Multi-agent models play a significant role in testing hypotheses about the unfolding of opinion dynamics in complex social networks. The model of the Argument Communication Theory of Bi-polarization (ACTB), developed by Maes and Flache (2013), shows that simple circulation of arguments among individuals in a group can determine strong diferentiation of opinions (bi-polarization efects) even with a small degree of homophily. The ACTB model and similar ones have nevertheless one limitation: given a topic of discussion, only direct pro and con arguments for it are considered. This does not allow to account for the topology of a more complex debate, where arguments may also interact indirectly with the topic at stake. This gap can be filled by using Quantitative Bipolar Argument Frameworks (QBAF). More specifically, by applying measures of argument strength for QBAFs in order to calculate the agents' opinion. In the present paper we generalize the ACTB measure of opinion strength to acyclic bipolar graphs and compare it with other measures from the literature. We then present a revised version of the ACTB model, where the agents' knowledge bases are structured as subgraphs of an underlying global knowledge base (described as a QBAF). We first test that the predictions of the ACTB model are confirmed when the underlying QBAF contains only direct pro and con arguments for a topic. We then explore more complex topologies of debate with two additional batches of simulations. Our first results show that changing the topology, while keeping the same number of pro and con arguments, has no significant impact on bi-polarization dynamics.
In social psychology, group polarization is commonly understood as a situation where the
opinions of individuals in a group tend to become more radical after discussing with peers [
A second type of models, most prominently the model of Argument Communication Theory
of Bi-polarization (ACTB) developed by [
In the ACTB model, both the set of all potentially available arguments (we call it the global
knowledge base) and the agents’ individual knowledge base are constituted by pro and con
arguments for a given topic. It is natural to frame such knowledge bases as directed graphs with
two diferent types of arrows for supports and attacks, i.e. bipolar argumentation frameworks
[
1 2 3 1 2
3 (a) 1 1 12 1 (b) 2 2 13 3 2 My doctor says I should get vaccinated. 3 I need the EU covid certificate.
A diferent knowledge base, say 2, may instead consist of the same con arguments 1, 2, 3, but a diferent set of pro arguments, namely: Pro’ 1 Vaccination is a social duty for everybody 12 Herd immunity has been reached in many cases for viruses that have now disappeared (e.g. smallpox). And even when this is still not the case, viruses have often disappeared in parts of the world where vaccination was significant (e.g. polio). 13 Vaccinated individuals are not totally immune, but the probability of getting infected, and therefore to infect others, is significantly lower.
1 and 2 correspond, respectively, to the graph in Figure 1a and Figure 1b. Both knowledge bases have three pro and three con arguments w.r.t. the topic , and therefore the ACTB measure cannot distinguish among them, so that the resulting opinion will be neutral (assuming that all arguments have equal weight). However, at an intuitive level, the opinion determined by 2 is more likely to support a favorable attitude towards .2 The generalized ACTB measures we introduce follow this intuition and, ceteris paribus, predict a higher opinion strength in the case of 2.
Given a larger variety of possible graph configurations and the new measures, it is then
an interesting question to test whether, given an equal number of pro and con arguments,
the topology of the underlying global knowledge base has an impact on the bi-polarization
dynamics predicted by the ACTB model. In particular, it is desirable to ascertain whether
augmenting the likelihood of a favorable attitude towards forces more positive consensus
among agents. Or alternatively, in cases where two subgroups end at the extreme poles of the
opinion spectrum, if this increases the cardinality of the group of agents with an absolute pro
opinion. To provide (partial) answers to these questions, we devised a revised version of the
original multi-agent ACTB model, written in Python, where global and individual knowledge
bases are encoded as bipolar graphs, and the opinion of the agents is obtained, at each step,
by measuring the strength of the topic in their individual knowledge base according to the
revised measures. We first test that results agree with those by [
The paper proceeds as follows. In Section 2 we introduce the basic notions concerning
Quantitative Bipolar Argumentation Frameworks (QBAF) and the standard ACTB measure. We
then show how to generalize it as a measure of argument strength for acyclic QBAFs and discuss
some of the properties of the new measure. In Section 3 we describe our revised version of
the ACTB model. In Section 3.2 we check that our model predicts bi-polarization efects that
are consistent with those predicted by the original model in [
The type of structures we deal with are instances of Quantitative Bipolar Argumentation Frameworks (QBAF) ([16], [15]), which are defined as follows: Definition 2.1 (QBAF [16]). A QBAF is a quadruple p, , , q consisting of a finite set of arguments, a binary (attack) relation on , a binary (support) relation on and a total function : from to a preordered set .
