Strength in coalitions: Community detection through argument similarity

1.Introduction

The identification of communities in social media and the detection of stances in Tweets has become increasingly important in recent times [15,18,19,26,28] as a result of the tangible effect that these platforms have on the public opinion. In this domain, identifying communities implies analyzing the position of contributing agents concerning a particular topic or their respective argumentative stance; several tools can be used for this purpose, for instance [19] describes an approximation solution based on a supervised classifier that finds stances, and classifies them, over a graph representation. Most of the explored work on identifying stances focuses on the classification of tweets as “in favor” (support), “against” (dispute), or “neutral” (comments or questions) regarding a previous tweet in a conversation [35,37]. Most of these methods focus on analyzing tweets to characterize the relationship between messages in a set.

In this work, we propose an argumentation-based method to analyze stances in a debate exchange and formally characterize the relationships between these stances using similarity, understanding the similarity as an attribute of the relationship, not just of the message. Thus, a similarity measure associated with arguments will allow us to find defined positions or communities in social media; to do that, we use argumentation to formalize the exchange of views. In a general sense, argumentation can be defined as the study of the interaction of arguments in favor and against a position or claim to determine which are acceptable. Formal Argumentation Theory provides several formalisms to model emerging behavior, creating different platforms to perform defeasible reasoning and solve several problematic situations [5–7,20,44]. In the context of our research, an argument and an argumentative stance will be considered synonyms.

In [16], Dung introduced Abstract Argumentation Frameworks (AFs), intending to create a tool for modeling situations where an agent considers arguments supporting a claim. The framework allows the representation of attack relations between abstract entities called arguments and provides different acceptability semantics that, in essence, characterize different criteria under which sets of arguments can be accepted together. Cayrol and Lagasquie-Schiex [12] extended Dung’s framework by considering two independent types of interactions between arguments: the relationships of attack and support. This formalism, called Bipolar Argumentation Frameworks (bafs), models situations where arguments may give support for other arguments, supplying additional reasons to believe in it. Furthermore, they extend Dung’s acceptability semantics, considering the support relationship between arguments.

Based on the bipolar formalism, in [11] the authors presented an approach to use a similarity degree measure between arguments to characterize the attack and support relations in a baf [2,12]. They consider gradual relations between arguments to define more flexible sets of acceptable arguments. An attractive application of this idea is that similar arguments linked by a support relation could represent a community characterized by a similar stance. The detection of communities in social media will be based on the support and similarity relations between arguments, while the classification of the communities will be done according to the similarity degree between the stances that conform to them. To do this, we will take advantage of the notion of coalition presented in [13] and the framework proposed in [11].

Briefly speaking, to represent a community’s strength, first, we analyze the arguments proposed by each community. Then, we consider the context where the argumentation discussion is put into play since the opinions of a particular community may vary according to the argumentative context. Finally, we perform a comparison procedure where arguments are analyzed considering the set of descriptors (a tag or a label describing an aspect to which an argument is connected) that have in common in combination with the defined context. Thus, the closer or more similar the arguments of a particular community are, the greater the strength of the position proposed by the community is. Note that we mainly refer to discursive communities. This clarification is necessary because it will allow us to regard communities as subgroups with cohesive thinking, inspiring us to find a cohesive measure to characterize their behavior.

In this regard, it is pertinent to find a way to determine how close, or cohesive, these communities are. Precisely, the cohesion associated with a community expresses how united its members are, how integrated, how fraternal and supportive they are to each other, how much they strive to think together, and how willing they are to work together to achieve collective goals and support a specific outcome.

The following example will illustrate the ideas involved in this research.

Example 1.

Figure 1 shows a set of opinions extracted from the ProCon website11 in favor of (pro) or against (con) the following proposition “Is Human Activity Primarily Responsible for Global Climate Change?”. We can roughly distinguish three communities that give opinions regarding the responsibility of humans for climate change: one of them supports the idea that human activity is responsible for climate change (arguments E and C), another confronts the previous one with the opposite position (arguments A, B, D and F), and lastly, there is one representing an intermediate posture between the other two (arguments G and H). Each of these communities aggregates different points of view, providing various aspects supporting the community’s overall stance.

Fig. 1.

Arguments in favor and against the possible causes of climate change.

By analyzing these well-defined general postures, we will obtain the details of the beliefs each community backs; but, by closely examining the opinions in each community, we can determine each community’s inherent strength.

We structured this presentation as follows: In Section 2, we briefly explore the notion of coalition in a baf, and we summarize the main ideas behind the Similarity Valued Argumentation Framework (s-baf) [11]; then, in Section 3, we examine the notion of community. In Section 4, we detail a mechanism to determine coalitions based on a similarity measure and, subsequently, communities from coalitions. Furthermore, we detail a mechanism to understand the attacks between them, while our introductory example is developed in Section 5. Related research is discussed in Section 6, while Section 7 is devoted to conclusions and future lines of further our investigations. Finally, in Appendix A, we include the necessary proofs of the theorems and propositions introduced in the main text; in Appendix B, we propose a basic algorithm to compute coalitions.

2.Background

In [12], Cayrol and Lagasquie-Schiex proposed an approach to model two different forms of interaction between arguments, one of these positive, when an argument provides support to another, and the other negative, as an argument performs an attack to another argument, thus extending Dung’s abstract argumentation frameworks by adding the support relation. This approach, known as bipolar argumentation, is characterized as an Bipolar Argumentation Framework or baf, and allows to represent an intelligent agentś “bipolar” behavior since reasons in favor of a claim can be regarded as supportive while reasons against it can be viewed as contrary. A baf is defined as a 3-tuple constituted of a set of atomic arguments and two binary relations representing the attack and support relationships between arguments. We begin by recalling the definition of baf as presented in [12].

