A summary of Community Detection from Coalitions through Argument Similarity

Abstract

We present a summary of an argumentation-based method for finding and analyzing communities in social media on the Web, where a community is regarded as a set of supported opinions that might be in conflict. First, we identify argumentative coalitions to define communities; then, we apply a similarity-based evaluation method over the set of arguments in the coalition to determine the level of cohesion inherent to each community, classifying them appropriately. Introducing conflict points and attacks between coalitions based on argumentative (dis)similarities to model the interaction between communities leads to considering a meta-argumentation framework where the set of coalitions plays the role of the set of arguments and where the attack relation between the coalitions is assigned a particular strength which is inherited from the arguments belonging to the coalition. Various semantics are introduced to consider attacks' strength to particularize the efect of the new perspective.

Keywords

• eol>Argument Similarity
• Communities
• Coalitions
• Strength of Attacks between communities

1. Introduction

The identification of communities in social media and the detection of stances in Tweets has

become increasingly important in recent times [ 1, 2, 3, 4, 5 ] as a result of the tangible efect 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 [ 1, 6, 7 ]. Most of these methods focus on analyzing tweets to characterize the relationship between messages.

The present work summarizes the argumentation-based method propose by Budán et al. [8]

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.

Although Formal Argumentation Theory provides several formalisms to model emerging behavior, e.g. [9, 10, 11, 12, 13]; this paper is based on the well-known Abstract Argumentation Frameworks

(AFs) proposed by Dung [ 14 ], and in the Cayrol and Lagasquie-Schiex [ 15 ] extended Dung’s framework that considers two independent types of interactions between arguments: attack and support. This formalism is called Bipolar Argumentation Frameworks (bafs), and models situations where arguments may give support for other arguments.

Based on the bipolar formalism, in [ 16 ] the authors presented an approach to use a similarity degree measure between arguments to characterize the attack and support relations in a baf [ 17, 15 ]. In [ 8 ], the detection of communities in social media is based on the support and similarity relations between arguments, while the classification of the communities was done according to the similarity degree between the stances that conform to them, taking advantage of the notion of coalition presented in [ 18 ] and the framework proposed in [ 16 ]. During the whole work, the context where the argumentation discussion is put into play is considered. 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.

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

opinions extracted from the ProCon website§ in favor of (pro) or against (con) the following proposition “Is Human Activity Primarily Responsible for Global Climate Change?” : A Con1: More than one thousand scientists disagree that human activity is primarily responsible for global climate change.

B Con2: The Cook review of 11,944 peer-reviewed studies found that 66.4% of the studies had no stated position on anthropogenic global warming, and while 32.6% of the studies implied or stated that humans are contributing to climate change, only 65 papers (0.5%) explicitly stated: “that humans are the primary cause of global warming.” C Pro1: The rise in atmospheric CO2 over the last century was caused by human activity, as it occurred at a rate much faster than natural climate changes could produce.

D Pro2: A National Climate Assessment report said human-caused climate changes, such as increased heat waves and drought, “are visible in every state”.

E Undef1: A 2012 Purdue University survey found that 47% of climatologists challenge the idea that humans are primarily responsible for climate change and instead believe that climate change is caused by an equal combination of humans and the environment (37%), mostly by the environment (5%), or that there’s not enough information to say (5%).

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 C and D), another confronts the previous one with the opposite position (arguments A, B), and lastly, there is argument E representing an intermediate posture between the other two. 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.

Given a system that represents knowledge as arguments and considers the existing conflicts and supports between these arguments, it is feasible to create maximal cohesive sets of arguments §See https://climatechange.procon.org. The ProCon website states that its goal is “To promote civility, critical thinking, education, and informed citizenship by presenting the pro and con arguments to debatable issues in a straightforward, nonpartisan, freely accessible way.” by taking advantage of the mechanism proposed in [ 16 ] to collect in a set as many conflict-free and related-by-support arguments as possible, ensuring coherence of the whole set.

