=Paper= {{Paper |id=Vol-2706/paper0 |storemode=property |title=Calibration of an agent-based model for opinion formation through a retweet social network |pdfUrl=https://ceur-ws.org/Vol-2706/paper1.pdf |volume=Vol-2706 |authors=Loretta Mastroeni,Maurizio Naldi,Pierluigi Vellucci |dblpUrl=https://dblp.org/rec/conf/woa/MastroeniNV20 }} ==Calibration of an agent-based model for opinion formation through a retweet social network== https://ceur-ws.org/Vol-2706/paper1.pdf

Calibration of an agent-based model for opinion
formation through a retweet social network
Loretta Mastroenia , Maurizio Naldib and Pierluigi Velluccia
a
Dept. of Economics, Roma Tre University, Via Silvio D’Amico 77 00145 Rome, Italy
b
Dept. of Law, Economics, Politics and Modern languages, LUMSA University, Via Marcantonio Colonna 19 00192 Rome,
Italy

Abstract
Calibration of agent-based models (ABM) for opinion formation is needed to set their parameters and
allow their employment in the real world. In this paper, we propose to use the correspondence between
the agent-based model and the social network where those agents express their opinions, namely Twitter.
We propose a calibration method that uses the frequency of retweets as a measure of influence and allows
to obtain the influence coefficients in the ABM by direct inspection of the weighted adjacency matrix
of the social network graph. The method has a fairly general applicability to linear ABMs. We report a
sample application to a Twitter dataset where opinions about wind power (where turbines convert the
kinetic energy of wind into mechanical or electrical energy) are voiced. Most influence coefficients (76%)
result to be zero, and very few agents (less than 5%) exert a strong influence on other agents.

Keywords
Opinion formation, Agent-based models, Twitter, Calibration

1. Introduction
Agent-based models (ABM) are increasingly used to analyse opinion formation (see the survey
in [1]), as an alternative to econophysics models [2]. Some examples with general applicability
are described in [3, 4, 5, 6]; they are also applied to study specific phenomena such as equality
bias [7] or personal finance decisions [8, 9].
However, models with no application to real world data may be too abstract. In the mathemat-
ical model describing the interactions among agents, we need to set the parameters governing
those interactions, i.e. to calibrate those models. Calibration allows us to obtain realistic ex-
pectations about the behaviour of a social group. Very few studies have attempted to calibrate
agent-based models. We can group them into two classes, where data are obtained respectively
through a laboratory experiment of from the observation of a real social network.
One of the first example of the former class is the celebrated Friedkin-Johnsen model [10],
which has been evaluated through data collected on small groups in a laboratory [11], where
the authors asked their subjects to estimate the extent to which each other group member
influenced their final opinion by means a mechanism based on a reward paid in poker chips.
Another example is provided in [12], where participants in the experiment expressed their
WOA 2020: Workshop “From Objects to Agents”, September 14–16, 2020, Bologna, Italy
" loretta.mastroeni@uniroma3.it (L. Mastroeni); m.naldi@lumsa.it (M. Naldi); pierluigi.vellucci@uniroma3.it
(P. Vellucci)
© 2020 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
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http://ceur-ws.org
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CEUR Workshop Proceedings (CEUR-WS.org)

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Loretta Mastroeni et al. 161–173

opinion about the best location for a new leisure center (on the line across two towns), and
the influence could be observed through the evolution of one’s opinion after examining the
opinions of others.
Influence in real contexts has instead been studied through social networks. An advice
network inside a manufacturing company was considered in [13] on the basis of the data
provided in [14]. The influence of any individual was assumed to be proportional to the number
of people who seek advice from him/her. A political context, namely the American Senate, was
instead investigated in [15], where co-sponsorship of bills was taken as a measure of influence.
The opinion value for each senator was assumed to be the fraction of votes when he/she was
present and voted with the majority. Finally, a co-habitation context, namely a university
student dormitory, was analysed in [16], where the Social Evolution dataset enclosed in [17]
was employed. A similar context is also the stage for the study contained in [17], where surveys
were conducted monthly on social relationships, health-related habits, on-campus activities and
other issues.
Here we wish to propose a method to calibrate agent-based models for opinion formation,
considering data extracted from an online social network, namely Twitter. This contrasts with
the contributions appeared in the literature so far, where just physical social networks have
been considered. The abundance of data appearing in online contexts makes them a natural
choice to look at for our goal. We measure the influence by the frequency of retweeting, which
makes our methods applicable to any social network where reposting of opinions is allowed.
Our major original contributions can be summarised as follows:

• we propose a calibration method based on the adjacency matrix in an online social
network;

• we provide a systematic assessment of its applicability, considering a taxonomy of agent-
based models;

• we demonstrate its application using Twitter opinions on wind power;

• for that specific context, we show that, though most actors exert an influence, very few
exert a strong influence (i.e., being retweeted more than once);

• for that specific context, we show that the influence of any agent is limited to one other
agent in most cases so that the matrix of influence coefficients is sparse.