The graph description introduced in Dung [16] is extended in baf, adding the representation of the support relation; thus, GΘ will denote a bipolar argumentation graph. As mentioned, Cayrol and Lagasquie-Schiex [12] presented the extensions of the attack and support notions introducing the supported and secondary defeats, combining a sequence of supports with a direct defeat; this move allows to explore the interaction between supporting and defeating arguments.

There exist other forms of attack that are considered in other interpretations of baf, some of them summarized in [55] as indirect attacks [39] that considers mediated attacks, extended attacks, and the tiered indirect attacks. However, in the present work, we consider only the simple forms of attack (defeat) presented in Definition 2. However, extending this proposal to consider a more refined baf version is possible. We will consider this extension in future work.

Following the Cayrol and Lagasquie-Schiex approach [12], in some sense, a set of arguments must keep a minimum of coherence to model one side of any reasonable dispute adequately. They propose that the coherence of an acceptable set of arguments can be kept internally by requiring the set not to contain an argument that attacks another one in the same set. Meanwhile, external coherence can be maintained by requiring that the set does not include both a supporter and an attacker of the same argument. Internal coherence can be obtained by extending the definition of conflict-free set proposed in [16], and external coherence can be captured by the notion of a safe set.

Conflict-freeness requires considering the direct, supported, and secondary attacks. Additionally, Cayrol and Lagasquie-Schiex show that the notion of a safe set is powerful enough to encompass the concept of conflict-freeness, i.e., if a set is safe, it is also conflict-free. The closure under Rs was introduced in [12] is a requirement that only concerns the support relation.

Succinctly, these are some salient features of this formalism:

In this work, we will introduce the concept of a coalition as it is revealed through considering similarity to help identify argumentative stances in a conversation and characterizing the relationships between these stances.

Example 2.

Next, we present a baf example described as Θ=⟨Args,Ra,Rs⟩, where:

Args={A,B,C,D,E,F,G,H,I,J,K}Ra={(B,D),(G,E),(H,I),(I,I),(F,J),(J,F)}Rs={(A,B),(C,B),(E,D),(E,F),(D,F),(G,H),(K,J)}

Fig. 2.

Representation of the attack and support relations in baf, the GΘ graph.

Conflict-freeness requires considering the direct, supported, and secondary attacks. Figure 2 shows the bipolar argumentation graph of this particular baf where we can identify the followings defeat relations: the argument G is a secondary defeater for F, while C and A are supported defeaters for argument D (See Fig. 3, where a secondary attack is highlighted in red, while a supported attack is emphasized in orange). Additionally, Cayrol and Lagasquie-Schiex show that the notion of a safe set is powerful enough to encompass the concept of conflict-freeness, i.e., if a set is safe, it is also conflict-free. The closure under Rs was introduced in [12] is a requirement that only concerns the support relation. Also, the argument K is a supported defeater for the argument F. Furthermore, G is a supported defeater for I, while I is a direct defeater for itself.

Fig. 3.

Representation of the secondary and supported attacks in baf.

This formalism can approach a representation of how human beings think by recognizing the bipolar nature of a debate when discussing a particular topic; however, it is not feasible to clearly distinguish stances on a specific subject. For this reason, in [13], the authors proposed a notion of coalitions between supporting arguments that will be discussed next.

2.1.Coalitions in bipolar argumentation framework

The capability of representing support among arguments available in Bipolar argumentation frameworks becomes relevant when it is necessary to reason with maximal and coherent sets of arguments that are collectively related through that relationship. Cayrol and Lagasquie-Schiex in [13] perform an in-depth analysis of the bipolar framework abstraction introducing the notion of coalition that aims to obtain the maximal set of coherent arguments which collaborate to justify a conclusion, i.e., arguments that do not attack each other directly or indirectly. The following definition formally introduces the notion of a coalition.

Definition 5

Definition 5(Coalitions in a baf).

Let Θ=⟨Args,Ra,Rs⟩ be a baf, and GΘ be a bipolar argumentation graph. A subset CΘ⊆Args is a coalition in Θ iff CΘ is a maximal conflict free set in Θ such that the subgraph GΘ′ induced by CΘ is connected only by support relations.

Note that if (i) A, B ∈ Args, (ii) A Rs B and (iii) there is no attack (direct, supported, or secondary attack) from A to B, then there exists a coalition over Θ which contains A and B.

A coalition represents a relationship on the set of arguments; therefore, the notion of attack between them introduces a meta-argumentation framework that provides a higher locus where to interpret and analyze the set of supported arguments and the attacks between those sets:

Definition 6

Definition 6(Attack between coalitions in baf).

Let Θ=⟨Args,Ra,Rs⟩ be a baf, and let C1 and C2 be two coalitions over Θ. If there exist A ∈C1 and B ∈C2 such that A Ra B, then the coalition C1 c-attacks (or just attacks) C2.

Example 3.

Continuing with Example 2, Fig. 4 shows the following four coalitions: C1={A,B,C}, highlighted green; C2={E,D,F}, highlighted purple; C3={G,H}, highlighted blue; and C4={J,K}, highlighted orange.

Fig. 4.

Coalitions in baf.

These are maximal conflict-free sets, and C1, C2, C3, and C4 are maximal sets closed under Rs. Additionally, we have that: C1 attacks C2, because B attacks D; the coalitions C3 and C4 attack the coalition C2, since J attacks F, and G attacks E; and finally, C2 attacks C4, as F attacks J.

The notions presented in [13] initially do not provide the tools to analyze how strong the coalitions, and the attacks between them, are. Also, note that argument I does not belong to any coalition, missing information about the discursive domain that would be important to consider. Furthermore, it would be possible that under certain conditions, a coalition assimilates another. We will address these observations in the following sections.