Here, for any P , pq is the base score of , intended as the weight of an argument previous to any impact from other arguments. In what follows we adopt the interval r0, 1s, with the natural ordering relation on real numbers, as our preordered set for all measures. Some useful notation is the following. pq t P | p, q P u denotes the set of direct attackers of , whereas pq t P | p, q P u is the set of its direct supporters. Following [16], it is useful to denote by pq (resp. pq) the set of efective attackers (resp. supporters) of .3 Furthermore, let us denote Y the union of both relations. Then, let pq be the set of all arguments such that there is path 0 . . . (with ¥ 1) that contains an odd number of . Let instead pq be the set of all arguments 3Depending on the modelling choice pq may either be equal to pq or to pqzt P pq | pq Ku , where K is the minimal element in the preorder and pq is the strength function defined below. That is, with the second option we discount attackers with null strength. The same holds for supporters. such that there is path 0 . . . (with ¥ 1) that contains an even number of . Intuitively, pq is the set of arguments with a negative influence on , and pq is the set of those with positive influence. 4 Here again, we set pq (resp. pq) as the set of arguments with an efective positive (resp. negative) influence on .5
The main idea behind gradual argumentation is to provide a semantics for the acceptability of
arguments in terms of a strength function : . The function pq is standardly provided
as a function of the argument’s base score pq and of the strength of all other arguments
afecting . If we only consider elements of pq and pq as the ones afecting , then we
obtain a local measure of argument strength. If we instead consider all ancestors in pq
and pq, then we have a global measure [
In the formal model of ACTB [
1 ‚ |,| 1 pq ,, where , is the set of relevant arguments for at time , and ,, is the relevance, either 0 or 1, of for at time . So the value of , ranges in the interval r 1, 1s. For our purposes, 1 it is easy to obtain an equivalent measure , ranging in the interval r0, 1s, by means of the following linear transformation:
1, 1 2, (2)
Despite their polarity, all relevant arguments in this calculation have an equal strength of 1
and therefore an equal and independent impact on the opinion about . Based on these features,
it is natural to represent the knowledge base of agent at time as a star tree like the one of
Figure 1a, where the node is the topic, the upper nodes are the relevant arguments and the
labelling of an edge from to identifies either as a pro argument (if the label is +) or a
con one (if the label is -). It then becomes natural to interpret , as a measure of strength
of the node . Given a generic node , this can be rewritten in the following form, using the
terminology of QBAFs we introduced:
pq
P pq pq P pq pq
| pq| | pq|
(1)
(3)
4More fine-grained distinctions about positive and negative influence can be found in the literature (see e.g.
[
5As before, we can either set pq as equal to pq or as pqzt P pq | pq Ku . Same for pq. which we normalize as
1 pq pq (4) 2 Again, pq is either pq or pq, and pq either pq or pq, depending on whether we adopt a local or a global measure. In the case of a star-tree graph, choice among the two options is clearly indiferent to calculate pq, since the set of ancestor nodes coincides with that of direct attackers and supporters. Clearly, Equation 4 cannot be employed to calculate the strength of the initial nodes nor of those with inefective ancestors, since we would get a division by 0. Typically, such cases are covered by postulating pq pq, so that we get the following definition by cases: pq # pq 1 pq 2 if pq Y pq H otherwise Now, this measure is well-defined for finite acyclic graphs, since it allows to calculate the strength of every node starting from the initial ones. Moreover, it is fully consistent with the original ACTB measure when we assume pq 1 for the initial nodes (i.e. maximal weight).