2. Background

In [15], Cayrol and Lagasquie-Schiex proposed an approach known as Bipolar Argumentation

Framework, to model the support and the attack between arguments:

Definition 1 (Bipolar Argumentation Framework ( baf)). A Bipolar Argumentation Framework is a 3-tuple Θ = ⟨Args, R, R⟩, where Args is a set of arguments, and R and R are two disjoint binary relations defined on Args called attack and support, respectively.

Cayrol et al. [15] presented the extensions of the attack and support notions introducing the

supported and secondary defeats, denoting with GΘ the bipolar argumentation graph: Definition 2 (Defeat in baf). Let Θ = ⟨Args, R, R⟩ be a baf, and A, B two arguments in Args. Then, it is said that: – A is a supported defeat for B if there exists a sequence A1 R1 . . . R A+1, with ≥ 1, where A1 = A and A+1 = B, such that R = R, 1 ≤ ≤ − 1, and R = R, A ∈ Args, 1 ≤ ≤ + 1. – A is a secondary defeat for B if there exists a sequence A1 R1 . . . R A+1, with ≥ 2, where A1 = A and A+1 = B, such that R1 = R, and R = R, 2 ≤ ≤ , A ∈ Args, 1 ≤ ≤ + 1.

Considering the simplest case of defeat in any baf, a sequence of two arguments A R B is also regarded as a supported defeat from A to B, i.e., a direct defeat is also a supported defeat.

A set of arguments must keep a minimum of coherence not containing an argument that

attacks another one in the same set [ 15 ]. And an external coherence requiring that the set does not include both a supporter and an attacker of the same argument: Definition 3 (Conflict-freeness and Safety Properties in BAF). Let Θ = ⟨Args, R, R⟩ be a baf, and S ⊆ Args be a set of arguments. We say that S is conflict-free if ∄ A, B ∈ S s.t. there is an attack (direct, or supported, or secondary) from A to B. We say that S is safe if ∄ A ∈ Args and ∄ B, C ∈ S s.t. there is an attack (direct, or supported, or secondary) from B to A, and either there is a sequence of support from C to A, or A ∈ S.

The closure under R introduced in [15] is a requirement that only concerns the support relation.

Definition 4 (Closure Property in BAF). Let Θ = ⟨Args, R, R⟩ be an baf. S ⊆ Args be a set of arguments. S is closed under R if ∀ A ∈ S, ∀ B ∈ Args if A R B then B ∈ S. Definition 5 (Coalitions in a baf). Let Θ = ⟨Args, R, R⟩ be a baf, and GΘ be a bipolar argumentation graph. A subset Θ ⊆ Args is a coalition in Θ if Θ is a maximal conflict free set in Θ such that the subgraph G′Θ induced by Θ is connected only by support relations.

C1 C2

A coalition represents a relationship on the set of arguments; therefore, the notion of attack

between them introduces a meta-argumentation framework: Definition 6 (Attack between coalitions in baf). Let 1 and 2 be two coalitions over Θ. If there exist A ∈ 1 and B ∈ 2 such that A R B, then the coalition 1 c-attacks (or just attacks) 2. Example 1. In the figure 1 we present a baf example described as Θ = ⟨Args, R, R⟩, where: Args = {A, B, C, D, E, F, G}; R = {(B, D)}; R = {(A, B), (C, B), (E, D), (E, F), (D, F)}. In the bipolar argumentation graph GΘ of this particular baf we have that the argument G is a secondary defeater for F, while C and A are supported defeaters for argument D. Moreover, we distinguish the following coalitions: 1 = {A, B, C}, highlighted green and 2 = {E, D, F}, highlighted purple. These are maximal conflict-free sets and 1, 2 are maximal sets closed under R. Additionally, we have that: 1 attacks 2, because B attacks D. 2.1. A Similarity Valued Argumentation Framework In [ 16 ] the authors presented a Similarity-based Bipolar Argumentation Framework (or s-baf), which is a mechanism for considering the context of the comparison between arguments, introducing enriched arguments, that is, arguments decorated with additional information. They assume a set of available descriptors corresponding to the discursive domain. Each descriptor has a set of values associated; for a descriptor ∈ , is the set of semantic values. Definition 7 (Enriched Argument). Let Θ = ⟨Args, R, R⟩ be a baf, A be an abstract argument in Θ, and be a set of descriptors. An enriched argument is a pair A = ⟨A, A⟩, where A is a finite non-empty set of pairs (, A ), where ∈ and ⊆ . The set of all enriched A arguments will be denoted as Args.