2.2.A similarity valued argumentation framework

Despite the representation capacity that bipolar frameworks offer, some argument’s characteristics should be taken into account to improve the performance of their semantics. For example, a very natural tool for argument-based reasoning is the notion of similarity among arguments: during an argumentation process, we sometimes tend to group arguments according to their shared characteristics or to the topics to which they refer. It can be argued that any comparison process requires defining a context in which such comparison can be meaningful [23,48]. These intuitions can be applied to arguments as follows: two arguments may be similar in a given context but may be entirely unrelated (or even incomparable) under different circumstances. Reasoning that considers similarities between arguments represents a natural form that is used in everyday human reasoning [11].

Budán et al. introduced a Similarity-based Bipolar Argumentation Framework (or s-baf) in [11] where the interested reader may find relevant examples and discussions illustrating the central ideas in the framework; in what follows, we will recall the definitions and notions concerning s-baf we need here. The authors described a mechanism for considering the context of the comparison between arguments, and that context is based on a set of descriptors the arguments being analyzed have in common, where a descriptor is a tag or a label describing an aspect to which an argument is connected. With the purpose of the comparison, they introduced enriched arguments, that is, arguments decorated with additional information.

We will make some notational conventions to facilitate the following definitions. We assume a set D of available descriptors corresponding to the domain where the argumentation is carried out. Each descriptor has a set of values associated; thus, for a descriptor d∈D, Vd will be the set of semantic values corresponding to descriptor d.

Definition 7

Definition 7(Enriched Argument).

Let Θ=⟨Args,Ra,Rs⟩ be a baf, A be an abstract argument in Θ, and D be a set of descriptors. An enriched argument is a pair A=⟨A,δA⟩, where δA is a finite non-empty set of pairs (d,VdA), where d∈D and VdA⊆Vd. The set of all enriched arguments will be denoted as Args.

Next, we introduce the notion of context of the argumentation.

Definition 8

Definition 8(Context).

Let D be a set of descriptors, a context C will be represented as C={(d,wd)|d∈D,wd∈[0,1]}, i.e., a context is a set of ordered pairs where d∈D is a descriptor and wd∈[0,1] is the weight associated with d. We denote with DC the set of descriptors mentioned in the context C, i.e., DC={d|(d,wd)∈C}.

Using the additional information provided by the context, it is possible to represent and determine similarities between arguments by introducing means to enrich the analysis of the relationships between them and distinguish between arguments that are weakly related to those with stronger relationships. In this direction, it is possible to compute an argument’s similarity degree between two arguments. To do that, we consider the descriptors that arguments have in common and the weight those descriptors have in the process comparison in a specific context comparison C. This context is defined as a subset of the description that specifies a point of analysis. Given a context C, for any argument X∈Args, we denote the descriptors in X that appear on the context C as DXC, i.e., DXC=DX∩DC.

Definition 9

Definition 9(Similarity coefficient for a descriptor).

Let Args be a set of enriched arguments, A=⟨A,δA⟩ and B=⟨B,δB⟩ two enriched arguments in Args, and C a context. We define the similarity coefficient for each descriptor d∈DAC∩DBC with weight wd, denoted Coefd(A,B), as follows:

Coefd(A,B)=|VdA∩VdB||VdA∩VdB‾|·wd if |VdA∩VdB‾|≠0wd otherwise

Intuitively, finding the similarity coefficient between two arguments for a particular descriptor requires finding the number of semantic values common to that descriptor in both arguments and dividing it by the number of semantic values for the descriptor that the arguments do not have in common, and then weighing the resulting value according to the relevance associated with the descriptor in the definition of the context considered [24,29,32].

Definition 10

Definition 10(Similarity degree between arguments).

Let Args be a set of enriched arguments, A=⟨A,δA⟩ and B=⟨B,δB⟩ be two enriched arguments in Args, and C be a context. The similarity degree between arguments in a context C, denoted SimC, is defined as a function SimC:Args×Args→[0,1], such that:

SimC(A,B)=αnif DAC∩DBC={d1,…,dn}≠∅0otherwise

where α1=Coefd1(A,B) and αi=⊙(αi−1,Coefdi(A,B))with2⩽i⩽n, and, the operator ⊙ should be either a T-norm satisfying the following properties: commutative, associative, monotonically increasing, and with 1 as its neutral element; or ⊙ should be a T-conorm, satisfying commutative, associative, monotonically decreasing with 0 as its neutral element.

Note that the order in which the descriptors are considered in computing SimC(·,·) is irrelevant since the operator ⊙ satisfies commutativity and associativity. Furthermore, considering the monotony property and the fact that T-norms and T-conorms are defined in the interval [0,1], we can ensure that the resulting value is non-negative.

The abstract concepts presented earlier will be illustrated in the following example.

Example 4.

Suppose that the arguments A and B of our abstract example represent the following opinions regarding students and their homework:

• A: Research published in the High School Journal indicated that “students who spent between 31 and 90 minutes each day on homework scored about 40 points higher on the SAT-Mathematics subtest than their peers, who reported spending no time on homework each day, on average.”

• B: Research by the Institute for the Study of Labor (IZA) concluded that “increased homework led to better GPAs and higher probability of college attendance for high school boys. In fact, students who attended college did more than three hours of additional homework per week in high school.”

Analyzing the arguments above, we observe that they have the following descriptors and values:

δA={(mentions_student,{yes});(refers_good_practice,{yes});(time_mention,{yes});(presents_results,{scored_better_Mathematics_subtest});(based_on_evidence,{yes})}.δB={(mentions_student,{yes});(refers_good_practice,{yes});(time_mention,{yes});(presents_results,{better_GPAs;higher_probability_college_atendance});(based_on_evidence,{yes})}.

Now, suppose that the context for the arguments comparison is the following:

C={(mentions_student,0.4);(refers_good_practice,0.4);(presents_results,0.2)},

Per each descriptor, we have:

• For the descriptor mentions_student: the two arguments have a single value in common and no different ones. So, the Coefd(A,B)=0.4.