It should be noticed that here, as in the ACTB measure, the strength of non-initial nodes is fully determined by the strengths of the afecting nodes. This way of measuring pq fully discounts the base score of , since it makes it depend only on the impact of its ancestors. To take this into account we may want to generalize our measure further to the following: 1pq 1 pq 2 pq (5) (6) where 1 2 1. Here 1 and 2 are parameters that determine a weighted average for updating the argument strength. Intuitively, the weight of 1 tells us how much to count the base score, while the parameter 2 says how much to weight the shift determined by the afecting nodes. We retrieve the ACTB measure by setting 1 0.6
The choice between a local and a global measure of argument strength can make a substantial diference in terms of opinion dynamics. This can be seen by considering a simple case of reinstatement as that of Figure 2. Here we assume that pq pq 1, and is always the topic at stake. For both the local and the global approach we obtain pq pq 1 and pq 0 (since has only one ancestor and it is an attacker with maximal strength). But then, the value of pq crucially depends on the choice. With a local measure, has only one node afecting it () with null strength. If we consider this as a limit case where pq Y pq H (see fn. 3 and Equation 5), then we get pq pq. Otherwise, pq 0.5. Diferently, in the global approach we cannot be in a limit case, since P pq. The value of pq then depends on whether or not counts as efective. If not, then pq 1, otherwise we have pq 0.75, since although being of null strength, mitigates the reinstatement efect of by having an impact on the denominator of pq.
6Note that setting 1 to 0 allows so-called big-jumps ([15]), i.e. the strength of any node can be easily brought to 0 by its attackers, even though has a high base score. Vice versa, a node with low base score can be brought to 1 by its supporters. Augmenting the weight of 1 mitigates this phenomenon.
Recent work in gradual argumentation ([16], [15]) explores general properties that can serve as desiderata for measures of argument strength. The most comprehensive list is provided by [16]. Given that these properties are provided for local measures, we can only test them on the local interpretation, i.e by reading pq as pq and pq as pq in Equation 3. It is however interesting to check that our measure pq difers from any other measure provided in the literature w.r.t. satisfaction of at least some of these properties (see [16] Sect. 5).
First of all, pq is not balanced, and a fortiori not strictly balanced ([16] Sect. 4). Indeed, it is not even the case that, when the set of attackers and supporters of have equal strength7 then pq pq. This is due to the fact that the strength of an argument is determined solely by its ancestors, and not by its base score, in non-limit cases. It is easy to check that among the properties implied by (strict) balance and listed by [16], only GP1 (aka stability) is satisfied. 8 For the abovementioned reason neither GP2 (weakening), GP3 (strengthening), GP4 (weakening soundness), nor GP5 (strengthening soundness) are guaranteed to hold.
On the other hand, pq is strictly monotonic ([16] Sect. 4). Monotonicity means, that if and are such that pq ⁄ pq, the set of supporters of is at least as strong as the set of supporters of , and the set of attackers of is at least as strong as the set of attackers of , then pq ⁄ pq.9 Strict monotonicity means that whenever we replace ‘at least as strong’ with ‘strictly more strong’ in one of the preconditions, then pq pq. By consequence, the properties GP6-GP11[16], implied by strict monotonicity, hold for this measure.
Our goal is to test whether the underlying topology of a debate has an impact on the
bipolarization dynamics predicted by the multi-agent ACTB model of [
8That is, If pq pq H then pq pq.
9Here, set is at least as strong as means that there is an injective map from to such that for any P , pq ⁄ p pqq. node (the topic ). In accordance with this, the individual knowledge base of any agent, at any point of the execution, is a subgraph of it, and it always contains . Then, in order to calculate the agent’s opinion at any point we adapt our measures of Section 2.2.
The multi-agent model consists of a society of interdependent agents, which simultaneously participate in an artificial influence process. Each agent is attributed an opinion , about the given issue at each time point . This is expressed by a numerical value that, in our case, ranges between 0 and +1. As in the original model, there is a limited number of potential arguments about the issue at stake, and they are divided into two sets of, respectively, pro and con arguments. As mentioned, this global knowledge base is structured not as a vector but as a connected and acyclic QBAF with as its terminal node. The structure of the graph also determines the polarity of the arguments: each argument in pq (see Section 2) is counted as a pro argument and each one in pq as a con argument.10 Each argument is attributed an initial base score pq such that 0 ⁄ pq ⁄ 1. As in the standard model, we assume that, at each time , the opinion of agent is based only on a subset , of relevant arguments, where |,| ⁄ . This is summarized, for each agent , by a relevance vector of elements. The relevance ,, of argument for agent at time is either 1 (relevant) or 0 (not relevant).
