arXiv CSopen access

A Framework for Reputation Aware Uninorm-driven Consensus Algorithms for Blockchain Networks

A Framework for Reputation Aware Uninorm-driven Consensus Algorithms for Blockchain Networks Bruno Ramos-Cruza , Javier Andreu-Pereza,b , David Richerbyb , Luis Martíneza a

Computer Science Department, University of Jaen, Jaen, 23071, Spain School of Computer Science and Electronic Engineering, University of Essex, Colchester, CO4 3SQ, United Kingdom

arXiv:2607.20700v1 [cs.DC] 22 Jul 2026

b

Abstract The operation of blockchain is governed by consensus algorithms (CA). Several consensus mechanisms require significant computational power, while others necessitate high amounts of stakes to select the participant to validate and verify the transactions in the block, leading to centralisation of power and participant exclusion. This paper proposes a novel methodology to address these issues in reputation-based consensus algorithms by studying the reputation behaviour of the validator using intuitionistic fuzzy sets (IFSs) and uninorm aggregation operations (UAOs). Our approach uses IFSs to express the ”reputation" because the reputation values in a consensus algorithm eventually imply uncertainty, and IFSs facilitate the representation of a lack of precise knowledge about reputation. Moreover, this methodology utilises uninorm aggregation operations to monitor reputation over time and reinforces the importance of negative and positive reputation. Consequently, this solution allows validators to rectify past failures in subsequent verification processes and foster an equitable consensus algorithm design. The proposed framework maintains linear computational complexity and does not introduce additional communication overhead beyond the underlying consensus protocol. Supported by experimental results, our methodology demonstrates improved performance and evaluation, promising advancements in blockchain ∗

Corresponding author: [email protected] (Javier Andreu-Perez) Email addresses: [email protected] (Bruno Ramos-Cruz), [email protected] (Javier Andreu-Perez), [email protected] (David Richerby), [email protected] (Luis Martínez)

network fairness and inclusivity. Keywords: Fuzzy sets, Intuitionistic fuzzy sets, Uninorm aggregation operators, Reputation-based consensus, Blockchain networks, Distributed ledger technology, Reputation management, Blockchain security 1. Introduction Blockchain technology has significantly transformed diverse industries and institutions, offering decentralised and transparent systems for transactions and data management [1]. However, despite its potential, some challenges persist within blockchain networks, particularly in the realm of consensus algorithms [2, 3]. These algorithms define rules for nodes on a distributed network, which ensures the validity and security of transactions, as well as maintaining trust and suitable network functioning. Consensus algorithms such as Proof of Work (PoW) [4, 5], Proof of Stake (PoS) [6], and Delegated Proof of Stake (DPoS) [7], among others, face hurdles related to the selection of nodes to carry out the block validation and verification process. For instance, PoW demands substantial computational power, PoS operates on a stake-based incentive model in which the nodes with the highest stake have a higher probability of being chosen, and DPoS allows network users vote for delegates to validate blocks, affecting decentralisation. Furthermore, for reputation-based consensus algorithms [6, 7], if a node loses its reputation, it can never again participate in the validation process. These challenges often lead to the concentration of power among a specific group of participants, consequently resulting in centralisation and the exclusion of smaller stakeholders, posing risks to the integrity and security of the network. Considering the challenges, this contribution focusses on studying the behaviour of the nodes involved in the block validation and verification process. To do so, a novel methodology is presented to monitor the reputation behaviour of the nodes in the consensus algorithm, integrating Intuitionistic Fuzzy Sets (IFSs) and Uninorm Aggregation Operations (UAOs) into blockchain networks. When discussing reputation in a consensus algorithm, the reputation values eventually imply uncertainty, and this proposal advocates the use of IFSs because the knowledge about reputation in a blockchain network cannot be complete. There is a relative uncertainty of the information about it; for this reason, IFSs facilitate the representation of a lack of knowledge about a 2