• For the descriptor refers_good_practice: the two arguments have a single value in common and no different ones. So, the Coefd(A,B)=0.4

• For the descriptor presents_results: arguments have different values for this descriptor, and no common value. So that, according to the similarity coefficient definition, the Coefd(A,B)=0.

Now, considering the bounded sum T-conorm, we have that the SimC(A,B)=0.8, given that:

α1:min(0.4+0.4,1)=min(0.8,1)=0.8α2:min(0.8+0,1)=min(0.8,1)=0.8

The similarity value obtained reflects that the both arguments refer to a good practice for the students, but each argument gives different reasons (results) for this assertion.

The next step is to define a cohesion degree between supporting arguments and a controversy degree between conflicting arguments. In the following definition, we introduce the enriched baf framework based on the original baf. Formally,

Definition 11

Definition 11((Induced) Enriched baf).

Let Θ=⟨Args,Ra,Rs⟩ be a baf, the enriched baf induced is defined as Θ‾=⟨Args,Rs,Ra⟩, where Args is the set of enriched arguments corresponding to arguments in Args, and Ra and Rs are the attack and support relationships in Args that are induced by Ra and Rs, respectively.

Given that the Enriched baf is based on the classic baf, the former contains the same arguments and induced relations that the classic baf, except that in the Enriched baf, the arguments are decorated with supplementary information. The additional information added to the arguments will be helpful later in extending the formalism.

Now, the cohesion degree of a set of supporting enriched arguments and the controversy degree associated with a set of attacking enriched arguments can be formally introduced.

Definition 12

Definition 12(Cohesion & Controversy degrees).

Let Θ‾=⟨Args,Ra,Rs⟩ be the enriched baf induced from the baf Θ=⟨Args,Ra,Rs⟩. Given a set of enriched arguments S⊆Args and a context C, let SimC be a similarity degree function for C, and RaS={(X,Y)∈Ra|X,Y∈S} be the subset of Ra restricted to the arguments of S and RsS={(X,Y)∈Rs|X,Y∈S} be the subset of Rs restricted to the arguments of S then we have:

• The cohesion degree for S, denoted as CohC(S), is defined as:

  CohC(S)=βnif RsS={(A1,B1),…,(An,Bn)}≠∅0otherwise
  where β1=SimC(A1,B1) and βi=⊕(βi−1,SimC(Ai,Bi)) with 2⩽i⩽n.

• The controversy degree for S, denoted as ContC(S), is defined as:

  ContC(S)=γnif RaS={(A1,B1),…,(An,Bn)}≠∅0otherwise
  where γ1=SimC(A1,B1) and γi=⊗(γi−1,SimC(Ai,Bi)) with 2⩽i⩽n.

Both CohC(·) and ContC(·) can be obtained independently using a recursive function instantiated with T-norms or T-conorms, essentially in the same manner as with the similarity function SimC, depending on the user modeling intentions. It is required that the functions chosen satisfy commutativity, associativity, and monotonicity and have an identity element. These properties ensure that the order of the calculations does not affect the result [27]. Note that both degrees deliver a non-negative real number; more precisely, both are defined as 2Args⟶[0,1].

Next, for simplicity, we present our abstract example where the similarity degree associated with each relationship (attack or support) was previously established (for more details, see [11]). Note that, in this formalism, both attacks and support are treated likewise. If an argument X attacks or supports another argument Y, the similarity measure is assigned to the relationship without differentiating the kind of relationship. Mainly it is because we want to be balanced in dealing with the positive and negative actions over a discussion.

Example 5.

We continue with our abstract example, the graph in Fig. 5 shows the similarity degree associated with the arguments in each relation. Intuitively, we can observe that the attack between the arguments B and D is weaker than the attack between G and E. Note that the attack between F and J have the same similarity degree that the attack from J to F since the similarity relation is symmetric. Additionally, we can also differentiate the weakest support relationship existing in the whole model: the one between G and H. Based on the similarity degree obtained in each relation, we compute the cohesion coefficient associated with the set of supporting arguments (considering a product T-norm) and the controversy coefficient associated with attacking arguments (considering a max T-conorm). Thus, we have that:

CohC({(E,D),(D,F)})=0.42CohC({(A,B)})=0.8CohC({(C,B)})=0.6CohC({(E,F)})=0.8CohC({(G,H)})=0.5CohC({(K,J)})=0.6ContC({(B,D)})=0.4ContC({(G,E)})=0.8ContC({(H,I)})=0.3ContC({(I,I)})=0.1ContC({(H,I),(I,I)})=0.03ContC({(F,J)})=0.5ContC({(J,F)})=0.5ContC({(F,J),(J,F)})=0.25

Fig. 5.

Similarity in the bipolar argumentation framework (Figs 2 and 4).

Observe that, in this particular case, the cohesion associated with the support relation is analyzed considering the support chain presented in the argumentation model (see Figure 5). At the same time, the controversy measure is obtained by analyzing the pairs of attacking arguments.

The enriched baf Θ‾ will be extended to include the degrees just defined.

Definition 13

Definition 13(Similarity-based baf).

Let Θ‾=⟨Args,Rs,Ra⟩ be an enriched baf and C a context, a Similarity-Based Bipolar Argumentation Framework (or s-baf) is defined as a tuple Φ=⟨Θ‾,SimC,CohCΘ‾,ContCΘ‾⟩, where SimC is a similarity degree function for enriched arguments in Args, and CohCΘ‾ and ContCΘ‾ are, respectively, the cohesion and controversy degree functions defined over Θ‾ in the context C.

When no confusion may arise, we will avoid mentioning the Θ‾ enriched baf as a superscript of the cohesion and controversy degree operators, writing instead ⟨Θ‾,SimC,CohC,ContC⟩, making the notation more straightforward.

