Applying Attribution Explanations in Truth-Discovery
                                Quantitative Bipolar Argumentation Frameworks
                                Xiang Yin1 , Nico Potyka2 and Francesca Toni1
                                1
                                    Imperial College London, UK
                                2
                                    Cardiff University, UK


                                                                         Abstract
                                                                         Explaining the strength of arguments under gradual semantics is receiving increasing attention. For
                                                                         example, various studies in the literature offer explanations by computing the attribution scores of
                                                                         arguments or edges in Quantitative Bipolar Argumentation Frameworks (QBAFs). These explanations,
                                                                         known as Argument Attribution Explanations (AAEs) and Relation Attribution Explanations (RAEs), com-
                                                                         monly employ removal-based and Shapley-based techniques for computing the attribution scores. While
                                                                         AAEs and RAEs have proven useful in several applications with acyclic QBAFs, they remain largely
                                                                         unexplored for cyclic QBAFs. Furthermore, existing applications tend to focus solely on either AAEs
                                                                         or RAEs, but do not compare them directly. In this paper, we apply both AAEs and RAEs, to Truth
                                                                         Discovery QBAFs (TD-QBAFs), which assess the trustworthiness of sources (e.g., websites) and their
                                                                         claims (e.g., the severity of a virus), and feature complex cycles. We find that both AAEs and RAEs can
                                                                         provide interesting explanations and can give non-trivial and surprising insights.

                                                                         Keywords
                                                                         Explainable AI, Quantitative Argumentation, Truth Discovery Application




                                1. Introduction
                                Abstract argumentation Frameworks (AFs) [1] are promising tools in the Explainable AI (XAI)
                                field [2] due to their transparency and interpretability, as well as their ability to support
                                reasoning about conflicting information [3, 4, 5]. Quantitative Bipolar AFs (QBAFs) [6] are
                                an extension of traditional AFs, which consider the (dialectical) strength of arguments and
                                the support relation between arguments. In QBAFs, each argument has a base score, and its
                                dialectical strength is computed by gradual semantics based on its base score and the strength
                                of its attackers and supporters [7]. QBAFs can be deployed to support several applications like
                                product recommendation [8], review aggregation [9] or stance aggregation [10].
                                   Another interesting application that has been considered recently are truth discovery net-
                                works [11, 12, 13]. Figure 1 shows an example of a Truth-Discovery QBAFs (TD-QBAF) to evaluate
                                the trustworthiness of sources and the reliability of claims made about an exhibition. We have
                                11 sources and 6 claims, each represented as an abstract argument. The nodes on the left
                                represent the 11 source arguments (𝑠0 to 𝑠10), while the ones on the right represents the 6
                                claim arguments. The claim arguments are categorized into three types — year, place, and theme

                                ArgXAI-24: 2nd International Workshop on Argumentation for eXplainable AI
                                $ x.yin20@imperial.ac.uk (X. Yin); potykan@cardiff.ac.uk (N. Potyka); ft@imperial.ac.uk (F. Toni)
                                 000-0002-6096-9943 (X. Yin); 0000-0003-1749-5233 (N. Potyka); 0000-0001-8194-1459 (F. Toni)
                                                                       © 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
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Xiang Yin et al. CEUR Workshop Proceedings                                                        56–66




Figure 1: Example of a TD-QBAF. (Nodes are arguments, where the 𝑠𝑖 and 𝑐𝑖 are identifiers for the
source and claim arguments, respectively (for ease of reference). Solid and dashed edges indicate attack
and support, respectively.)