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

Definition 8 (Context). Let be a set of descriptors, a context C will be represented as C = {(, ) | ∈ , ∈ [ 0, 1 ]}, i.e., a context is a set of ordered pairs where ∈ is a descriptor and ∈ [ 0, 1 ] is the weight associated with . We denote with C the set of descriptors mentioned in the context C, i.e., C = { | (, ) ∈ C}.

Given a context C, for any argument X ∈ Args, we denote the descriptors in X that appear

on the context C as XC, i.e., XC = X ∩ C.

Definition 9 (Similarity coeficient for a descriptor). Let Args be a set of enriched arguments, A = ⟨A, A⟩ and B = ⟨B, B⟩ two enriched arguments in Args, and C a context. We define the similarity coeficient for each descriptor ∈ AC ∩ BC with weight , denoted Coef(A, B), as follows: ⎧ A B ⎪ | ∩ B | · ⎪⎪ A Coef(A, B) = ⎨ | ∩ | ⎪ ⎪⎪⎩ if | ∩ B |̸= 0

A otherwise

Intuitively, to determine the similarity between two arguments based on a specific descriptor, you count the shared semantic values in that descriptor, divide by the non-shared values, and adjust it based on the descriptor’s relevance in the given context, as mentioned in several references [19, 20, 21].

Definition 10 (Similarity degree between arguments). Let Args be a set of enriched arguments, A = ⟨A, A⟩ and B = ⟨B, B⟩ 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) = {︃ 0 if AC ∩ BC = {1, . . . , } ̸= ∅ otherwise where 1 = Coef1 (A, B) and = ⊙ ( − 1, Coef (A, B)) with 2 ≤ ≤ , 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.

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

Example 2. Suppose that we are analyzing the arguments A and C of our example, which have the following descriptors and values: A = {(climate_change, {});(refers_evidence, {})}; C = {(climate_change, {});(refers_evidence, {});(human_responsability , {})}. Now, suppose that the context for the arguments comparison is the following: C = {(climate_change, 0.4);(human_responsability , 0.4);(non_human_causes, 0.2)}. For climate_change descriptor, we have that the two arguments have a single value in common and no diferent ones. So, the Coef(A, C) = 0.4; for human_responsability , arguments have diferent values for this descriptor, and no common value. So that, according to the similarity coeficient definition, the Coef(A, C) = 0; and for non_human_causes, it is not mentioned in the arguments under analysis. Now, considering the bounded sum T-conorm, we have that the SimC(A, C) = 0.4, given that: (0.4 + 0, 1) = 0.4. The similarity value obtained reflects that the both arguments refer to climate change, but each argument in diferent way.

The following definition introduces the enriched baf framework based on the original baf:

Definition 11 ((Induced) Enriched baf). Let Θ = ⟨Args, R, R⟩ be a baf, the enriched baf induced is defined as Θ = ⟨Args, R, R⟩, where Args is the set of enriched arguments corresponding to arguments in Args, and R and R are the attack and support relationships in Args that are induced by R and R, respectively. 0.8 0.4 0.6 0.6

C

Now, we introduce the cohesion degree of a set of supporting enriched arguments and the controversy degree associated with a set of attacking enriched arguments.

