesize that users will be prone to enjoy reading such
stories and thus be more likely to visit the site.
Another rewarding reinforcement for social news website users is when individuals post a story
and receive positive feedback (votes) on that story from other community users. We capture this
effect using the average of votes received on stories posted by a user in the previous time
window. Online time variable is the amount of time user is active on Balatarin in each day and
controls for the fact that the number of visits online may change due to spending more/less time
online the previous day. Day of the week has a significant effect on users activity in the website
(higher activity in weekend compared to working days), which we take into account using
dummy variables. We also add fixed effects () for users to control for variations in individual
motivation and availability. Equation (3) specifies the regression model formally:
=++ ×+×+
 =,×... (4)
Table 2 provides regression coefficients for model (3) (Conditional R-squared: 0.4014092,
number of data points: 8531):
Estimated Value
(Standard Deviation)
0.6784 (0.02797) 0.00
1.78E-05 (1.58E-06) 0.00
1.26E-04 (4.89E-05) 0.00993
3.16E-04 (5.55E-04) 0.56928
Table 2. Regression coefficients, standard errors and p-values for variables in model (3)
All coefficients except the one for “average number of votes user’s posted stories received” are
statically significant. Days of week dummy variables are all statistically significant. In short,
individuals are sensitive to the feedback they receive from reading other stories, which increases
their likelihood to go online.
3.2.3. How often individuals post stories?
Number of stories users post per day is another decision rules critical to the working of an online
community. Building on similar learning arguments as in the previous estimation, we use another
Poisson regression to capture the impact of following independent variables on that rate:
proximity of opinion between user and stories (equation 4), average number of votes each of
posted stories receive, and number of posted stories in the previous time window. Again we add
fixed effects ( ) on users to expel the control for the effect of individual motivation, and
dummy variables for days of week. The following equation represents the formal regression
=++ ×+
Table 3 provides regression coefficients for model (Conditional R-squared: 0.5567877, number
of data points: 3767):
Estimated Value
(Standard Deviation)
-3.325 (0.08663) 0.00
-0.068 (0.01555) 1.11E-05
0.00138 (14E-4) 0.00
0.0011 (0.0010) 0.2861
Table 3. Regression coefficients, standard errors and p-values for variables in model (5)
Again all the variables except “average number of votes user’s posted stories received” are
statically significant. Days of week dummy variables are all statistically significant. The results
point to a saturation effect (posting many stories in the previous day is likely to reduce the stories
an individual will post today) and reading stories that are consistent with an individual’s opinion
increase the chances of posting more stories.
3.2.4. How do users vote?
Different factors affect users’ votes to stories. Given the procedure we used for estimating
opinions, it is natural to expect the proximity of user’s opinion to a story to be an important
explanatory variable for explaining the voting patterns (proximity is calculated using equation 4).
Other factors may also play a role. First, much research in diffusion and social influence suggest
that the popularity of an item is likely to increase its chances of getting a vote because stories
that have already received many votes signal quality and social support. Stories may be from
categories inherently more attractive for a user. There is also potential for strategic voting, in
which users vote for stories posted by others in a reciprocal tit for tat fashion. Finally, stories
location on the website pages could influence their attractiveness. Using the information we
obtained from mapping users and stories opinion and reconstruction of user navigation patterns
we are able to measure the impact each of these factors on users’ voting behavior. We estimate a
logistic regression model that predicts the probability of voting for each story if it is exposed to a
user (,is more than zero (or assumed weight for missing values)):
Number of votes is a good proxy for stories’ popularity at the time that the story is exposed to
the user. Place of a story is measured based on which page and row it is placed in and is a good
indicator of its visibility. In Balatarin each story is assigned to a category (such as political,
social, sport, and art) and since users have different interests and concerns, category of the story
is a potentially relevant factor for user’s voting. We choose the category that contains the user’s
most voted stories in the past as her favorite, and use a dummy variable (equals to 1 if the story
belongs to that category and 0 otherwise) to capture mentioned impact. There is a subtle
complication in using the proximity measures for assessing the closeness of story in the opinion
space for the current regression. Specifically, there is circularity in the logic if the opinion values
are estimated using the same votes that we attempt to predict in the current regression. To
circumvent this problem we first set-aside 20% of user-story vote data (and set the corresponding
exposure weights to a small value (0.05)). We estimate the opinion space measures using the
remaining data, and then estimate the logistic regression using the 20% of data not used in the
estimation of opinion space. Table 4 provides the result (McFadden's R-squared: 0.250039,
number of data points: 31372)1 :
Estimated Value
(standard deviation)
-2.26 (0.0137) 0.00
9.98E-04 (2.33E-04) 1.86E-05
2.06 (0.0457) 0.00
0.695 (0.0163) 0.00
-1.63E-03 (4.50E-05) 0.00
3.39 (0.0288) 0.00
Table 4. Regression coefficients, standard errors and p-values for variables in model (6)
Fixed effects are embedded in individual and story opinion vectors and thus do not need to be
separately captured. All of the effects are statistically significant. Not only the proximity
measure is very significant, but also individuals are more likely to vote for stories that already
have more votes. Users have a very strong tendency to engage in reciprocal voting. Favorite
categories receive significantly more votes, and stories better placed on the page (lower page
number and towards the top of the page) are more likely to receive votes.