2. The retweet network
We base our calibration on Twitter data. In this section, we report some basic information about
Twitter and using retweets to calibrate an agent-based model.
Twitter is a popular messaging service, with 330 million monthly active users (see https:
//www.oberlo.com/blog/twitter-statistics), where messages (aka tweets) are no longer than 140
characters [18]. Message receiving uses an opt-in mechanism: you decide to follow somebody
and receive all his/her updates. Messages may include hashtags, i.e., # symbol terms that associate

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Loretta Mastroeni et al. 161–173

14
𝐵 𝐶
30

13
25
3 𝐴 2
4
5

6
𝐸 𝐷

Figure 1: Toy retweet network

a topic to the message: all messages concerning a specific topic (as identified by the hashtags
included in those messages) can be retrieved at once by searching for that hashtag.
Twitter allows retweeting a message, i.e., reposting somebody else’s tweet. This is a sign
of support for somebody else’s opinion, stronger than just following him/her because it is an
uncritical sharing of his/her opinions (retweeting through the Retweet button does not allow
to add comments). It is also specific because it concerns a single message. Being retweeted is
a sign of influence. We are using retweets to measure the influence of somebody’s opinion
on other participants in a social network. In particular, we build a retweet network. Retweet
networks have been designated (see Chapter 4 of [19]) as a major tool to analyse the influence
of twitterers. In our network each node represents a twitterer (an agent in the corresponding
agent-based model) and an edge from node 𝐴 to node 𝐵 means that 𝐴 has been retweeted by
𝐵. The weight associated to that edge is the number of retweets: the weighted out-degree of
a node is the measure of the associated agent’s influence. In the end, we obtain a weighted
directed network. It is to be noted that this network allows us to quantify the influence of each
agent on each other agent, which is what we need in our any-to-any agent-based model, rather
than the overall influence of a twitterer (as in [20]).
In Figure 1, we show a toy retweet network made of five twitterers to illustrate this concept.
We see that 𝐶 has been retweeted 13 times by 𝐴, and 14 times by 𝐵, but has never retweeted
either 𝐴 or 𝐵. Retweeting is not symmetric in general.

3. Calibration methodology
Our aim is to set the parameters defining an agent-based model for opinion formation, i.e., to
calibrate the model, exploiting the data retrieved from Twitter. In this section, we describe our
calibration method and highlight its applicability.
For the time being, without loss of generality, we refer to an agent-based model such as
described in [5], where the generic agent 𝑖 features a quantity 𝑥𝑖 (𝑡) representing its opinion at
time 𝑡. Then we denote the opinions of all the agents at time 𝑡 by the vector 𝑥(𝑡); its evolution

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over time follows the state updating equation:

𝑥(𝑡 + 1) = 𝑆𝑥(𝑡), (1)

with 𝑆 being the matrix of influence coefficients. Namely, the element 𝑠𝑖𝑗 of 𝑆 describes the
effect of the opinion of the agent 𝑗 on the opinion of the agent 𝑖.
The process leading from Twitter activity to the matrix of influence coefficients is made of
the following phases:
1. scraping Twitter data;

2. building the retweet network and extracting the adjacency matrix;