Definition 12 (Cohesion & Controversy degrees). Given a set of enriched arguments S ⊆ Args and a context C, let SimC be a similarity degree function for C, and RS = {(X, Y) ∈ R | X, Y ∈ S} be the subset of R restricted to the arguments of S and RS = {(X, Y) ∈ R | X, Y ∈ S} be the subset of R restricted to the arguments of S then we have: – The cohesion degree for S, denoted as CohC(S), is defined as:

where 1 = SimC(A1, B1) and = ⊕ ( − 1, SimC(Ai , Bi )) with 2 ≤ ≤ . – The controversy degree for S, denoted as ContC(S), is defined as:

CohC(S) = ContC(S) = {︃

0 {︃ 0 if RS = {(A1, B1), . . . , (A, B)} ̸= ∅ otherwise if RS = {(A1, B1), . . . , (A, B)} ̸= ∅ otherwise where 1 = SimC(A1, B1) and = ⊗ ( − 1, SimC(Ai , Bi )) with 2 ≤ ≤ .

Both CohC(· ) and ContC(· ) can be obtained independently using a recursive function instantiated

with T-norms or T-conorms, in the same manner as with the similarity function SimC, depending on the user modeling intentions [ 22 ]. Next, we present our example where the similarity degree associated with each relationship was previously established (for more details, see [ 16 ]). Example 3. We continue with our abstract example, the graph in Figure 2 shows the similarity degree associated with the arguments in each relation. Intuitively, we can observe that the weakest support is between the arguments C and B and D and F. Based on the similarity degree obtained in each relation, we compute the cohesion coeficient associated with the set of supporting arguments (considering a product T-norm) and the controversy coeficient associated with attacking arguments (considering a max T-conorm). Thus, we have that: CohC({(E, D), (D, F)}) = 0.42; CohC({(A, B)}) = 0.8; CohC({(C, B)}) = 0.6; CohC({(E, F)}) = 0.8; and ContC({(B, D)}) = 0.4. 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 2). 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 (Similarity-based baf). Let Θ = ⟨Args, R, R⟩ 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 ∈ [ 0, 1 ] will be considered in the specification of the type of attack being analyzed. Consequently, the attacks in an s-baf will be of two types: () strong, when the cohesion and controversy values associated with the attack are greater than the threshold ; in this situation, we have strong-direct attack, strong-supported attack, and strong-secondary attack, or () weak, if at least one of the values is less than ; then, in this case, we have weakly-direct attack, weakly-supported attack, and weakly-secondary attack: Definition 14 (Conflict-freeness and Safety properties in a s-baf). Let Φ = ⟨Θ, SimC, CohC, ContC⟩ be a s-baf, where Θ = ⟨Args, R, R⟩ is the enriched baf, and ∈ [ 0, 1 ] be a given threshold. Then: – S is a strongly-conflict-free set if there is no A, B ∈ S such that there exists a strong or weak attack from A to B. – S is a -conflict-free set if there is no A, B ∈ S such that there exists a strong attack from

A to B and ContC(S) > . – S is a weakly-conflict-free set if there is no A, B ∈ S such that there exists a strong attack from A to B. – S is a strongly-safe set if 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 -safe set if 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}) > , and either there is a sequence of support from C to A such that CohC({C, . . . , A}) > , or A ∈ S. – S is weakly-safe set if 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}) > , or A ∈ S.

In the following step, in [16] the authors extended the notions of defense for an argument with respect to a set of arguments.

Definition 15. Let S ⊆ Args be a set of arguments, and A ∈ Args an argument. Then: – S is a strong defense for A if for all B ∈ Args such that if B is a strong or weak attacker of

A then there exists C ∈ S where C is a strong attacker of B. – S is a weak defense for A if for all B ∈ Args such that if B is a strong or weak attacker attacker of A then there exists C ∈ S where C is a weak attacker attacker of B.

C1

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

Example 4. We continue analyzing the Example 3 presented in Figure 2, introducing a threshold = 0.48. With that addition we obtain: a weakly-direct attacks, with controversy coeficient lower than , are from B to D; a weakly-supported attacks are from C to D (since CohC({(C, B)}) ≥ and ContC({(B, D)}) < ), from A to D (because CohC({(A, B)}) ≥ and ContC({(B, D)}) < ). Additionally, we have: 1 = {A, B, C}, is strongly-conflict-free , strongly-safe, because there are no elements in the set that simultaneously support and attack external arguments.