4. Simulating an Empirically Estimated Online Community
Equipped with estimates for the key actions users take online, we now turn our attention to
building and simulating an agent-based model that is grounded in Balatarin data. Specifically, we
consider users of the website who visit the site, post stories, read and vote for stories, and change
their opinions due to reading different types of stories. Over time opinions and the motivations of
individuals change, leading to different levels of activity by different users.
Another key feature of the simulation model replicates the core feature of social news websites,
in which the site promotes popular stories (those with votes more than a specified threshold) to
its first page. Promoted stories stay on the website for five days but those that couldn’t make it to
the first page will be removed after only one day. Therefore, we define two states for stories in
the model (New Page and First Page) with different life cycles (Figure 3.a). User visiting the
website could choose between reading and voting on first page stories or the recently posted
stories (New Page). Figure 3.b shows the different states and transition defined for users.
1 Due to collinearity, we residualized proximity over other variables before using it to fit the model.
a. Stories’ state charts and transitions
b. Users’ state charts and transitions
Figure 3. Simulation states and transitions
We simulate a population of 1500 users (close to the core group of active users on Balatarin) and
do not consider new entries into the system for simplicity. Incidentally, due to technical issues
that Balatarin faced which lead to stopping the acceptance of new users for a long time, this
scenario is close to Balatarin’s experience, but different entry rates could be easily incorporated.
We consider a two dimensional opinion space which is consistent with our statistical analysis.
Besides the decision rules specified in the previous section, several other parameters are directly
estimated from the data or from the regression results and residual distributions. These are
summarized in Table 5 and include the distribution of users’ opinion and their activity, diversity
of story opinion from its publisher, distribution of stories’ attractiveness, average time users
spend on the website in each online session, rate of reading stories, probability of reading first
page stories compared to recently published ones, and number of active users per day. We then
run the simulation to project the opinion dynamics in Balatarin based on the empirically
estimated decision rules.
Parameter Estimated value
Distribution of users’ opinion on 1
dimension Normal(mean=0.26,sd= 2.23)
Distribution of users’ opinion on 2
dimension Normal(mean=-0.24,sd=2.38)
Distribution of users’ activity Normal(mean=3.24,sd=5.53)
Deviation of story from its publisher’s opinion in 1
dimension Normal(mean=0,sd=1.15)
Deviation of story from its publisher’s opinion in 2
dimension Normal(mean=0,sd=1.36)
Distribution of stories’ attractiveness Normal(mean=-6.71,sd=1.23)
Average time of online session 34 minutes
Rate of reading stories 44/hour
Probability of reading first page 41%
Table 5. Parameter values estimated from data
Figure 4 presents the timeline of changes in agents’ opinion in each dimension for a single
simulation run over 10 years. The individual metrics are color coded based on initial opinions of
agents in each dimension separately.
a. Distribution of agents’ opinion on 1st dimension over 3650 days (10 years)
b. Distribution of agents’ opinion on 2nd dimension over 3650 days (10 years)
Figure 4. Distribution of agents’ opinion
Simulation results show majority consensus shapes in the 1st dimension after nearly 5 years and
in 2nd dimension after only 2 years. In both opinion dimensions people close to the majority
opinion (with middle-ground initial opinions) converge quickly as they post stories that support
each others’ opinion and reinforce a single majority position. The convergence process includes
two stages, early on the individuals change their opinion only very slowly because the stories in
their confidence bound are not many. As they get closer to the population mean, they are
exposed to an increasing pressure towards conformity, and thus their speed of convergence
increases. The convergence speed ultimately goes down due to the incremental opinion updating
process. Those with minority views (at very low or high levels of initial opinion) may take a very
long time to converge to the majority. For one thing, exposed to contrary items, their motivation
is likely to go down leading to fewer visits to the website and thus a reduction in their exposure.