3. mapping the edge weights on the influence coefficients.
The first phase consists in retrieving the list of tweets/retweets concerning a specific hashtag (or
any combination of hashtags and keywords). This phase is often referred to as Twitter Scraping
and can be easily accomplished using Twitter’s API (Application Programming Interface),
accessible upon opening a Twitter developer account. Tweets were retrieved using the R package
twitteR [21]. The search index has a 7-day limit, which means that only tweets posted in the
latest seven days will be retrieved. Our inspection interval falls in the week ending on December
9, 2019.
As to Phase 2, we build the retweet network as in Section 2, i.e. a social network where the
nodes are the twitterers, who represent the agents in our ABM. That social network is described
by its weighted adjacency matrix 𝑀 , whose generic element 𝑚𝑖𝑗 is the number of times that
the twitterer 𝑖 has been retweeted by the twitterer 𝑗.
For the computation of influence coefficients (Phase 2), we assume that the set of twitterers
is the set of agents. If the order of elements in the two sets is not the same, a permutation on
either set is needed before applying the calibration method. Since the weights of the edges
in the retweet network represent the influence exerted by the twitterers (i.e., the agents) on
one another, we obtain the matrix of influence coefficients in the ABM by the simple equation
𝑆 = 𝑀 , if no normalization is needed. However, it is to be noted that the adjacency matrix
elements 𝑚𝑖𝑗 ∈ N. A further stage is therefore needed if the influence coefficients belong to a
different domain. We can just examine the cases where the domain of the influence coefficients
is either N or R+ . We introduce the set 𝑉 of values acceptable for the influence coefficients. For
example, in [5], we have 𝑉 = [0, 1]. Let’s consider first the case where 𝑉 ⊂ R+ . Assuming that
𝑉 always includes the value 0 to describe the case of no influence, defining 𝑣 as the upper bound
of 𝑉 and 𝑚 = max 𝑚𝑖𝑗 we can map the values of the adjacency matrix 𝑀 into influence
𝑖,𝑗
coefficients by the following linear scaling
𝑣
𝑠𝑖𝑗 = 𝑚𝑖𝑗 𝑖, 𝑗 = 1, 2, . . . , 𝑛. (2)
𝑚
If 𝑉 ⊂ N, we have instead either a contraction mapping or a dilation mapping according to
whether we have 𝑚 > 𝑣 or 𝑚 < 𝑣 respectively.
For the applicability of the method, we recall the following categories adopted to classify
agent-based models in the survey [1]:

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Loretta Mastroeni et al. 161–173

• opinion domain;
• interaction direction;
• interacting agents;
• updating equation;
• updating frequency;
• utility function.
Opinion domain. Though an opinion is intrinsically a qualitative and potentially multi-
faceted feature, we have to describe it by a numeric variable and choose the domain where that
variable can lie. The following three domains have been surveyed in [1]:
• discrete;
• continuous over a bounded interval;
• continuous over R.
In the discrete case, the agent may choose its opinion within a limited set. Discretization lends
itself well to represent a qualitative feature through a proper mapping. For example, if we
consider the simplest discrete case where we have a binary opinion variable, the two values
may represent respectively a positive versus a negative opinion.
If the domain is instead a continuous but bounded interval, a common choice is the [0, 1]
interval.
The method proposed here is generally applicable to any opinion domain, since the applica-
bility condition concerns the values of the influence coefficients. However, the closure property
may impose some conditions on the influence coefficients. By closure property we mean that
the set 𝑉 is closed with respect to the application of the state updating equation. For example,
in [5], where 𝑉 = [0, 1] and the agents belong to one of 𝑐 classes (𝑛𝑖 representing the number
of agents in class 𝑖), and similarly in [4] for the pairwise case, the closure property requires that
𝑐
∑︁ 𝑐
∑︁
𝑠𝑖𝑖 (𝑛𝑖 − 1) + 𝑠𝑖𝑘 𝑛𝑘 ≤ 1 𝑖 = 1, 2, . . . , 𝑛𝑗 (3)
𝑘=1 𝑗=1
𝑘̸=𝑖

Interaction direction. That feature considers which way the agents influence each other.
again, we refer to the classification established in [1]. In a bilateral interaction, any two agents
always influence each other mutually. We have instead a unilateral influence when an agent
may influence another agent without being influenced by it. Even in the bilateral case, the
influence may not be perfectly symmetrical, since we can have different weights in the opinion
updating equations, signalling that the impact of agent 𝑋 on agent 𝑌 is different from that in
the reverse direction.
The method is applicable when the interaction direction is bilateral, non symmetric since
retweeting may take place in either direction.
Interacting agents. As to the number of agents that interact at each step, the classification
adopted in [1] considered the following three:

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Loretta Mastroeni et al. 161–173

• pairwise;

• any-to-any;

• closest neighbours.