In the next, we explore how these arguments can be organized into communities or coalitions.

3. Communities from Valuated-Similarity Coalitions

The community term definition is a complex task that is being approached from diferent perspectives [ 23, 24, 22, 25, 26 ]. For us, a community is a group of agents presenting diferent postures through a set of arguments expressing supporting and conflicting positions in a setting akin to a debate (see Figure 3). Support signifies a relationship based on common opinions, forming coalitions that represent communities. Additionally, we measure the internal cohesion of a community. Conflict between communities indicates the degree of controversy on specific statements. In a knowledge system that represents arguments and their conflicts and supports, we can find cohesive sets of arguments by employing a mechanism from Budan et al. [ 16 ] to gather conflict-free and support-related arguments. We also introduce a degree of controversy by considering the addition of attacks while maintaining coherence. This results in sets of arguments or stances forming a community—a group with consistent stances on a topic. We define a threshold with dual significance: it represents the maximum controversy a community can have without losing coherence or the minimum coherence required for a community to have a solid position. To identify communities, we use a bipolar argumentation graph where arguments are labeled with similarity degrees between related arguments.

Definition 16 (S-valued bipolar argumentation graph). Given Φ = ⟨Θ, SimC, CohC, ContC⟩, an s-baf where Θ = ⟨Args, R, R⟩ is the underlying bipolar argumentation framework. An s-valued Bipolar Argumentation Graph, denoted GΦ, is the argumentation graph where the nodes are the elements of Args and the arcs between nodes depict the R (dashed arcs) and R (full arcs) relationships, where the arcs are decorated with the similarity degree SimC.

Now, it is necessary to revisit the concept of coalitions [15] to extend it by formalizing how a similarity degree can influence the support relations.

Definition 17 (S-coalitions ). Given an s-baf Φ = ⟨Θ, SimC, CohC, ContC⟩, where Θ = ⟨Args, R, R⟩ is the underlying bipolar argumentation framework, let GΦ be the s-valued bipolar interaction graph over Φ, and ⊆ Args be a set of enriched arguments. Then, we say that is an s-coalition if it is a maximally strongly-conflict-free set such that the sub-graph G′Φ induced by is connected only by support relations. We will denote as Φ the set of coalitions obtained from Φ.

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).

Proposition 1. Each enriched argument, which is not self-attacking, belongs to an s-coalition.

Given that a 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. Definition 18 (Types of s-coalitions). Let Φ = ⟨Θ, SimC, CohC, ContC⟩ be an s-baf, ∈ Φ be a coalition obtained from Φ, and ∈ [ 0, 1 ] be a threshold. Then: – is a strong-coalition if CohC() = 1. – is a -coalition if ≤ CohC() < 1.

– is a weak-coalition if 0 ≤ CohC() < .

Intuitively, in a strong-coalition the pieces of knowledge refer to the same aspects of the

argumentation process, without conflict. In a -coalition, the opinions allude to the same values for each descriptor considered but contain some descriptors whose values difer. Lastly, in a weak-coalition, although the arguments do not contradict each other, they can refer to the same aspects diferently, or some of them might refer to diferent aspects of the issue. Example 5. Continuing our Example 4, using a product T-norm to obtain the cohesion value, considering a = 0.48, and analyzing the abstract argumentation framework represented in Figure 3, we have that the coalitions 1 = {A, B, C}, 2 = {E, D, F} are weak because the CohC(1) = 0.2 < , CohC(2) = 0.34 < . However, if we choose a diferent function to obtain the cohesion of the sets, the max T-conorm for instance, we have that: CohC(1) = 0.8 > ; the same occurs with 2. Under this perspective, all the communities are -coalitions. At the level of semantic analysis, by using a product T-norm to obtain the cohesion value, we find that the 1 and 2 are sd-fragile-communities, while in the second interpretation that relies on a max T-conorm, we conclude that 1 and 2 are sd-moderate-communities.