of the exhibition — each distinguished by different colors. For pairs of contradictory claims,
where different values are asserted for the same object, a bi-directional attack relationship
is introduced between the claims. For each report (one for each pair of source and claim), a
bi-directional support relationship is established between the source and the claim. Following
[13], we use a base score of 0.5 for source argument (we are initially indifferent about the
trustworthiness of a source), and a base score of 0 for claims (we do not believe claims without
evidence). We compute the dialectical strength of arguments using the Quadratic Energy (QE)
gradual semantics [14], and the final strengths of arguments are displayed on their side in Figure
1. While the strength values seem plausible, it can be challenging to understand why certain
claims and sources receive higher or lower trust scores.
   To address this problem, attribution explanations (AEs) have been proposed. Specifically, given
an argument of interest (topic argument) in a QBAF, AEs can explain the impact of arguments
on the topic argument. AEs can be broadly categorized into Argument Attribution Explanations
(AAEs) (e.g., [15, 16, 17]) and Relation Attribution Explanations (RAEs) (e.g., [18, 19]). AAEs
explain the strength of the topic argument by assigning attribution scores to arguments: the
greater the attribution score, the greater the argument’s contribution to the topic argument.
Similarly, RAEs assign the attribution scores to edges to measure their contribution. Removal-
based and Shapley-based techniques are commonly used for computing the attribution scores.
   However, most existing studies focus on explaining acyclic QBAFs rather than cyclic ones,
leaving a gap in understanding the complexities of the latter. In addition, current research
typically examines only one type of attribution — either AAEs or RAEs — without providing a



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comprehensive comparison of both methods. In this paper, we aim to address these gaps by
investigating the applicability of removal and Shapley-based AAEs and RAEs in the context of
cyclic TD-QBAFs. Furthermore, we offer a comprehensive comparison between them to better
understand the applicability of these AEs.


2. Preliminaries
2.1. QBAFs and the QE Gradual Semantics
We briefly recall the definition of QBAFs and the QE gradual semantics [14].

Definition 1 (QBAF). A Quantitative Bipolar Argumentation Framework (QBAF) is a quadru-
ple 𝒬 = ⟨𝒜, ℛ− , ℛ+ , 𝜏 ⟩ consisting of a finite set of arguments 𝒜, binary relations of attack
ℛ− ⊆ 𝒜×𝒜 and support ℛ+ ⊆ 𝒜×𝒜 (ℛ− ∩ℛ+ = ∅) and a base score function 𝜏 : 𝒜 → [0, 1].

The base score function in QBAFs assigns an apriori belief to arguments. QBAFs can be
represented graphically (as in Figure 1) using nodes to represent arguments and edges to show
the relations between them. Then QBAFs are said to be (a)cyclic if the graphs representing them
are (a)cyclic.
   In this paper, we use the QE gradual semantics [14] to evaluate the strength of arguments in
QBAFs. Like most QBAF semantics, it computes strength values iteratively by initializing the
strength value of each argument with its base score and repeatedly applying an update function.
Let us represent the strength of arguments in the 𝑖-th iteration by a function

                                         𝜎 𝑖 : 𝒜 → [0, 1],

where 𝜎 0 (𝛼) = 𝜏 (𝛼) for all 𝛼 ∈ 𝒜. In order to compute 𝜎 𝑖+1 from 𝜎 𝑖 , the update function first
computes the energy 𝐸𝛼𝑖 of attackers and supporters of each argument 𝛼 defined by
                                  ∑︁                      ∑︁
                     𝐸𝛼𝑖 =                  𝜎 𝑖 (𝛽) −               𝜎 𝑖 (𝛽).
                            {𝛽∈𝒜|(𝛽,𝛼)∈ℛ+ }           {𝛽∈𝒜|(𝛽,𝛼)∈ℛ− }

It then computes the strength in the next iteration via
                              ⎧
                              ⎨𝜏 (𝛼) − 𝜏 (𝛼) · (𝐸𝛼𝑖 )2             𝑖𝑓 𝐸𝛼𝑖 ≤ 0;
                                                1+(𝐸𝛼𝑖 )2
                  𝜎 𝑖+1 (𝛼) =                             𝑖 2
                              ⎩𝜏 (𝛼) + (1 − 𝜏 (𝛼)) · (𝐸𝛼 )𝑖 2      𝑖𝑓 𝐸𝛼𝑖 > 0.
                                                        1+(𝐸𝛼 )


The final dialectical strength of each argument 𝛼 is then defined as the limit lim𝑡→∞ 𝜎 𝑡 (𝛼). In
cyclic graphs, the strength values may start oscillating and the limit may not exist [20]. In all
known cases, the problem can be solved by continuizing the semantics [14, 13]. However, we
do not have space to discuss these issues in more detail here and will just restrict to examples
where the strength values converge.
  To better understand the QE gradual semantics, let us look at an example.