In the pairwise case, just two agents interact at any single opinion updating round. Since the
pairs may change at each round, any agent may interact (influencing or being influenced by)
with any other agent in the long run. The any-to-any interaction case is obviously the case
where all the agents change their opinion at each time step, since they are influenced by the
opinions of the other agents at the previous time step. Finally, in the closest neighbour case,
any agent interacts just the closest agents (where the notion of closest involves the use of some
distance metric, which is natural in a social network).
In our method, since the interaction is associated with retweeting, and any twitterer can
retweet any other twitterer, the calibration we propose applies to all three categories. For the
case of pairwise interaction, the weights will be employed in pairs at each round, though they
have been estimated considering the embedding social network as a whole. As to the closest
neighbour case, if the distance employed for agents is that established on the embedding retweet
network, there is actually no difference, since the closest neighbour are those retweeting and
actually the only ones exhibiting a non-zero weight.
Updating equation. The updating equation is the function that relates the opinion of an
agent to the opinions of the other agents. Here, the classification proposed in [1] is quit simple,
considering a linear vs a nonlinear model.
Here we have described the method considering just linear updating equations so far, but
it could also be applied to non-linear updating equations, though requiring a more complex
mapping from 𝑀 to 𝑉 .
Updating frequency. This parameter can be considered as the speed of the opinion for-
mation process. The survey in [1] considers periodic and aperiodic updating. In the periodic
setting, each time step involves a change of opinions for all the agents. On the other hand, we
fall in the aperiodic case when just a couple of agents changes their opinion at each time step
(and it is not known in advance when their turn comes again), or opinions are updated just after
a triggering event, or opinion change takes place for a random selection of agents at each time.
Our calibration method is agnostic to the choice of updating frequency, since it can be applied
as many times as desired.
Utility function. Again, our calibration method is agnostic to the choice of utility function,
as long as that does not impact on the influence coefficients.

4. The dataset
As recalled in the Introduction, for our calibration method we have chosen an application
example concerning the influence of people’ opinions about wind power. In this section, we
describe our dataset and the procedure we have adopted to build it.
The procedure goes along the following four phases:

1. tweet retrieval;

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Loretta Mastroeni et al. 161–173

2. duplicate removal;

3. selection of relevant tweets;

4. retweet network building.

In order to retrieve all relevant tweets, we exploit the Twitter API by searching for all the tweets
containing either of the following word combinations:

• wind AND power;

• wind AND energy.

We deem those words to be fairly representative of the tweets associated to wind power for our
calibration demo. Of course, if our aim were to go beyond a demo, we could devise an ampler
set of words to obtain a dataset as exhaustive as possible. In this paper, we consider the tweets
posted in the week ending on December 9, 2019. Again, the calibration could be made more
accurate by extending the analysis horizon over several weeks or even longer periods.
Since many tweets contain both the above combinations, our basket after the retrieval phase
may contain duplicates. In the second phase of our procedure, we remove all duplicates. The
unique tweets after Phase 2 are 36539.
We must however recognize the possibility of including tweets that, despite binding the word
wind to power or energy terms, are not relevant to our actual theme, i.e. the use of wind to get
electrical power. Examples of such tweets are shown in Figure 2.
We need to eliminate as many as possible non-relevant tweets. In order to arrive at a set
of relevant tweets, we employ a semi-automatic procedure, based on hashtags and the co-
occurrence principle. Our procedure, which makes up Phase 3 of the overall procedure mentioned
above, goes through the following steps:

1. select the 𝑘 most frequent hashtags in the dataset of interest;

2. identify the hashtags that are surely relevant with out topic and form a group with them
(say Group X);

3. form a group with all the other hashtags (say Group Y);

4. examine all tweets containing Group Y hashtags but not Group X ones, and move their
hashtags to Group X if those tweets are relevant;

5. assign to Group X all the hashtags co-occurring with Group X hashtags (this is not done
iteratively, but just once for each Group X hashtag).

The last step of the procedure is equivalent to:

1. building the network of hashtags, where an edge is drawn between two hashtags if those
two hashtags co-occur in at least one tweet;

2. identifying Group X hashtags;

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Loretta Mastroeni et al. 161–173

(b) Tweet 2

(a) Tweet 1

3. adding neighbours of Group X hashtags to Group X.

A subset of the hashtag network is shown in Figure 3, where the nodes (hashtags) belonging
to Group X are highlighted in dark colour.
At this point we have the Group X of relevant hashtags. We can consider a tweet as relevant
if it contains any hashtag included in Group X and build the retweet network on the basis of
those tweets as described in Section 3. The resulting network is built out of 4739 tweets and is
made of 3528 nodes and 3617 edges.