concept that, in this case, will be reputation. Moreover, in consensus algorithms where the reputation fluctuates, increasing and decreasing [8], classical algorithms typically reduce the reputation without providing a mechanism for recovery. In contrast, this methodology introduces a reputation recovery mechanism based on UAOs, reinforcing nodes that demonstrate positive behaviour while penalising those exhibiting negative behaviour within the network. UAO are mathematical functions used in fuzzy logic to combine multiple input values into a single output value. These operators present very interesting properties and play a crucial role in synthesising information from various sources. They are particularly useful for handling uncertain or imprecise data. The proposed methodology uses UAO to reinforce both the evolution of positive and negative reputations over time, computing a reputation weight for each node in the consensus algorithm that takes into account current and previous results in the validation process. Additionally, using UAO, validators’ reputation is monitored over time. This dynamic approach allows participants to rectify past mistakes, regain reputation, and promote a more equitable and inclusive consensus algorithm design. The main novelties of this paper are: • An innovative methodology for reputation aware uninorm-driven consensus algorithms for blockchain. • An intuitionistic framework for managing reputation uncertainty. • A reputation recovery mechanism that reinforces both positive and negative reputation evolution over time. • An analysis to define the suitable functions used in both IFSs and UOAs for the consensus algorithm in blockchain. Moreover, through empirical validation, an illustrative example is presented to show the performance of the proposed approach, showcasing its potential to offer a new paradigm for enhancing resilience, security, and diversification on the blockchain. The paper is structured as follows: Section 2 provides background on blockchain, intuitionistic fuzzy sets, and uninorm aggregation operations. Section 3 reviews related work. Section 4 details the methodology. Section 5 defines implementation features and presents key results. Finally, Section 6 and Section 7 discuss the future work and the conclusions, respectively. 3

V Vτ vi t SuccV R(vi , t) RepD(vi , t) w(vi , t) SV (vi , t) A µA (x) νA (x) πA (x) e U (x, y) T (a, b) S(a, b)

Nomenclature Set of validators Set of validators > τ i-th Validator Current round Successful Validation Rate Reputation Degree Reputation Weight Successful Validation Intuitionistic fuzzy set Membership Function Non-Membership Function Intuitionistic Index Neutral element Uninorm aggregated operator Triangular norm Triangular conorm

Table 1: This table shows the nomenclature used in this article.

2. Background This methodology is proposed to study the reputation behaviour in consensus algorithms for blockchain networks, and it is based on intuitionistic fuzzy sets and uniform aggregation operations. Then, this section briefly describes blockchain, intuitionistic fuzzy sets, and uninorm aggregation operations. Before defining the main concepts, the nomenclature used in this work is presented in Table 1. This table provides definitions for key terms to ensure clarity and consistency throughout the text. 2.1. Blockchain Blockchain technology was introduced by the pseudonymous SatoshiNakamoto in the context of Bitcoin [5]. A blockchain is a decentralised, immutable, and distributed ledger technology that enables a secure and transparent way to record transactions and verify data through a public, private, or hybrid network of interconnected nodes or computers [9]. Blockchains can be seen as distributed databases consisting of a continuously growing list of records, stored in a chain of blocks, which, through cryptographic algorithms, 4

are linked and timestamped to ensure the integrity of the information. The decentralised design ensures that no single entity has control over the entire network, providing resistance to tampering and fault tolerance, preventing the centralisation of power and mitigating the risks of single points of failure. A crucial concept within the blockchain is the consensus algorithms [10]. The decentralised nature of this technology requires a consensus mechanism to maintain trust and appropriate network functioning. These algorithms also enable distributed networks of stakeholders to achieve consensus or agreement on the state of the blockchain despite the lack of trust between participants. Moreover, this consensus ensures that only valid and verified transactions are added to the blockchain while preventing malicious actions, such as double-spending [11]. There are several types of consensus algorithms, each suited to different use cases and environments. The two most well-known consensus algorithms are PoW [5] and PoS [6]. The former was popularised by Bitcoin and requires participants, known as miners, to solve complex mathematical puzzles to validate transactions and create new blocks [5]. The latter was popularised by Ethereum, where the validators are chosen to create new blocks based on the number of ethers they hold and are willing to "stake" as collateral [6]. PoW is known for its security but is criticised for its energy consumption and scalability limitations, as the computational effort required increases with network growth [12]. PoS is more energy-efficient than PoW, and both lack diversification, so only a selected group of participants with higher computational power or higher stake is chosen to validate and verify the block transaction [3]. In addition to PoW and PoS, other consensus mechanisms have emerged, such as Delegated Proof of Stake (DPoS) [13] and Byzantine Fault Tolerance (BFT) [14], each designed to address specific scalability, security, or decentralisation challenges. Consensus algorithms are the backbone of blockchain technology, ensuring the trustworthiness and reliability of distributed ledger systems, and their continued development and innovation are key to the evolution of blockchain ecosystems. The development and optimisation of these consensus algorithms remains critical to the widespread adoption and future growth of blockchain applications across various industries, from finance and supply chain management to healthcare and governance [15].