Additionally, in s-baf, the support and attack relations will have a particular interpretation since a threshold τ∈[0,1] will be considered in the specification of the type of attack being analyzed. The strong-support relation in s-baf occurs when the cohesion associated with the relationship is greater than the threshold τ and when this value is less than the τ, we will say we have a weak-support relation. Consequently, the attacks in an s-baf will be of two types: (i) strong, when the cohesion and controversy values associated with the attack are greater than the threshold τ; in this situation, we have strong-direct attack, strong-supported attack, and strong-secondary attack, or (ii) weak, if at least one of the values is less than τ; then, in this case, we have weakly-direct attack, weakly-supported attack, and weakly-secondary attack.

Now, and considering the elements introduced above, the authors in [11] redefine the classical notions of conflict-free and safe sets in a baf that are the basis for a new family of semantics (see op. cit. for more details).

Definition 14

Definition 14(Conflict-freeness and Safety properties in a s-baf).

Let Φ=⟨Θ‾,SimC,CohC,ContC⟩ be a s-baf, where Θ‾=⟨Args,Rs,Ra⟩ is the enriched baf, and τ∈[0,1] be a given threshold. Then:

• S is a strongly-conflict-free set iff there is no A,B∈S such that there exists a strong or weak attack from A to B.

• S is a τ-conflict-free set iff there is no A,B∈S such that there exists a strong attack from A to B and ContC(S)>τ.

• S is a weakly-conflict-free set iff there is no A,B∈S such that there exists a strong attack from A to B.

• S is a strongly-safe set iff there is no A∈Args and no pair B,C∈S such that there exists a strong or weak attack from B to A, and either there is a sequence of support from C to A, or A∈S.

• S is τ-safe set iff there is no A∈Args and no pair B,C∈S such that there exists a strong attack from B to A, ContC(S∪{A})>τ, and either there is a sequence of support from C to A such that CohC({C,…,A})>τ, or A∈S.

• S is weakly-safe set iff there is no A∈Args and no pair B,C∈S such that there is a strong attack from B to A and either there is a sequence of support from C to A such that CohC({C,…,A})>τ, or A∈S.

In the following step, in [11] the authors extended the notions of defense for an argument with respect to a set of arguments. Furthermore, the paper introduces different definitions of admissibility, from the most general and strong to the most specific and weak. The most general is based on the classical notion of admissibility, where only the attack relations are considered, both the strong and the weak ones.

Definition 15.

Let S⊆Args be a set of arguments, and A∈Args an argument. Then:

• S is a strong defense for A iff for all B∈Args such that if B is a strong or weak (direct, supported, or secondary) attacker of A then there exists C∈S where C is a strong (direct, supported, or secondary) attacker of B.

• S is a weak defense for A iff for all B∈Args such that if B is a strong or weak attacker (direct, supported, or secondary) attacker of A then there exists C∈S where C is a weak attacker (direct, supported, or secondary) attacker of B.

Then, this notion is extended by considering external coherence and under different attack and support degrees among arguments. Finally, external coherence is strengthened by requiring the closure under the support relation (Rs).

Definition 16.

Let Φ=⟨Θ‾,SimC,CohCΘ‾,ContCΘ‾⟩ be a s-baf with the underlying enriched bipolar argumentation framework Θ‾=⟨Args,Rs,Ra⟩, and S⊆Args be a set of enriched arguments. Then:

• S is d-strongly-admissible if S is strongly-conflict-free and strongly-defends all its elements.

• S is d-τ-admissible if S is τ-conflict-free and there exists a strong or weak defense for all its elements.

• S is d-weakly-admissible if S is weakly-conflict-free and there exists a strong or weak defense for all its elements, or S is strongly-conflict-free and weakly-defends all its elements.

• S is s-strongly-admissible if S is strongly-safe and strongly-defends all its elements.

• S is s-τ-admissible if S is τ-safe and there exists a strong or weak defense for all its elements.

• S is s-weakly-admissible if S is weakly-safe and there exists a strong or weak defense for all its elements, or S is strongly-safe and weakly-defends all its elements.

• S is c-strongly-admissible if S strongly-conflict-free, closed under Rs and strongly-defends all its elements.

• S is c-τ-admissible if S τ-conflict-free, closed under Rs and there exists a strong or weak defense for all its elements.

• S is c-weakly-admissible if S weakly-conflict-free, closed under Rs and there exists a strong or weak defense for all its elements, or S strongly-conflict-free, closed under Rs and weakly-defends all its elements.

In this manner, in [11] it is argued that admissibility becomes a characteristic of a set of arguments that can be perceived from different perspectives. The most restrictive admissible sets do not admit conflicts and defend all their elements with values of controversy greater than the given threshold. A more flexible admissibility property is when a certain level of controversy associated with the set, which is limited by the threshold, is acceptable. In this case, the arguments’ defense can oscillate between strong and weak. Finally, the most flexible set allows the existence of conflicts where the controversy associated with them is strictly less than the threshold; i.e., the controversy is analyzed individually for each pair of conflicting arguments. In the last two cases, the arguments’ defense can fluctuate between strong and weak.

From the notions of coherence (internal and external) and admissibility, it is possible to introduce different acceptability semantics. In [11], Budan et al. introduced a more fine-grained definition of preferred extensions as follows:

Definition 17.

Let Φ=⟨Θ‾,SimC,CohCΘ‾,ContCΘ‾⟩ be a s-baf with the underlying enriched bipolar argumentation framework Θ‾=⟨Args,Rs,Ra⟩, and S⊆Args be a set of enriched arguments. Then:

• S is a d-strongly-preferred (resp. s-strongly-preferred, c-strongly-preferred) extension of Φ if S is ⊆-maximal among the d-strongly-admissible (resp. s-strongly-admissible, c-strongly-admissible) subsets of Args.

• S is a d-τ-preferred (resp. s-τ-preferred, c-τ-preferred) extension of Φ if S is ⊆-maximal among the d-τ-admissible (resp. s-τ-admissible, c-τ-admissible) subsets of Args.