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Xiang Yin et al. CEUR Workshop Proceedings                                                      56–66




Figure 2: Example of a QBAF structure for computing the QE gradual semantics.


Example 1. Consider the QBAF in Figure 2, where the base scores are given as 𝜏 (𝛼) = 0.8, 𝜏 (𝛽) =
0.6, 𝜏 (𝛾) = 0.9, and 𝜏 (𝛿) = 0.7. Since 𝛽 and 𝛾 have no parents, we have 𝐸𝛽𝑖 = 𝐸𝛾𝑖 = 0 for all 𝑖
and thus 𝜎(𝛽) = 𝜏 (𝛽) = 0.6 and 𝜎(𝛾) = 𝜏 (𝛾) = 0.9. For 𝛿, we have 𝐸𝛿𝑖 = 𝜎 𝑖 (𝛾) − 𝜎 𝑖 (𝛽) = 0.3
for all 𝑖, hence 𝜎(𝛿) = 𝜏 (𝛿) + (1 − 𝜏 (𝛿)) · 0.32 /(1 + 0.32 ) = 0.72. For 𝛼, we have 𝐸𝛼𝑖 = 𝜎 𝑖 (𝛾) +
𝜎 𝑖 (𝛿) − 𝜎 𝑖 (𝛽) = 1.02 for all 𝑖 ≥ 1. Hence, 𝜎(𝛼) = 𝜏 (𝛼) + (1 − 𝜏 (𝛼)) · 1.022 /(1 + 1.022 ) = 0.90.

  In the remainder, unless specified otherwise, we assume as given a generic QBAF 𝒬 =
⟨𝒜, ℛ− , ℛ+ , 𝜏 ⟩ and we let ℛ = ℛ− ∪ ℛ+ We will often need to restrict QBAFs to a subset of
the arguments or edges, or change the base score function, as follows.

Notation 1. For 𝒰 ⊆ 𝒜, let 𝒬|𝒰 = ⟨𝒜 ∩ 𝒰, ℛ− , ℛ+ , 𝜏 ⟩. Then, for any 𝛼 ∈ 𝒜, we let 𝜎𝒰 (𝛼)
denote the strength of 𝛼 in 𝒬|𝒰 .

Notation 2. For 𝒮 ⊆ ℛ, let 𝒬|𝒮 = ⟨𝒜, ℛ− ∩ 𝒮, ℛ+ ∩ 𝒮, 𝜏 ⟩. Then, for any 𝛼 ∈ 𝒜, we let 𝜎𝒮 (𝛼)
denote the strength of 𝛼 in 𝒬|𝒮 .

Notation 3. For 𝜏 ′ : 𝒜 → [0, 1] a base score function, let 𝒬|𝜏 ′ = ⟨𝒜, ℛ− , ℛ+ , 𝜏 ′ ⟩. Then, for
any 𝛼 ∈ 𝒜, we let 𝜎𝜏 ′ (𝛼) denote the strength of 𝛼 in 𝒬|𝜏 ′ .

2.2. Truth Discovery QBAFs (TD-QBAFs)
TD-QBAFs allow reasoning about truth discovery problems using quantitative argumentation.
Truth discovery problems can be described concisely as truth discovery networks (TDNs) [11].
Formally, a TDN is a tuple N = (𝒮, 𝒪, 𝒟, 𝒫) consisting of a finite set of sources 𝒮, a finite set
of objects 𝒪,⋃︀a set 𝒟 = {𝐷𝑜 }𝑜∈𝑂 of domains of the objects, and a set of reports 𝒫 ⊆ 𝒮 × 𝒪 × 𝑉 ,
where 𝑉 = 𝑜∈𝑂 𝐷𝑜 , and for all (𝑠, 𝑜, 𝑣) ∈ 𝒫, we have 𝑣 ∈ 𝐷𝑜 , and there is no (𝑠, 𝑜, 𝑣 ′ ) ∈ 𝒫
with 𝑣 ̸= 𝑣 ′ . Given a TDN N , we are interested in a truth discovery operator that assigns a
trust score to each source and each claim [11].
   Singleton suggested to reason about TDNs using bipolar argumentation frameworks, where
we have bi-directional support edges between sources and their claims (trustworthy sources
make claims more believable, and, conversely, believable claims make sources more trustworthy)
and contradictory claims attack each other [12]. TD-QBAFs implement this idea with QBAFs,
where sources have a base score of 0.5 (we are initially indifferent about the trustworthiness of
sources) and claims have a base score of 0 (we do not believe anything without evidence).