Moreover, being in the extreme positions, they are likely not to read many stories that they find
credible enough to change their opinion (i.e. few stories in their confidence bound). In fact a
handful of users will not adjust their opinion to the majority during the 10 years of simulation as
they find very few credible stories and are generally inactive.
We include the possibility of users posting stories not embedding the exact opinion as the
individual holds. This is different from previous theoretical models (Friedkin and Johnsen 1990;
Hegselmann and Krause 2002; Van Alstyne and Brynjolfsson 2005) but is motivated by a couple
of observations. First, in many cases individuals may not find stories that they fully agree with.
Moreover, they may perceive the inherent opinion of that story different from how others
perceive it. In fact we find clear evidence from our estimation of opinion space that confirms this
intuition. The resulting deviations are captured in the parameters “Deviation of story from its
publisher’s opinion” in Table 5 and calculated by empirically comparing the opinions of posted
stories and the users who posted them.
The impact of this natural noise is potentially important. On the one hand, the community will
never converge to a single opinion because there is always a random noise around the mean
community opinion sustained by randomness in expression of opinions. Moreover, no user can
sustain an opinion very different from the core group for a very long time horizon, because even
the extremely distant individuals are infrequently exposed, by chance, to a few stories that are
not far from their opinion but are biased towards the community consensus. These perspective
over the long run change the distant users’ opinions towards the consensus, albeit very slowly,
until they get close enough that the majority falls close to their confidence bound and then they
are quickly attracted to the consensus group.
In fact, a similar mechanism makes it harder to observe the emergence of competitively
polarized communities, where the members converge to two or more smaller sub-communities.
The randomness in posted stories’ opinions creates a mechanism for communication between
these sub-communities and triggers a slow migration from smaller to larger communities. Figure
5 shows this effect: if there is no deviation in the opinions expressed in the stories, compared to
those of the posting user, the community fragments to many small sub-communities, each with
slightly different, but fully homogenous opinions. In the absence of randomness in expressed
opinions, individuals converge to the exact same opinion values and thus stop hearing from other
sub-communities which are outside of groups’ confidence bound. Disconnected from each other,
each sub-community in this hypothetical case will continue to post and be impacted by stories
that are fully consistent with their opinion and ignoring the rest of the community.
a. Distribution of agents’ opinion on 1st dimension over
100 days without deviation of stories
b. Distribution of agents’ opinion on 2nd dimension over
100 days without deviation of stories
Figure 5. Distribution of agents’ opinion without deviation of stories
In fact, within the parameter ranges specified for motivation, posting, and other effects, the
model is quite sensitive to the deviation distribution, and is able to generate different trajectories
by only altering the standard deviation value. Figure 6 show different formation including
polarization (6-a), plurality (6-b), diversity (6-c), and full consensus (6-d) predicted by the model
with different values of standard deviation in deviation distribution. Such sensitivity shows the
importance of misunderstanding in interaction on future opinion formation.
a. Polarity (1st dimension SD=0.65)
b. Plurality (2nd dimension SD=0.55)
c. Diversity (1st dimension SD=5)
d. Consensus (2nd dimension SD=3)
Figure 6. Different opinion formation generated by deviation distribution
5. Discussion and conclusion
The current paper combines statistical estimation and dynamic modeling to better understand the
dynamics of online communities. Specifically, we provided an automated method for extracting
user opinions in online communities based on their interaction pattern, used that method to
estimate the online opinion changes, and showed how individual opinions change as a result of
exposure to stories not too far from them. We also estimated the underlying decision rules that
guide individual participation in online communities including visiting the website, posting
stories, and voting for stories. Using these findings we then built an agent-based model that is
fully specified based on realistic parameter ranges. Analysis of this model showed that the
dominant mode of operation in the social news website we analyzed is majority domination,
where the majority of participants converge to a single region of the opinion space, and the
outliers become relatively inactive in the system. We also show that other modes of behavior,
including competitively polarized, diversity, and complete consensus are also feasible. An
important parameter in determining the dominant mode of behavior is the distance between the
posted stories embedded opinion and the opinion held by the individual who posts the story.