• S is a d-weakly-preferred (resp. s-weakly-preferred, c-weakly-preferred) extension of Φ if S is ⊆-maximal among the d-weakly-admissible (resp. s-weakly-admissible, c-weakly-admissible) subsets of Args.

Next, we analyze our running example to obtain the different types of acceptable argument sets, where the properties of conflict-freeness and safety are considered.

Example 6.

We continue analyzing the Example 5 presented in Fig. 5, introducing a threshold τ=0.48. With that addition we obtain:

• Weakly-direct attacks, with controversy coefficient lower than τ, are from B to D, H to I, and from I to I.

• Strongly-direct attacks, with a controversy coefficient greater than τ, are from F to J, J to F, and from G to E.

• Weakly-supported attacks are from C to D (since CohC({(C,B)})⩾τ and ContC({(B,D)})<τ), from A to D (because CohC({(A,B)})⩾τ and ContC({(B,D)})<τ), from E to J (given that CohC({(E,D),(D,F)})<τ and ContC({(F,J)})⩾τ), and from G to I (due to CohC({(G,H)})⩾τ and ContC({(H,I)})<τ).

• Strongly-supported attacks, with the controversy and cohesion coefficients greater than τ, are from E to J, and from K to F.

• A strongly-secondary attack, with the controversy and cohesion coefficients greater than τ, is from G to F.

• A weakly-secondary attack is from B to F because ContC({(B,D)})<τ) and CohC({(D,F)})⩾τ.

Additionally, we have:

• S1={A,B,C}, is strongly-conflict-free, strongly-safe, because there are no elements in the set that simultaneously support and attack external arguments. This set does not receive attacks, therefore is d-strongly-admissible, s-strongly-admissible and c-strongly-admissible sets (because is closed under support relation), however it is not a maximal set (see Fig. 6).

• S2={D,E,F} is strongly-conflict-free and strongly-safe. However, the set does not defend D from the attack from B. Furthermore, there is a conflict cycle between the arguments F and J (see Fig. 6).

• S3={H,G} is strongly-conflict-free and strongly-safe. This set does not receive attacks, therefore is d-strongly-admissible, s-strongly-admissible, and c-strongly-admissible set. However, it is not a maximal set (see Fig. 6).

• S4={K,J} is strongly-conflict-free and strongly-safe. However, there is a conflict cycle between the arguments F and J (see Fig. 6).

• S5={A,B,C,H,K,J,G} is a strongly-conflict-free, strongly-safe set and it is closed under support relation. The set S5 strongly-defends J from F attacks; therefore, it is a d-strongly-admissible, a s-strongly-admissible, and a c-strongly-admissible set. This is a maximal set, therefore is a d-strongly-preferred, a s-strongly-preferred, and a c-strongly-preferred extension (see Fig. 7).

• S6={A,B,C,H,K,J,G,I} is a τ-conflict-free, strongly-safe set and it is closed under support relation. The set S6 strongly-defends J from F attacks; therefore, it is d-τ-admissible, a s-strongly-admissible, and a c-τ-admissible set. This is a maximal set, therefore it is a d-τ-preferred extension and a c-τ-preferred extension (see Fig. 7).

Fig. 6.

Analysis of admissibility in s-baf.

Fig. 7.

Preferred extensions in s-baf.

As we can see, the analysis of a baf enriched with the added notion of threshold becomes complex, but it provides more information to obtain a set of arguments with the property of conflict-freeness or safety. Moreover, different levels can be defined, where the notion of conflict-free and safety can lead to weakening allowing the acceptance of arguments that otherwise would not have been accepted. Next, we will analyze how these arguments can be grouped in communities using the notion of a coalition and how they are related in an argumentation domain.

3.Conceptualizing communities in argumentation

A human community is a social unit considered a discrete constituent of society with shared norms, values, customs, or identity. Communities may share a sense of being placed in a given geographical area (e.g., country, village, town, or neighborhood) or a virtual space through digital platforms, including associations expanding outside direct genealogical relations, which also define a sense of community, becoming essential to the identity, practice, and roles in the various usual social institutions such as family, workplace environment, governmental organizations, or any other social construct to which individuals consider themselves as participants [25]. As we can see, formulating a definition of the community term is a complex task that is being approached from different perspectives [9]. It is possible to find several notions about this kind of organization, like the ones that follow:

From the perspective of Social Psychology, a person is attracted to a group in which they can serve as inspiration or give an opinion according to the culture of the social organization [34]. This characteristic has a significant effect on the cohesion of a community. However, another critical concept exists when we refer to these groups: the consensual validation that represents the uniformity and conformity in the community. It is important to note that the members of the community share feelings, opinions, beliefs, priorities, or goals. In other words, the community has structural and semantic aspects [14]. From a different standpoint, Sarason in [45] argues that a psychological sense of community is the perception of being similar to others. There is an acknowledged interdependence with others, a willingness to maintain it by giving to or doing what others expect from them, including the feeling that one is part of a larger, dependable, stable structure.

Adopting a practical stance, McMillan and Chavis in [34] identify four elements of the “sense of community” involving the four aspects of membership, sense of influence, the fulfillment of needs, and a shared emotional connection: (i) the feeling of belonging to a group or, in other words, a sense of membership, (ii) the feeling of being essential to the group or having influence in this group, (iii) the members of the community are integrated into it, and they can fulfill their necessities leading to the reinforcement of that feeling, and (iv) the group members share an emotional connection represented by the belief that they have and will continue having a shared history and places, spending time together, and partaking of comparable experiences. Concerning this, a “sense of community index (SCI)” was developed by Chavis and colleagues22 to assess the sense of community in neighborhoods, and the index is used to characterize schools, the workplace, and a variety of types of communities [38].

All the observations above outline an essential characteristic of a community: its members have a shared context unique to them. The context informs all the activities inside the community and provides the background information necessary for reasoning and acting by its members.

Fig. 8.

Communities in bipolar argumentation frameworks.