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Xiang Yin et al. CEUR Workshop Proceedings                                                    56–66


Definition 2 (TD-QBAF induced from a TDN). The TD-QBAF induced from the TDN N =
(𝒮, 𝒪, 𝒟, 𝒫) is defined as 𝑄 = (𝒜, ℛ− , ℛ+ , 𝜏 ), where 𝒜 = 𝒮 ∪{(𝑜, 𝑣) | ∃𝑠 ∈ 𝒮 : (𝑠, 𝑜, 𝑣) ∈ 𝒫},
ℛ− = {(𝑐, 𝑐′ ) ∈ 𝒜2 ∩ 𝐶 2 | obj(𝑐) = obj(𝑐′ ), val(𝑐) ̸= val(𝑐′ )}, ℛ+ = {(𝑠, (𝑜, 𝑣)), ((𝑜, 𝑣), 𝑠) |
(𝑠, 𝑜, 𝑣) ∈ 𝒫}. 𝜏 (𝑠) = 0.5 for all 𝑠 ∈ 𝒮 and 𝜏 (𝑐) = 0 for all 𝑐 ∈ 𝐶.
Every QBAF semantics gives rise to a truth discovery operator that is defined by associating
each source and claim with its final strength under the semantics. The semantical properties of
QBAF semantics like balance and monotonicity directly translate to meaningful guarantees for
the derived trust scores.

2.3. Argument Attribution Explanations
In order to explain trust scores in TD-QBAFs, we recall the removal-based and Shapley-based
AAEs. AAEs aim at evaluating the impact of an argument on a given topic argument. The
removal-based AAEs proposed by [15] measure how the strength of the topic argument changes
if an argument is removed.
Definition 3 (Removal-based AAEs). Let 𝛼, 𝛽 ∈ 𝒜. The removal-based AAE from 𝛽 to 𝛼
under 𝜎 is:
                            𝜙𝛼𝜎 (𝛽) = 𝜎(𝛼) − 𝜎𝒜∖{𝛽} (𝛼).
   The Shapley-based AAEs [16, 21] use the Shapley value from coalitional game theory [22]
to assign attributions. Each argument in a QBAF is seen as a player that can contribute to the
strength of the topic argument. Intuitively, Shapley-based AAEs look at all possible ways how
the argument could be added to the QBAF and average its impact on the topic argument.
Definition 4 (Shapley-based AAEs). Let 𝛼, 𝛽 ∈ 𝒜. The Shapley-based AAE from 𝛽 to 𝛼
under 𝜎 is:
                      ∑︁    (|𝒜 ∖ {𝛼}| − |𝒰 | − 1)! |𝒰|! [︀
            𝜓𝜎𝛼 (𝛽) =
                                                                               ]︀
                                                           𝜎𝒰 ∪{𝛽} (𝛼) − 𝜎𝒰 (𝛼) .
                                    |𝒜 ∖ {𝛼}|!
                      𝒰 ⊆𝒜∖{𝛼,𝛽}