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.

Proposition 2. Let Φ = ⟨Θ, SimC, CohC, ContC⟩ be an s-baf, where Θ is the underlying bipolar argumentation framework Θ = ⟨Args, R, R⟩, and let A, B ∈ Args two enriched arguments such that (A, B) ∈ R, and R does not contain (A, B) or (B, A). If A R B is either a strong-support or weak-support relation, then there exists at least a weak-coalition containing both A and B.

Now, it is necessary to introduce an attack relationship between conflicting coalitions:

Definition 19 (Internal attacks between s-coalitions). Given the s-baf Φ = ⟨Θ,SimC,CohC, ContC⟩, where Θ = ⟨Args, R, R⟩ is the underlying bipolar argumentation framework, let Φ be the set of s-coalitions obtained from Φ, and , ′ ∈ Φ be two s-coalitions. We will say that there exists an attack point from to ′ if there are two enriched arguments A ∈ and B ∈ ′ such that (A, B) ∈ R. We will denote as R[,′] the set of all attacks points between and ′.

Intuitively, it is possible to say that if there is an attack between two arguments that belong to

two diferent coalitions, then it is natural to raise this conflict to the coalition level and define now an attack between these coalitions.

Definition 20 (S-coalitions Attacks). Let Φ = ⟨Θ, SimC, CohC, ContC⟩ be an s-baf, where Θ = ⟨Args, R, R⟩ is the underlying bipolar argumentation framework, let Φ be the set of s-coalitions obtained from Φ. We will define the attack relation between s-coalitions derived from Φ, denoted RΦ , as

RΦ = {(, ′) | , ′ ∈ Φ and R[,′] ̸= ∅}

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:

Definition 21 (Strength of attack between s-coalitions). Let Φ be the set of s-coalitions obtained from Φ, , ′ ∈ Φ be two s-coalitions, and R[,′] = {(A1, B1), . . . , (A, B)} ⊆ R be the set of all attack points between and ′ with R[,′] ̸= ∅. The attack strength, or attack degree between and ′, denoted StrΦ(, ′), is defined as:

StrΦ(, ′) = , where is defined as 1 = SimC(A1, B1) and = ⊗ ( − 1, SimC(Ai , Bi )) with 2 ≤ ≤ .

The attack degree can be obtained by instantiating the SimC(· , · ) similarity function with T

norms or T-conorms, considering the user modeling preferences. 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 diferent semantics by using diferent forms of acceptability.

Definition 22 (Classification of attacks between s-coalitions). Given an s-baf Φ = ⟨Θ, SimC, CohC, ContC⟩, with Θ = ⟨Args, R, R⟩ as the underlying bipolar argumentation framework, let Φ be the set of s-coalitions obtained from Φ, , ′ ∈ Φ be two s-coalitions such that (, ′) ∈ RΦ , and ∈ [ 0, 1 ] be a threshold. We say that: - strongly-attacks ′ if CohC() ≥ and StrΦ(, ′) ≥ , - weakly-attacks ′ if CohC() < or StrΦ(, ′) < .

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. Now, we can formalize a new meta-argumentation framework to analyze a new kind of semantics concerning the set of communities.

Definition 23 (Meta-argumentation framework). Given Φ = ⟨Θ, SimC, CohC, ContC⟩, an s-baf with Θ = ⟨Args, R, R⟩ as the underlying bipolar argumentation framework, we define the meta-argumentation framework associated with Φ, as a 3-tuple Ω = ⟨Φ, RΦ , StrΦ⟩, where Φ is the set of s-coalitions obtained from Φ, RΦ is an attack relation between s-coalitions derived from Φ, StrΦ is the attack strength function defined over Φ.