Now, considering our specific application domain, Porter in [40] defines a virtual community as an aggregate of individuals or business partners (in connection with one or more organic communities) that interact on a shared (or complementary) interest and in which a common language implements the interaction and eventually a possible common paralanguage, led by some protocols or shared norms. Taking as a basis this definition, Prodnik in [41] establishes a virtual community as a social construction where the language is the basis of its organization, and the technology in general and the internet, in particular, have a predominant role. Given the importance of the language in virtual communities, arguments can be helpful in detecting them, reflecting the thoughts of the organization’s members in a given discussion. Thus, arguments may have an opinion for or against a specific statement, representing communities interacting in a particular argumentative discourse.

In this work, when referring to a community, it will be understood as a group of agents presenting different postures through a set of arguments expressing supporting and conflicting positions in a setting akin to a debate (see Fig. 8). Support can be interpreted as a relationship among the group members through common opinions; consequently, coalitions in baf will represent a community. Furthermore, the support relation will have associated a measure of internal cohesion of the community following [11]; thus, this measure can be considered an SCI in the argumentation domain, reflecting to a certain degree the four characteristics mentioned above. Finally, the conflict relationship between different communities represents how different positions on a specific statement are brought into play. Measuring the degree of conflict between different positions is essential in determining how strong the relationship is, considering the degree of controversy between communities.

4.Communities from valuated-similarity coalitions

Given a system that represents knowledge as arguments and considers the existing conflicts and supports between these arguments, a primary goal is to find sets of arguments that can be kept together by handling the conflicts appropriately while fulfilling relevant properties. It is feasible to create maximal cohesive sets of arguments by taking advantage of the mechanism proposed in [11] to collect in a set as many conflict-free and related-by-support arguments as possible, ensuring coherence of the whole set. Nevertheless, it is also interesting to include some degree of controversy by considering the addition of attacks and maintaining a coherence threshold in a dialogue or debate. Thus, it is possible in the proposed framework to find sets of arguments or stances that conform to a community, where for the present work, a community is a set of consistent stances in favor (or against) a specific topic. The threshold has two different meanings in the valuation proposed here. On the one hand, the threshold is the maximal degree of controversy that a community can admit without losing the essence that identifies it as a discursive and coherent community. On the other hand, the threshold represents the minimal level of coherence required by a set of opinions to be considered at least as a community with a moderately solid and consolidated position among its members. There might be many possible threshold settings; in each case, it is essential to determine the most appropriate one to use. This threshold setting is a methodological issue involving the semantics of the domain. The question could be tackled by devising experiments using examples where the desired conclusion is well known or by performing tests using the cognitive evaluation of human subjects to approximate their assessment of the valuations obtained after their interactions. Furthermore, according to [46], it could be challenging to find the correct value for a heuristic threshold; a complete discussion of the generality of this choice for representing uncertain information can be found in [50]. This issue exceeds the scope of our work. However, given the practical usefulness of this parameter, we plan to return to this question in future works.

When analyzing social media conversations as an exchange of arguments, it is natural to find many arguments in favor of a conclusion; generally, these arguments are similar but have some nuances in their meanings. To recognize communities, we propose considering an argumentation graph where the arguments are decorated with labels that will allow us to determine how similar the supported and attacked arguments are. Aiming at that, we introduce a bipolar argumentation graph whose arcs are labeled with a similarity degree between the related arguments, as follows:

Now, it is necessary to revisit the concept of coalitions given by Cayrol and Lagasquie-Schiex in [12] to extend it by formalizing how a similarity degree can influence the support relations. Formally:

Note that self-attacking arguments are disregarded in this approach according to the classic definition of a coalition where no attacks are permitted (Definition 5). In other words, an opinion that contradicts itself cannot be part of a discourse community. However, in future research, the attack might spur different coalition classes by weakening the strong-conflict-free condition by admitting certain conflicting opinions within a community.

The following result follows naturally from the definition of S-coalitions.

Once the set of coalitions is obtained, we can use the internal coherence of each element in this set to characterize an s-coalition. Note that, as an s-coalition is a set of enriched arguments, we can use the cohesion function established in Definition 12 to determine a cohesion measure associated with that s-coalition.

Intuitively, a strong-coalition does not admit conflict between its arguments, assuring that these pieces of knowledge refer to the same aspects of the argumentation process, i.e., the opinions allude to precisely the same values for each considered descriptor. In a τ-coalition, even though the arguments do not contradict each other, they essentially refer to the same aspects, i.e., the opinions allude to the same values for each descriptor considered but contain some descriptors whose values differ. Lastly, in a weak-coalition, although the arguments do not contradict each other, they can refer to the same aspects differently, or some of them might refer to different aspects of the issue, i.e., either the opinions mainly allude to each considered descriptor differently, or each opinion refers to different descriptors. The threshold is a sensitive value to determining what type the s-coalition is a group of supported opinions. A higher threshold value allows more flexibility in the descriptors’ values for each considered descriptor.

A coalition can be regarded as an argument at a meta-level built from argumentation stances that deal with contextual features with different degrees of strength. In an everyday discussion, even when stances do not contradict each other, the different members of the coalition do not have the same strength. So, we can associate a coalition to discourse communities. In this direction, a possible way to detect and classify these discourse communities is to find s-coalitions. Furthermore, the following criteria for distinguishing discourse communities from coalitions can be considered:

Those criteria are defined taking into account: (i) the collective or consensual validation as a distinctive feature in an argumentative process, (ii) the necessity of modeling the community as a computable concept, and (iii) the requirement to distinguish the connection level (emotional or any other kind of support), as well as any other distinctive features in the discourse communities.

According to the characteristics of a community, an sd-robust-community represents a maximum level of consensual validation and strong support (possibly emotional) connection. In contrast, an sd-fragile-community shows a minimum level of consensus and support connection inside the organization. Next, we introduce an example to clarify the preceding ideas.

Example 7.