2.4. Relation Attribution Explanations
RAEs are similar to AAEs, but measure the impact of edges rather than the impact of arguments.
Analogous to the idea of removal-based AAEs [15], we consider the removal-based RAEs.
Definition 5 (Removal-based RAEs). Let 𝛼 ∈ 𝒜 and 𝑟 ∈ ℛ. The removal-based RAE from
𝑟 to 𝛼 under 𝜎 is:
                            𝜆𝛼𝜎 (𝑟) = 𝜎(𝛼) − 𝜎ℛ∖{𝑟} (𝛼).
  Shapley-based RAEs [18, 19] share the same idea with Shapley-based AAEs, but the attribution
objects are changed from arguments to edges.
Definition 6 (Shapley-based RAEs). Let 𝛼 ∈ 𝒜 and 𝑟 ∈ ℛ. The Shapley-based RAE from 𝑟
to 𝛼 under 𝜎 is:
                           ∑︁ (|ℛ| − |𝒮| − 1)! |𝒮|! [︀
                 𝜑𝛼𝜎 (𝑟) =
                                                                          ]︀
                                                       𝜎𝒮∪{𝑟} (𝛼) − 𝜎𝒮 (𝛼) .
                                      |ℛ|!
                          𝒮⊆ℛ∖{𝑟}




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3. Explaining TD-QBAFs with AAEs and RAEs
3.1. Settings
To compare the different AEs, we explain the strength of argument 𝑐5 in Figure 1. Since there
are 17 arguments and 32 edges in Figure 1, computing Shapley-based AAEs and RAEs exactly
is prohibitively expensive. We therefore apply the approximation algorithm from [19] that
approximates the Shapley values using sampling (we set the sample size to 1000).
   We report the removal and Shapley-based AAEs and RAEs in Figure 3 and 4 1 . In addition,
to provide intuitive explanations for argument 𝑐5, we visualize the removal and Shapley-
based AAEs and RAEs as shown in Figure 3 and 4, where blue/red arguments or edges denote
positive/negative AAEs or RAEs. The darkness of the color of arguments and the thickness of
the edges denote the magnitude of the their AAEs and RAEs, respectively2 .

3.2. Results and Analysis for AAEs
Figure 3 shows the results of removal and Shapley-based AAEs.
   For the removal-based AAEs, we observe that 𝑠7, 𝑠8, 𝑠9, and 𝑠10 have noticeably positive
influences on 𝑐5, followed by minor positive influences from 𝑠3, 𝑐1, and 𝑐2. This is because 𝑠7
to 𝑠10 are direct supporters for 𝑐5, whereas 𝑠3, 𝑐1, and 𝑐2 indirectly support 𝑐5. Specifically, 𝑐2
supports 𝑠3, 𝑠3 supports 𝑐1, 𝑐1 supports 𝑠7, and then 𝑠7 supports 𝑐5, meaning 𝑠3, 𝑐1, and 𝑐2 all
indirectly support 𝑐5. These indirect influences also explain why the AAEs of 𝑠3, 𝑐1, and 𝑐2 are
much smaller than those of 𝑠7 to 𝑠10. Besides, since 𝑠7 is supported by 𝑐1, its AAE is slightly
larger than those of 𝑠8 to 𝑠10, which have consistent AAEs due to their symmetrical structure
to 𝑐5. In contrast, 𝑠0, 𝑠1, 𝑠2, and 𝑐0 have minor negative influences on 𝑐5 because 𝑐0 attacks
𝑐1, an indirect supporter for 𝑐5. Furthermore, 𝑠0 to 𝑠2 support 𝑐0, and thus they have negative
influences on 𝑐5 as well. However, their negative influences are not obvious due to the indirect
influences. Finally, the remaining arguments have AAEs close to 0, indicating their negligible
influences on 𝑐5.
   When considering the Shapley-based AAEs, the results are similar to those of removal-based
AAEs, where 𝑠7 to 𝑠10 still have significant influences on 𝑐5. Unlike removal-based AAEs,
however, we notice that both 𝑐4 and 𝑠6 have minor negative influences on 𝑐5. This is because
𝑐4 directly attacks 𝑐5, while 𝑠6 indirectly attacks 𝑐5 by supporting 𝑐4, although the QE strength
of 𝑐4 is very small (close to 0). Also, the negative influences of 𝑠0 to 𝑠2 and 𝑐0 and positive
influence of 𝑐2 are relatively negligible compared with those of in removal-based AAEs due to
their indirect connection to 𝑐5.
   In this case study, both removal and Shapley-based AAEs can effectively capture the main
influential arguments despite having some tiny differences in those low contributing arguments.
This is mainly because of their different mechanisms of computing the AAEs. Another important
reason is probably due to the approximation algorithm used for Shapley-based AAEs, leading
to different AAEs even with the same sample size for the coalitions. We also noticed that
the qualitative influence (the sign) of those Shapley-based AAEs close to 0 is sensitive when