In the standard model , is determined by Equation 1. Here, , is calculated by means of the measure described in Equation 5. More in detail, we first determine the subgraph , of the global knowledge base, constituted by all arguments relevant for at time and the edges among them (as determined by the global knowledge base). Then, we calculate pq for all nodes starting from the initial ones down to the terminal .11 We set pq pq and pq pq (see Section 2.2). The main issue, though, is that the graph , can easily be disconnected (see below), and therefore we need to decide whether pq and pq are evaluated w.r.t. or ,. Choice between the two options is a parameter of the model.12 The value of pq thus calculated is our ,. The acyclicity assumption ensures that this process terminates.
As in the ACTB model, each agent is attributed a recency vector of elements, each one having a value ranging from 0 to ,, where a higher value indicates that the corresponding argument has been taken into account more recently, and where the value of 0 indicates that the argument is not relevant (because it has never entered the database or because it is too old and therefore disregarded). For each argument , we denote its recency for agent at time by the number ,, The recency vector is then updated at each step following the same mechanism 10In the general case of an acyclic QBAF, it is possible for an argument to fall in both sets pq and pq. This does not constitute a problem when implementing our measures. However, for our present purpose, we decided to initialize our graphs as rooted in-trees, so to ensure that t pq, pqu forms a partition of the set of all arguments (See Section 3.2). So, the graph contains 1 | pq| | pq| 1 nodes.
11This can be done either with a local or a global interpretation of pq (see Section 2.2). Choice between interpretations is set as a parameter of the model.
12Both options are intuitively grounded according to diferent interpretations of one agent’s background
knowledge. Choice between them can make a substantial diference w.r.t. the resulting opinion dynamics. Indeed, when
pq and pq refer to ,, disconnected arguments, no matter how strong, have no impact on the calculation
of ,.
described by [
Exactly as in the original model, the opinion of each agent evolves as the result of a sequence
of events, each one corresponding to one interaction between two agents. Each interaction goes
in two sequential phases. (i) A selection phase where one agent is randomly picked and then a
partner is selected with a probability proportional to the similarity of its opinion with that of
(opinion homophily).13 (ii) A social influence phase, where the opinion of agent is updated
as a result of the interaction with . Here again, we implement the exact same mechanism of
the original model: one of ’s relevant arguments is picked, and is then adopted by .14 Then,
updates its recency vector as described. This mechanism ensures that one argument is added
and another is discarded, and therefore the number of arguments in the knowledge base of is
kept constant. As in the original model, each run iterates events until equilibrium is reached,
and there are two kinds of equilibria: perfect consensus and maximal bi-polarization. In perfect
consensus all agents hold the same opinion based on the same set of arguments. In maximal
bi-polarization there are two maximally distinct subgroups, where members agree with each
other in the same way (i.e. same opinion based on the same set of arguments). Both equilibria,
and only them, are stable situations, as explained by [
We implemented three diferent configurations of a global knowledge base, as in Figure 3. All scenarios have the same number of pro and con arguments but diferent topologies, to the efect that the strength pq of the topic node increases from 0.5 (Scenario 1) to 0.75 (Scenario 3).15 This enables to answer our initial question as to whether providing stronger reasons for a pro attitude towards has an influence on bi-polarization dynamics, e.g. by inducing more general consensus for or, in case of a group split, by determining larger clusters of agents with a favorable attitude.
The first configuration consists of a star-tree with equal number of pro and con arguments.
This configuration reduces to the vector configuration of the original ACTB model, and therefore
should give similar results. To test this, we initialized a QBAF consisting of 41 arguments, i.e.
the topic node , 20 direct attackers (con arguments) and 20 direct supporters (pro) of , all with
maximal base score 1, as in Figure 3a, so that pq 0.5 in the global knowledge base. We
impose a strong level of homophily for the selection phase [ℎ=9] (See [
14By running the model, we observed that the directionality of the exchange has a strong impact on the resulting bi-polarization dynamic. Indeed, by setting as the speaker and as the receiver, the rate of bi-polarizations in simulation runs drops dramatically.
15Here, pq can be regarded as measuring the opinion of an omniscient agent.
16More precisely, each agent is randomly assigned a pair p, q such that . Then, pro arguments 1 . . .
20 21
. . .
(a) Scenario 1. pq 21 . . . . . . 30 . . .
10 . . .
