of the exhibition — each distinguished by diferent colors. For pairs of contradictory claims,
where diferent 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
[
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., [
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 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 ofer a comprehensive comparison between them to better understand the applicability of these AEs.
We briefly recall the definition of QBAFs and the QE gradual semantics [
Definition 1 (QBAF). A Quantitative Bipolar Argumentation Framework (QBAF) is a
quadruple = ⟨, ℛ− , ℛ+, ⟩ consisting of a finite set of arguments , binary relations of attack
ℛ− ⊆ × and support ℛ+ ⊆ × (ℛ− ∩ℛ+ = ∅) and a base score function : → [
In this paper, we use the QE gradual semantics [
∑︁ { ∈|(, )∈ℛ+} ( ) −
∑︁ { ∈|(, )∈ℛ− }
( ).
⎧ ( )2 +1( ) = ⎨ ( ) − ( ) · 1+( )2
≤ 0; ( )2 ⎩ ( ) + (1 − ( )) · 1+( )2 > 0.
The final dialectical strength of each argument is then defined as the limit lim→∞ ( ). In
cyclic graphs, the strength values may start oscillating and the limit may not exist [
To better understand the QE gradual semantics, let us look at an example.
Example 1. Consider the QBAF in Figure 2, where the base scores are given as ( ) = 0.8, ( ) = 0.6, ( ) = 0.9, and ( ) = 0.7. Since and have no parents, we have = = 0 for all and thus ( ) = ( ) = 0.6 and ( ) = ( ) = 0.9. For , we have = ( ) − ( ) = 0.3 for all , hence ( ) = ( ) + (1 − ( )) · 0.32/(1 + 0.32) = 0.72. For , we have = ( ) + ( ) − ( ) = 1.02 for all ≥ 1. Hence, ( ) = ( ) + (1 − ( )) · 1.022/(1 + 1.022) = 0.90.
In the remainder, unless specified otherwise, we assume as given a generic QBAF = ⟨, ℛ− , ℛ +, ⟩ and we let ℛ = ℛ
− ∪ ℛ the arguments or edges, or change the base score function, as follows.
+ We will often need to restrict QBAFs to a subset of denote the strength of in | .
Notation 1. For ⊆ , let | = ⟨ ∩ , ℛ− , ℛ+, ⟩. Then, for any ∈ , we let ( ) denote the strength of in | .
Notation 2. For ⊆ ℛ , let | = ⟨, ℛ− ∩ , ℛ + ∩ , ⟩. Then, for any ∈ , we let ( ) any ∈ , we let ′ ( ) denote the strength of in | ′ .
Notation 3. For ′ : → [
where = ⋃︀
∈
TD-QBAFs allow reasoning about truth discovery problems using quantitative argumentation.
Truth discovery problems can be described concisely as truth discovery networks (TDNs) [
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 [
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.
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 [
under is:
( ) = ( ) − ∖{ }( ).
The Shapley-based AAEs [
Let , ∈ . The removal-based AAE from to
under is: Let , ∈ . The Shapley-based AAE from to ( ) =
∑︁ ⊆∖{ , } (| ∖ { }| − | | − | ∖ { }|! 1)! | |! [︀ ∪{ }( ) − ( )︀] .
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 [
to under is:
() = ( ) − ℛ∖{}( ).
Shapley-based RAEs [
Let ∈ and ∈ ℛ. The removal-based RAE from
to under is:
Let ∈ and ∈ ℛ. The Shapley-based RAE from () =
∑︁ ⊆ℛ∖{ } (|ℛ| − || − |ℛ|! 1)! ||! [︀
∪{}( ) − ( )︀] . 3. Explaining TD-QBAFs with AAEs and RAEs
To compare the diefrent AEs, we explain the strength of argument 5 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 [
We report the removal and Shapley-based AAEs and RAEs in Figure 3 and 4 1. In addition, to provide intuitive explanations for argument 5, we visualize the removal and Shapleybased 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.
Figure 3 shows the results of removal and Shapley-based AAEs.