   1
       The numerical AAEs and RAEs can be found in the Appendix
   2
       The code of all experiments is available at https://github.com/XiangYin2021/TD-QBAF-AAE-RAE.



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                      Removal-based AAEs                                     Shapley-based AAEs
(Strengths) Sources                           Claims               Sources                             Claims          (Strengths)
   0.66       s0                           c0: Year = 1958           s0                            c0: Year = 1958        0.70

   0.66       s1                                                     s1

   0.66       s2                           c1: Year = 1962           s2                            c1: Year = 1962        0.47

   0.81       s3                                                     s3

   0.70       s4                           c2: Place = Bath          s4                            c2: Place = Bath       0.83

   0.70       s5                                                     s5

   0.50       s6                       c3: Place = London            s6                           c3: Place = London       ~0

   0.83       s7                                                     s7

   0.72       s8                       c4: Theme = History           s8                       c4: Theme = History          ~0

   0.72       s9                                                     s9

   0.72      s10                           c5: Theme = Art           s10                           c5: Theme = Art        0.90




Figure 3: Removal and Shapley-based AAEs for the topic argument 𝑐5 of TD-QBAF in Figure 1.
(Blue/red/grey nodes denote positive/negative/negligible AAEs, respectively. The darkness of nodes
represents the magnitude of their AAE values.)


applying the approximation algorithm, thus we do not visualize those close to 0. However, this
should not be a concern since their influence is negligible.

3.3. Results and Analysis for RAEs
Figure 4 shows the results of removal and Shapley-based RAEs.
   Let us first discuss the removal-based RAEs. We see that (𝑠7, 𝑐5) has the largest positive
impact on 𝑐5. Following closely are (𝑠8, 𝑐5), (𝑠9, 𝑐5), and (𝑠10, 𝑐5), which also have notably
positive influences on 𝑐5 because they are direct incoming supports to 𝑐5. There are also
four outgoing supports from 𝑐5, namely (𝑐5, 𝑠7), (𝑐5, 𝑠8), (𝑐5, 𝑠9), and (𝑐5, 𝑠10), with positive
influences but their RAEs are greatly smaller than that of the previous four as they are indirect
supports. For instance, 𝑐5 first supports 𝑠7, and then 𝑠7 supports 𝑐5, indicating the indirect
positive influence of (𝑐5, 𝑠7). Additionally, (𝑐5, 𝑐4) also contributes positively to 𝑐5 because
𝑐5 attacks its attacker 𝑐4, thus the attack from 𝑐4 to 𝑐5 is weakened. We can also observe



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Xiang Yin et al. CEUR Workshop Proceedings                                                                                  56–66


                      Removal-based RAEs                                        Shapley-based RAEs
(Strengths) Sources                             Claims                Sources                             Claims         (Strengths)
   0.66       s0                             c0: Year = 1958            s0                             c0: Year = 1958      0.70

   0.66       s1                                                        s1

   0.66       s2                             c1: Year = 1962            s2                             c1: Year = 1962      0.47

   0.81       s3                                                        s3

   0.70       s4                            c2: Place = Bath            s4                            c2: Place = Bath      0.83

   0.70       s5                                                        s5

   0.50       s6                           c3: Place = London           s6                           c3: Place = London      ~0

   0.83       s7                                                        s7

   0.72       s8                           c4: Theme = History          s8                           c4: Theme = History     ~0

   0.72       s9                                                        s9

   0.72      s10                            c5: Theme = Art             s10                           c5: Theme = Art       0.90




Figure 4: Removal and Shapley-based RAEs for the topic argument 𝑐5 of TD-QBAF in Figure 1.
(Blue/red/grey edges denote positive/negative/negligible RAEs, respectively. The darkness of edges
represents the magnitude of their RAE values.)