The core feedback mechanisms we empirically estimate are simple, but powerful. First,
individuals change their opinions as they consume media. In turn, they produce media for the
consumption of others (in our case through posting stories they find from the internet) which is
closely related to their own opinion. As a result the majority tends to convert more of the
community members, further reducing the diversity of opinions expressed in this community. A
second mechanism relates to erosion of motivation among the outliers of the community.
Observing few attractive stories and getting little support, these individuals are more likely to
leave the community, which will further reduce the heterogeneity of the opinions expressed.
Qualitative evidence in support of these mechanisms are abound, however, measuring them
empirically has not been simple. Empirically we find the first feedback to be pretty strong:
individual opinions change in time scales that can be measured within one week of exposure to
different stories. The change in motivation was also significant, but less so, and more likely
unfolds over months rather than days. We expect to extend the analysis to a longer time horizon
that captures these longer-term effects.
From the emergence of cultural norms to evolution of public opinion, the reinforcement of
prejudices and the construction of tolerance, opinion dynamics are central to many questions
researchers and policy makers care about. With the increasing proliferation of digital
interactions, new questions and opportunities for research emerge. Mapping the opinions in
online communities using the readily available digital traces of user actions provides a unique
opportunity to empirically assess how people change their opinions, how they express
themselves, and how their motivations change. One can map the path to formation of new norms
in a community, and see how opinions which were initially considered extreme manage to
become the norm in a large parts of the society. While our analysis did not focus on the role of
specific individuals, much heterogeneity may exist in the opinion adjustment speeds, and that
may create novel perspectives into the role of opinion leaders and outliers in changing
community dynamics.
Current online community structures leverage two alternative designs to prioritize information
sharing and avoid overload: Filtering and ranking systems. Ranking systems promote the
majority’s point of view (like or and leads to majority dominant
online environment. Filtering methods create a bubble around similar opinions by personalized
filtering (e.g. Netflix). This research can also be utilized to design online communities that are
better with respect to some social utility measures. For example the filtering of stories presented
to individuals in a social news website could be personalized and take into account the impact on
the polarization, motivation, and long-term diversity of the community. In fact experiments
insides a simulation model that is well calibrated can be used for filtering the best design
modifications, which could then be tested empirically and assessed using the metrics and
methods we demonstrated. This approach can extend the current filtering and ranking methods
significantly and provide opportunities not currently available.
Yet, converging on a single preferred dynamic pattern is hard and likely premature. Studies on
opinion formation indicate that each reference mode has its own advantages and pitfalls. For
example, consensus decreases the conflict but leads to lack of variation in opinions and
encouraged group-think, bias, and discrimination of minority. On the other hand, polarized
communities increase the interaction among competing groups and force them to challenge
beliefs and assumptions, but increase the conflict which reduces user motivation, derives out the
less outspoken members, and is likely not stable over long horizons. The dynamics within a
single platform, such as Balatarin, would also interact with the other platforms. For example
individuals who are dissatisfied with one platform may leave it for one which is more consistent
with their opinions or design preferences. Therefore the sustainability of any new design is also
constrained by the economics of the media platform.
6. References
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7. Appendix A
Exposure weighting formula:
 (7)
Where , is the corresponding probability of story being exposed to user . In our case
formula (7) means how likely the user seen the stories in between each pairs he voted on a same
page. Here and define the slope of decrease in weight by getting far from pair of voted
stories. We set == 0.01 and consider a small weight (0.05) for missing values. We put
 = 1 for each pair of (,) that have  = 1, (voted stories definitely exposed to the voter).
is the distance (number of stories) between story and previous voted story in that page, and
has the same definition for current voted story. There are number of alternative ways to deal
with missing data (Pan, Zhou et al. 2008; Balali, Zahraie et al. 2012; Jalali and Ghoddusi 2013),
but based on our available data on Balatarin we proposed a behavior reconstruction process that
gives us a good estimation on such binary missing data.
... The opinion exchange models in the HK and DW models account for these bounded confidence notions. The work in [19] addresses statistical estimation of the bound of confidence. In [20] agents are treated as Bayesian decision-makers and Bayes' risk error has been used to estimate the bounds. ...
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