For the removal-based AAEs, we observe that 7, 8, 9, and 10 have noticeably positive influences on 5, followed by minor positive influences from 3, 1, and 2. This is because 7 to 10 are direct supporters for 5, whereas 3, 1, and 2 indirectly support 5. Specifically, 2 supports 3, 3 supports 1, 1 supports 7, and then 7 supports 5, meaning 3, 1, and 2 all indirectly support 5. These indirect influences also explain why the AAEs of 3, 1, and 2 are much smaller than those of 7 to 10. Besides, since 7 is supported by 1, its AAE is slightly larger than those of 8 to 10, which have consistent AAEs due to their symmetrical structure to 5. In contrast, 0, 1, 2, and 0 have minor negative influences on 5 because 0 attacks 1, an indirect supporter for 5. Furthermore, 0 to 2 support 0, and thus they have negative influences on 5 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 5.
When considering the Shapley-based AAEs, the results are similar to those of removal-based AAEs, where 7 to 10 still have significant influences on 5. Unlike removal-based AAEs, however, we notice that both 4 and 6 have minor negative influences on 5. This is because 4 directly attacks 5, while 6 indirectly attacks 5 by supporting 4, although the QE strength of 4 is very small (close to 0). Also, the negative influences of 0 to 2 and 0 and positive influence of 2 are relatively negligible compared with those of in removal-based AAEs due to their indirect connection to 5.
In this case study, both removal and Shapley-based AAEs can efectively capture the main influential arguments despite having some tiny diferences in those low contributing arguments. This is mainly because of their diferent mechanisms of computing the AAEs. Another important reason is probably due to the approximation algorithm used for Shapley-based AAEs, leading to diferent 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 1The numerical AAEs and RAEs can be found in the Appendix 2The code of all experiments is available at https://github.com/XiangYin2021/TD-QBAF-AAE-RAE. Removal-based AAEs
Shapley-based AAEs 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.
Figure 4 shows the results of removal and Shapley-based RAEs.
Let us first discuss the removal-based RAEs. We see that (7, 5) has the largest positive impact on 5. Following closely are (8, 5), (9, 5), and (10, 5), which also have notably positive influences on 5 because they are direct incoming supports to 5. There are also four outgoing supports from 5, namely (5, 7), (5, 8), (5, 9), and (5, 10), with positive influences but their RAEs are greatly smaller than that of the previous four as they are indirect supports. For instance, 5 first supports 7, and then 7 supports 5, indicating the indirect positive influence of (5, 7). Additionally, (5, 4) also contributes positively to 5 because 5 attacks its attacker 4, thus the attack from 4 to 5 is weakened. We can also observe some marginal influences, such as the positive influences provided by (1, 7), (3, 1), and (2, 3) on 5, while the negative influences from (0, 1), (0, 0), (1, 0), and (2, 0). The remaining edges have RAEs close to 0, showing their negligible influence on 5.
When it comes to the Shapley-based RAEs, which have similar efects to removal-based RAEs, the four incoming supports to 5 are still the major contributors, and the four outgoing supports from 5 have minor RAEs. Diferent from removal-based RAEs, Shapley-based RAEs capture some diferent negligible influences, such as the negative influence by (4, 5) and (6, 4). However, Shapley-based RAEs also disregard some tiny influences, like (0, 0), (1, 0), and (2, 0), which are shown by removal-based RAEs.
In this case study, both removal and Shapley-based RAEs have a consistent ranking for the main influential edges despite having some tiny diferences 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 arguments are 7 to 10, while in both RAEs, the outgoing edges from these arguments ((7, 5), (8, 5), (9, 5), and (10, 5)) also rank in the top-4. In addition, 0 to 4 and 0 to 2 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 diferent 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.
Since most existing applications of AAEs and RAEs focus on acyclic QBAFs, this paper investigated 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 1 and 5 because they are in diferent topics ( Year=1962 and Theme=Art), but AEs demonstrate that 1 can contribute to 5 through 7. 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 ofer 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 diferent gradual semantics influence 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.
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, 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.