some marginal influences, such as the positive influences provided by (𝑐1, 𝑠7), (𝑠3, 𝑐1), and
(𝑐2, 𝑠3) on 𝑐5, while the negative influences from (𝑐0, 𝑐1), (𝑠0, 𝑐0), (𝑠1, 𝑐0), and (𝑠2, 𝑐0). The
remaining edges have RAEs close to 0, showing their negligible influence on 𝑐5.
   When it comes to the Shapley-based RAEs, which have similar effects to removal-based
RAEs, the four incoming supports to 𝑐5 are still the major contributors, and the four outgo-
ing supports from 𝑐5 have minor RAEs. Different from removal-based RAEs, Shapley-based
RAEs capture some different negligible influences, such as the negative influence by (𝑐4, 𝑐5)
and (𝑠6, 𝑐4). However, Shapley-based RAEs also disregard some tiny influences, like (𝑠0, 𝑐0),
(𝑠1, 𝑐0), and (𝑠2, 𝑐0), which are shown by removal-based RAEs.



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   In this case study, both removal and Shapley-based RAEs have a consistent ranking for the
main influential edges despite having some tiny differences in those low contributing edges.
The reasons are the same as we discussed above.
   Let us further compare the results of AAEs and RAEs. In this case study, we observe some
connections between AAEs and RAEs. For example, in both AAEs, the top-4 influential argu-
ments are 𝑠7 to 𝑠10, while in both RAEs, the outgoing edges from these arguments ((𝑠7, 𝑐5),
(𝑠8, 𝑐5), (𝑠9, 𝑐5), and (𝑠10, 𝑐5)) also rank in the top-4. In addition, 𝑠0 to 𝑠4 and 𝑐0 to 𝑐2 have
minor influences in the removal-based AAEs, while their incoming or outgoing edges also
have minor influences in the removal-based RAEs. A similar phenomenon can be found in
the Shapley-based AAEs and RAEs. While it is expected that the RAEs for outgoing edges
of important arguments are relatively high, the consistency observed across different sets of
arguments and edges is noteworthy. Besides, we found that removal or Shapley-based AAE of
an argument does not necessarily equate to the sum of RAEs of all its incoming and outgoing
edges, which goes against a reasonable expectation. We will leave the investigation of their
formal relationships for future work.


4. Conclusion
Since most existing applications of AAEs and RAEs focus on acyclic QBAFs, this paper inves-
tigated their applicability in cyclic QBAFs. First, we found that AAEs and RAEs can provide
intuitive explanations. By displaying the ranking of arguments or edges, it is easy to identify
the most influential arguments or edges in the QBAF without delving into the complex (cyclic)
structure of the QBAFs, particularly in TD-QBAFs where the number of arguments is typically
large and the connections between source arguments and claim arguments are bi-directional.
Second, AAEs and RAEs can provide interesting or even surprising explanations. For example,
in the case study provided earlier, one might overlook the influence between claim arguments
𝑐1 and 𝑐5 because they are in different topics (Year=1962 and Theme=Art), but AEs demonstrate
that 𝑐1 can contribute to 𝑐5 through 𝑠7. Third, RAEs provide more fine-grained explanations
than AAEs. This is because when computing AAEs, such as removal-based AAEs, removing an
argument means removing all the incoming and outgoing edges associated with that argument,
whereas RAEs offer a more detailed insight by processing every incoming and outgoing edge
individually. One can choose between them depending on the granularity for practical use.
   For future work, it would be worthwhile to investigate how different gradual semantics influ-
ence AAEs and RAEs, because the property satisfaction of semantics have an influence on the
property satisfaction of explanations. Additionally, the formal relationship between AAEs and
RAEs requires further exploration. However, we believe AAEs and RAEs can complement each
other, providing a deeper and more comprehensive understanding of the internal mechanisms
of QBAFs, particularly the interactions between arguments and edges in complex QBAFs.