Note that in the new meta-argumentation framework, the set Φ of coalitions plays the role of

the argument set, and the relation RΦ represents the set of attacks. Henceforth, we will describe this meta-argumentation framework Ω = ⟨Φ, RΦ , StrΦ⟩ through a weighted directed graph

GΦ , 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 diferent types of attacks are analyzed to specify how contradictory they are.

Definition 24 (Controversy degree for a s-coalition set). Given a meta-argumentation framework Ω = ⟨Φ, RΦ , StrΦ⟩, where Φ is a set of s-coalitions, RΦ is an attack relation, StrΦ is the attack strength function, ⊆ Φ be a set of s-coalitions, and R = {(1, 2), . . . , (− 1, )} ⊆ RΦ . The controversial measure for , denoted ContΦ (), is defined as:

ContΦ () = 0

otherwise {︃ if R ̸= ∅ where 1 = StrΦ(1, 2) and = ⊗ ( − 1, StrΦ(− 1, )) with 2 ≤ ≤ .

The instantiation of the controversy degree function is a design decision. Two possible choices are the T-norms and T-conorms.

Proposition 3. Let ⊆ Φ be a set of coalitions, and S ⊆ involved in , then ContC(S) = ContΦ ().

Args be the enriched arguments

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 [ 16 ], we introduce the notions of conflict-free s-coalition sets in our metaargumentation framework Ω. Thus, it is possible to determine the set of communities that can coexist within an argumentative model.

Definition 25 (Conflict-freeness in Ω ). Given a meta-argumentation framework Ω = ⟨Φ, RΦ , StrΦ⟩, where Φ is the set of s-coalitions, RΦ is an attack relation between s-coalitions, and StrΦ the strength of attack function. Given a controversy degree function ContΦ defined over Ω . Let ⊆ Φ be a subset of coalitions, and be a threshold. Then: – is a strongly-conflict-free set if there is no 1, 2 ∈ such that there exists a strong or weak attack from 1 to 2. – is a -conflict-free set if there is no 1, 2 ∈ such that there exists a strong attack from 1 to 2, and ContΦ

() ≤ . – is a weakly-conflict-free set if there is no 1, 2 ∈ such that there exists a strong attack from 1 to 2.

The following proposition establishes the semantic connections between the metaargumentation framework dealing with coalitions of arguments and the subjacent similaritybased argumentation framework.

Proposition 4. Let Ω = ⟨Φ, RΦ , StrΦ

⟩ be the meta-argumentation framework associated with Φ, ContΦ a controversy degree function defined over Ω , {1, . . . , } be a finite set of coalitions, and ∈ [ 0, 1 ] be a threshold. Then: ) {1, . . . , } is strongly-conflict-free for Ω if 1 ∪ · · · ∪ is strongly-conflict-free for Φ. ) {1, . . . , } is strongly-conflict-free for Ω if 1 ∪ · · · ∪ is strongly-safe for Φ. ) If 1 ∪ · · · ∪ is -conflict-free for Φ, then {1, . . . , } is -conflict-free for Ω . ) If 1 ∪ · · · ∪ is at least -safe for Φ then {1, . . . , } is -conflict-free for Ω . ) {1, . . . , } is weakly-conflict-free for Ω if 1 ∪ · · · ∪ is weakly-conflict-free for Φ. ) {1, . . . , } is weakly-conflict-free for Ω if 1 ∪ · · · ∪ is at least weakly-safe for Φ.

The following example exercises the concepts just introduced:

Example 6. Continuing the analysis of Example 5, and recalling that the threshold set is = 0.48 we have that there is a conflict point between the s-coalitions 1 and 2: the pair (B, D). In this case, StrΦ

(1, 2) = 0.4 < , therefore, 1 weakly-attacks 2.

The characterization of the attack relationship between coalitions and considering the associated strength of attacks allows us to establish the following property.

Proposition 5. Let 1, 2 ∈ Φ be two s-coalitions. If 1 and 2 are two disjoint s-coalitions that are connected by the attack relation, then there exists at least a weak-attack between 1 and 2.