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Loretta Mastroeni et al. 161–173

Figure 3: Network of hashtags

5. Experimental calibration results
After proposing the calibration method in Section 3, in this section we apply it to the dataset
described in Section 4.
In Figure 4, we show the resulting retweet network. In order to avoid excessive garbling, we
have drawn just the nodes with degree larger than one and have arranged it by degree, so that
the most central nodes are located in the inner core of the graph. We see that there are just a
few very much retweeted twitterers.
Aside from the sheer demonstration of the applicability of our calibration method, we exploit
this application to investigate the following research questions (RQ):

1. how frequent is the retweet phenomenon?

2. how widespread is the influence of any single agent upon the community?

3. how heavy is the influence of any single agent on another specific agent?

As to RQ1, we already have a partial answer from Figure 4. There are wide imbalances in the
connectivity of individual nodes. The sparsity of the adjacency matrix is 99%; this means that
the network is extremely far from a fully-meshed network where every twitterer retweets all
his/her fellows. However, this should be better investigated over time, since we considered
a single week, and retweeting relationship accrue over time. In fact, though older tweets are
often quickly forgotten, most users tend to “live in the present”, forgetting or abandoning topics
they followed just some hours or days after, the retweeting relationship lasts over time if the
retweeting user is a convinced follower/supporter of the retweeted one.
In order to answer RQ2, we show the distribution of nodes by their degree in Figure 5(a).
We see that 89.6% of the agents influences just another agent (the corresponding nodes have
degree 1), though there is a very small number of agents exerting their influence on a large

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Loretta Mastroeni et al. 161–173

Figure 4: Retweet network arranged by degree centrality

number of other agents, even more than 10. An overall measure of the imbalance in influence is
given by the graph centralization measure (see Chapter 5.3.1 of [22]), which takes values in the
[0,1] range, with 0 corresponding to a network where all the agents have the same influence,
and 1 corresponding to a star network (maximum possible imbalance). The centrality measure
we consider is based on the degree (i.e., the number of links incident upon a twitterer in the
embeddding retweet network). Mathematically, if 𝑔𝑣 is the degree of node 𝑣, and 𝑣 * is the node
exhibiting the highest degree the graph centralization is:
𝑛
1 ∑︁
𝐺 := [𝑔𝑣* − 𝑔𝑣𝑖 ] . (4)
(𝑛 − 1)(𝑛 − 2)
𝑖=1

In our case, the graph centralization measure achieves an intermediate value, namely 0.4275.
Though this could appear lower than expected, given the emergence of a few dominant twitterers,
those large imbalances that inflate the sum in Equation (4) are probably countered by the large
size of the network (hence, large value of 𝑛 that enlarges the denominator in Equation (4)).
As to RQ3, in the correspondence between the retweeting network and the agent-based
model, we have set the influence coefficients proportional to the edge weights. A measure of
the level of influence exerted by twitterers on their fellows is therefore the weight: if some
edge is associated to a large weight, that relationship bears a heavy influence. If we look at the
distribution of edge weights in Figure 5(b), we see that most twitterers (94.8%) exert just a small
influence on others (i.e., the weight of their edges is just 1), though there are a small minority
that are retweeted more frequently (even 7 times over the week of observation) and therefore
exert a heavier influence.

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3,161
3,000

No. of agents
2,000

1,000

201
50 30 13 15 9 7 2 8 4
0

1 2 3 4 5 6 7 8 9 10 11
degree

(a) Distribution of nodes by their degree
3,430

3,000
No. of edges

2,000

1,000

145 29
6 1 3 2 0
0

1 2 3 4 5 6 7 8
weights

(b) Distribution of edges by their weight
Figure 5: Influence by agents

6. Conclusions
Our paper deals with a critical issue in the development of agent-based models, i.e., setting
the parameters that govern the model and allow to apply such models in the real world (what
we call the calibration of the model). Our method can be applied whenever the agents act in a
social network. Though we showed an example using Twitter data, any social network allowing
an opinion reposting mechanism can be used. Also, the class of agent-based models to which
it can be applied is fairly large. The only significant limitation of the current approach is the
linear form of the equations that govern the interaction among agents. The possible limitations
on the correspondence between the ranges of influence coefficients in the agent-based model
on one side and edge weights in the social network on the other side could be addressed easily
by suitable operations, e.g., by translation and rescaling to realign the two ranges or possibly
by nonlinear transformations.
We therefore envisage this calibration model to fill the gap between the theoretical analysis

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of an agent-based model and its applicability in a real world context. We wish to address its
limitations as to the applicability to nonlinear models and different parameter ranges in our
future work.

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