Acknowledgments
This research was partially funded by the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (grant agreement No. 101020934,



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ADIX) and by J.P. Morgan and by the Royal Academy of Engineering under the Research Chairs
and Senior Research Fellowships scheme. Any views or opinions expressed herein are solely
those of the authors.


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Additional Results for AAEs and RAEs

Table 1
Comparison of removal-based AAEs and Shapley-based AAEs (in descending order) for the argument 𝑐5
of TD-QBAF in Figure 1. Note that they are in different scales.
            Argument      Removal-based AAE          Argument    Shapley-based AAE
                s7            0.084304029                s7          0.285373360
                s8            0.066738248                s9          0.259206533
                s9            0.066738248               s10          0.257762474
               s10            0.066738248                s8          0.256392126
                s3            0.006673061                c1          0.020544840
                c1            0.006635552                s3          0.007852405
                c2            0.002938110                c2          0.001789997
                s4            0.000076913                s5          0.000810191
                s5            0.000076913                s4          0.000789093
                s6           -0.000008421                s0         -0.000593859
                c3           -0.000008421                s2         -0.000917430
                c4           -0.000008423                s1         -0.001164825
                s0           -0.002444482                c3         -0.001309005
                s1           -0.002444482                c0         -0.001711158
                s2           -0.002444482                c4         -0.010154039
                c0           -0.002476209                s6         -0.010892383




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Table 2
Comparison of removal-based RAEs and Shapley-based RAEs (in descending order) for the argument 𝑐5
of TD-QBAF in Figure 1. Note that they are in different scales.
              Relation    Removal-based RAE         Relation    Shapley-based RAE
               (s7, c5)       0.083473613            (s7, c5)       0.255513421
               (s8, c5)       0.066745475           (s10, c5)       0.247761961
               (s9, c5)       0.066745475            (s8, c5)       0.245927825
              (s10, c5)       0.066745475            (s9, c5)       0.238930772
               (c5, s7)       0.022507211            (c5, s7)       0.024524066
               (c5, s8)       0.014968725            (c5, s9)       0.022019375
               (c5, s9)       0.014968725           (c5, s10)       0.020724751
              (c5, s10)       0.014968725            (c5, s8)       0.020059231
               (c5, c4)       0.014703252            (c5, c4)       0.015153831
               (c1, s7)       0.006793938            (c1, s7)       0.009541657
               (s7, c1)       0.006577898            (s3, c1)       0.006460594
               (s3, c1)       0.006576020            (s7, c1)       0.005974419
               (c2, s3)       0.002946488            (s3, c2)       0.001282426
               (c1, s3)       0.000805779            (s4, c2)       0.001206074
               (c1, c0)       0.000695966            (c2, s3)       0.001140980
               (s3, c2)       0.000212036            (s5, c2)       0.001083422
               (s4, c2)       0.000085191            (c2, c3)       0.000928671
               (s5, c2)       0.000085191            (c1, s3)       0.000881815
               (c3, c2)      -0.000007913            (c1, c0)       0.000834337
               (s6, c3)      -0.000007913            (c2, s5)       0.000670102
               (c3, s6)      -0.000007913            (c2, s4)       0.000631869
               (s6, c4)      -0.000007913            (c0, s2)       0.000586827
               (c4, s6)      -0.000007913            (c0, s1)       0.000558687
               (c4, c5)      -0.000007915            (c0, s0)       0.000558250
               (c2, c3)      -0.000115977            (c3, c2)       0.000164534
               (c2, s4)      -0.000120350            (s2, c0)       0.000144884
               (c2, s5)      -0.000120350            (s0, c0)      -0.000017152
               (c0, s0)      -0.000641070            (s1, c0)      -0.000036753
               (c0, s1)      -0.000641070            (s6, c3)      -0.000276491
               (c0, s2)      -0.000641070            (c3, s6)      -0.000465298
               (s0, c0)      -0.002436075            (c4, s6)      -0.001147533
               (s1, c0)      -0.002436075            (c0, c1)      -0.001659705
               (s2, c0)      -0.002436075            (c4, c5)      -0.028383722
               (c0, c1)      -0.002598473            (s6, c4)      -0.029099303




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