Now, we will introduce the notions of defense for coalitions by extrapolating from the defense relationship between the arguments gathered in the coalitions.

Definition 26. Let Ω = ⟨Φ, RΦ , StrΦ⟩ be the meta-argumentation framework associated with Φ, ContΦ a controversy degree function defined over Ω , ⊆ Φ be a set of coalitions over Φ, and 1 ∈ Φ a s-coalitions. Then: – The set is a strong defense for 1 if for all 2 ∈ Φ such that if 2 is a strong or weak attacker of 1 then there exists 3 ∈ where 3 is a strong attacker of 2. – The set is a weak defense for 1 if for all 2 ∈ Φ such that if 2 is a strong or weak attacker of 1 then there exists 3 ∈ where 3 is a weak attacker of 2.

Example 7. Continuing with the running example, we observe that 1 does not receive any attack. Furthermore, there is no defense for the attacks of 1 to 2.

The concept of a coalition introduced provides a valuable framework for defining communities

within argumentation-supported debates. It is worth noting that the principles of conflict-freeness and safety can also be applied to these communities. These characteristics are particularly useful for analyzing intricate debates, such as those often encountered on social networks. This expanded formal argumentation theory equips us with tools to enhance the analysis of debates.

4. Related Works, Conclusion and Future Work

Puertas et al. [ 27 ] used Twitter data to detect social communities. They employed expert knowledge, computational linguistics, and AI techniques to extract vocabulary-based community features, and explore language-related relationships within the social network. One notable difference from our work is that the s-coalition detection method here doesn’t focus on individual opinions. Instead, it considers relationships among opinions to find and characterize communities or coalitions. However, both approaches involve language-related features. Puertas et al. employ term frequency techniques, while our method relies on enriched arguments using descriptors. Lenine [ 28 ] propose a work to categorize coalition into three types: conceptual (math-based), quasi-conceptual (deductive empirical), and extrapolative (statistical). Our work falls into the first category, building on s-baf [ 16 ]. Another notable approach by Vassiliades et al. [ 29 ] introduces an Abstract Argumentation Framework (AF) with domain-specific arguments, allowing the determination of argument acceptance scope. It difers from Budan et al. [ 16 ] in considering only attack relations, and using diferent argument modeling tools. Bistarelli at al. [ 30 ] detail a set of semantics based on weighted defences. A threshold obtained from collective attacks received by an argument is used to define a threshold representing the conflict permitted in the AF semantics without alter its coherence, providing flexibility. In our approach, the threshold is a given value that fulfills the same role, but it is applied both direct attacks and attacks that involve the support relationships necessary to express the coalition strength.

The detection of communities has broad applications today, such as identifying common research areas in collaboration networks, targeting like-minded users for marketing, or predictions in political areas. This work introduces a novel approach to identify meta-structures (coalitions) based on the similarity between supported arguments. We utilize similarity to characterize attacks between coalitions and assess controversy. However, these methods rely on specialized argument mining techniques and argument descriptors, and computational costs vary based on tree traversal. Future research directions include implementing this approach for coalition detection and community modeling, particularly in decision support systems with user preferences. Additionally, applying the proposed conceptualization to enhance argument schemes, especially those involving analogies represents an intriguing avenue for development.

Acknowledgments

This work was supported by funds provided by Universidad Nacional de Santiago del

Estero (UNSE) under grant 23/C201PIP-2023, Universidad Nacional del Sur (UNS) under

grants PGI 24/N046 and PGI 24/ZN057), Agencia Nacional de Promoción Científica y

Tecnológica under grants PICT-2018-0475 (PRH-2014-0007) and PICT-2020-SERIEA-01481,

and CONICET under grants PIP 11220200101408CO and PIP 11220170100871CO. The authors also acknowledge support by the Spanish project PID2022-139835NB-C21 funded by MCIN/AEI/10.13039/501100011033, MCIN/AEI/ 10.13039/501100011033 and by “European Union

NextGenerationEU/PRTR”.

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