40
(b) Scenario 2. pq
initialized our model so that, when calculating the agents’ opinion, the sets pq and pq
are evaluated w.r.t. , so that arguments disconnected from in the individual knowledge
base are still counted as relevant for opinion formation.17 We then ran the model as described
in Section 3.1. More precisely, halt conditions are triggered (a) when all agents have opinion
value either 0 or 1, and therefore maximal bi-polarization is bound to obtain18; (b) when the
number of arguments considered as relevant by at least some of the agents is equal to , which
implies perfect consensus; and (c) after 6M events for space limits (which rarely happens, in
this configuration, before (a) or (b) are triggered). Out of 500 simulation runs, we obtained
bi-polarization 424 times with subgroups of equal cardinalities (an average of 10,02 con-oriented
individuals and 9.98 pro-oriented), consistently with the results of [
As a second and third setup, we instead organized our graph as a rooted in-tree with nodes
at a maximum distance of 2 from the root. As in the previous case, 20 pro and 20 con nodes
are present. However, in the second setup we have 20 direct attackers, 10 direct supporters
and 10 defenders (attackers of attackers), as in Figure 3b. In the third setup we instead have 20
direct attackers and 20 defenders (see Figure 3c). When we calculate the opinion strength by
evaluating pq and pq relative to , these choices guarantee that pq is higher in the
global knowledge base of Scenario 2 than in Scenario 1 (pq 0.625), and even stronger in
Scenario 3 (pq 0.75). In both the second and third setup all remaining parameters are kept
the same. Again, we ran 500 simulations per setup and did not observe significant diferences
w.r.t. Scenario 1 in terms of numbers of bi-polarizations, nor concerning the cardinalities of
subgroups of pro and con oriented agents. Indeed, we obtained 433 and 415 bi-polarizations
respectively (Figure 4(a)), both with a slight average variation in time steps for obtaining group
split: 4098 for Scenario 1 against 4140 and 3655 for Scenario 2 and 3 (see Figure 4(b)). The
average of pro-oriented agents after group split is 10.33 in Scenario 2 and 10.40 in Scenario 3
and con arguments are randomly selected and attributed diferent values from 1 to in the agent’s recency vector.
17For this star-tree configuration, this choice is indiferent, but it will be not for our second and third setup.
18This because the probability of communication between agents at the opposite poles is 0 as by [
In this paper we generalized the ACTB measure of opinion into a measure of argument strength for QBAFs, and show how to integrate QBAFs into a multi-agent model of opinion dynamics. This opens up for the study of how topological features of a debate may influence consensus and bi-polarization efects. At the present stage, our results do not witness any significant impact. However, the generality of the model opens up for the exploration of a large parameter space and many questions can be framed via simulative experiments. As a first step, we need to test our preliminary observations on the three scenarios of Section 3.2 by varying the initialized parameters. Then, in order to look for robust results about the initial questions we asked, we need to analyze more scenarios and diferent ways of implementing the generalized ACTB measure of opinion, as provided in Section 2.3. It can be of further interest to also implement other measures from the literature in our model, to check whether their general properties, mentioned in Section 2.4, have an impact on the opinion dynamics of our model. There is then another interesting and probably more relevant question that our framework allows to ask, and it concerns the relevance update mechanism of the original ACTB model that we used. Indeed, once we are able to calculate the strength of the arguments in an individual knowledge base, it becomes natural to investigate how preferential communication of strong arguments or discarding of weaker ones may influence our opinion dynamics. These are only few of the possible venues for further research on a simulative basis.
The authors wish to thank the GARR Consortium which has provided the infrastructure for the simulative part of this work. [12] P. Baroni, M. Romano, F. Toni, M. Aurisicchio, G. Bertanza, Automatic evaluation of design alternatives with quantitative argumentation, Argument & Computation 6 (2015) 24–49. [13] A. Rago, F. Toni, M. Aurisicchio, P. Baroni, Discontinuity-free decision support with quantitative argumentation debates (2016). [14] P. Baroni, G. Comini, A. Rago, F. Toni, Abstract games of argumentation strategy and game-theoretical argument strength, in: International Conference on Principles and Practice of Multi-Agent Systems, Springer, 2017, pp. 403–419. [15] L. Amgoud, J. Ben-Naim, Evaluation of arguments in weighted bipolar graphs, International
Journal of Approximate Reasoning 99 (2018) 39–55. [16] P. Baroni, A. Rago, F. Toni, From fine-grained properties to broad principles for gradual argumentation: A principled spectrum, International Journal of Approximate Reasoning 105 (2019) 252–286.