Continuing our Example 6, using a product T-norm to obtain the cohesion value, considering a τ=0.48, and analyzing the abstract argumentation framework represented in Figure 9, we have that the coalitions C1={A,B,C}, C2={E,D,F} are weak because the CohC(C1)=0.2<τ, CohC(C2)=0.34<τ. On the other hand, C3={G,H} and C4={K,J} are τ-coalitions since the τ⩽CohC(C3)=0.5<1 and τ⩽CohC(C4)=0.6<1. However, if we choose a different function to obtain the cohesion of the sets, the max T-conorm for instance, we have that: CohC(C1)=0.8>τ; the same occurs with C2. Under this perspective, all the communities are τ-coalitions. At the level of semantic analysis, by using a product T-norm to obtain the cohesion value, we find that the C1 and C2 are sd-fragile-communities, while in the second interpretation that relies on a max T-conorm, we conclude that C1 and C2 are sd-moderate-communities. Finally, C3 and C4 are sd-moderate-communities in either analysis.

Fig. 9.

Communities in bipolar argumentation frameworks.

The difference between an sd-robust-community and an sd-moderate-community is a design choice. Still, we believe the intuition behind an s-coalition is that the stances (arguments) in it must be fully supported, taking into account the aspects they refer to. It is possible to follow a simple procedure to find and classify s-coalitions computationally from an s-valued bipolar argumentation graph: first, consider the paths following the support relation; then, calculate the coherence value for each path found considering the threshold τ given; and finally, determine the corresponding s-coalition type based on these coherence values.

The nature of the support relation allows us to establish some properties. For example, when we find a strong support relation inside a coalition, it is natural to think that the cohesion associated with the supported arguments would be high.

Now, it is necessary to introduce an attack relationship between conflicting coalitions. The characterization of these new attacks considers the attacks between the arguments that are part of these coalitions as formalized in the following definition.

Intuitively, it is possible to say that if there is an attack between two arguments that belong to two different coalitions, then it is natural to raise this conflict to the coalition level and define now an attack between these coalitions.

Furthermore, it is interesting to study the strength of the attack from one coalition to another by considering the strength of the attacks that define the existing points of conflict. Formally:

The attack degree can be obtained by instantiating the SimC(·,·) similarity function with T-norms or T-conorms, considering the user modeling preferences. This measure returns a non-negative real number in [0,1], i.e., it is defined as StrCΦ:2Args→[0,1].

Once the attacks between s-coalitions are identified, and their strength is computed, we begin by using the attack degree to distinguish between strong and weak attacks. This classification can be employed to define different semantics by using different forms of acceptability; for instance, conflicting s-coalitions could be part of a set of acceptable s-coalitions when the attack degree is not strong enough to be considered a defeat.

The previous definition formalizes the intuition that a strong attack considers two necessary elements: the strength of attack and the s-coalition internal cohesion measure applied to the set of the enriched arguments in the s-coalition. Once the s-coalitions and associated attacks are obtained from the s-baf, we can formalize a new meta-argumentation framework to analyze a new kind of semantics concerning the set of communities.

Note that in the new meta-argumentation framework, the set CΦ of coalitions plays the role of the argument set, and the relation RaCΦ represents the set of attacks. Henceforth, we will describe this meta-argumentation framework ΦC=⟨CΦ,RaCΦ,StrCΦ⟩ through a weighted directed graph GCΦ, called meta-argumentation graph, with a unique kind of edge representing attacks between coalitions. Furthermore, each edge is assigned a weight representing the strength behind the attack it represents under the interpretation of attack strength.

Next, we will introduce the measure of controversy associated with a set of s-coalitions, where the different types of attacks are analyzed to specify how contradictory they are.

The instantiation of the controversy degree function is a design decision, i.e., users can choose what they consider more appropriate for the problem at hand. Two possible choices are the T-norms and T-conorms. This measure returns a non-negative real number in [0,1], i.e., it is defined as ContCΦ:2Args→[0,1].

Given that the controversy associated with a set of coalitions is the same as the controversy associated with the set of enriched arguments involved, the previous result establishes a common point between the s-baf and the meta-argumentation framework.

Now, based on the semantic analysis done in [11], we introduce the notions of conflict-free s-coalition sets in our meta-argumentation framework ΦC. Thus, it is possible to determine the set of communities that can coexist within an argumentative model.

The following proposition establishes the semantic connections between the meta-argumentation framework dealing with coalitions of arguments and the subjacent similarity-based argumentation framework.

Previous examples have examined how to obtain the coalitions associated with an s-baf, classifying them according to their degree of cohesion. The following example exercises the concepts just introduced by analyzing attacks between coalitions and the degree of controversy associated with them.

Example 8.

Continuing the analysis of Example 7, and recalling that the threshold set is τ=0.48 and considering the meta-argumentation graph presented in Figure 10, we have that:

• There is a conflict point between the s-coalitions C1 and C2: the pair (B,D). In this case, StrCΦ(C1,C2)=0.4<τ, therefore, C1 weakly-attacks C2.

• There is a conflict point between the s-coalitions C3 and C2: the pair (G,E). In this case, StrCΦ(C3,C2)=0.8>τ, and CohC(C3)>τ. Thus, we can establish that C3 strongly-attacks C2.

• There is a conflict point between the s-coalitions C2 and C4: the pair (F,J) representing a weak attack, given that StrCΦ(C2,C4)=0.5>τ but CohC(C2)<τ. However, in this case, there is a conflict point between the s-coalitions C4 and C2 too: the pair (J,F) identified a strong attack because StrCΦ(C4,C2)=0.5>τ and CohC(C4)>τ.

Fig. 10.

Attacks between coalitions in a meta argumentation framework.

The characterization of the attack relationship between coalitions and considering the associated strength of attacks allows us to establish the following property. This result will be relevant to characterize how an s-coalition “absorbs,” or “assimilates” other s-coalitions.