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2.2. Intuitionistic fuzzy sets Fuzzy logic offers a powerful framework for dealing with uncertainty and imprecision [16] by allowing for the representation of vague or ambiguous information using linguistic variables and fuzzy sets. Intuitionistic fuzzy sets are introduced by Atanassov [17] as an extension of Zadeh fuzzy sets [16], where in addition to the membership function µA defined for fuzzy set A, there exists a non-membership function νA , such that µA + νA ∈ [0, 1]. When µA + νA = 1, the intuitionistic fuzzy set behaves like a classical fuzzy set, as the degree of membership is the complement of the degree of nonmembership. The quantity 1 − µA − νA , known as the intuitionistic index, degree of non-determinacy or degree of uncertainty, is also relevant. A formal definition for the IFS is given in Definition 1. Definition 1. [17] Let X be a fixed set. An intuitionistic fuzzy set (IFS) A over a set X is defined as A = {(x, µA (x), νA (x)) : x ∈ X},

(1)

µA (x) : X → [0, 1]

(2)

νA (x) : X → [0, 1]

(3)

where the functions and define the degree of membership and the degree of non-membership of the element x ∈ X, respectively, and for every x ∈ X: 0 ≤ µA (x) + νA (x) ≤ 1.

(4)

πA (x) = 1 − µA (x) − νA (x)

(5)

The value of is called the intuitionistic index, degree of non-determinacy or degree of uncertainty of the element x ∈ X to the intuitionistic fuzzy set A. This methodology uses IFSs to express variable reputation and computes a reputation degree for the nodes, indicating their honesty and providing insights into their reliability. IFSs allow us to separately consider evidence that a node is honest against evidence that is dishonest. For example, consider the case where µA = 0.8 and νA = 0.1, representing an individual for whom substantial evidence indicates honest behaviour, in contrast to µA = 0.08 and 6

νA = 0.01, which correspond to an individual about whom little evidence is available, despite most of it suggesting honesty. IFSs offer a valuable framework for handling uncertain and vague information with a more semantic approach compared to traditional fuzzy sets [18]. IFSs introduce a function that takes into account the degree of nonmembership and the intuitionistic index. 2.3. Uninorm aggregation operations Uninorm aggregation operators are a generalisation of the t-norm and tconorm. Yager [19, 20] states that the t-norm and t-conorm can characterise the and and or operators used in fuzzy logic, and they were defined as follows. Definition 2. [19, 20] A triangular norm or t-norm T is a mapping T : [0, 1] × [0, 1] → [0, 1]

(6)

having the following properties for all a, b, c, d ∈ [0, 1] 1. Commutativity: T (a, b) = T (b, a) 2. Associativity: T (T (a, b), c) = T (a, T (b, c)) 3. Monotonicity: T (a, b) ≤ T (c, d) if a ≤ c and b ≤ d 4. Neutral element: T (a, 1) = a. Definition 3. [19, 20] A triangular conorm or t-conorm S is a mapping S : [0, 1] × [0, 1] → [0, 1]

(7)

having the properties 1-3 in Definition 2 plus S(a, 0) = a. The uninorm aggregation operations share the initial three properties with both the t-norm and t-conorm but offer greater flexibility concerning the fourth property. t-norms and t-conorms have 1 and 0, respectively, as the neutral element. Uninorms may have any element e in the unit interval as the neutral element. The following definition captures the concept of uninorm.

7

Definition 4. [20] An uninorm U is a mapping U : [0, 1] × [0, 1] → [0, 1]

(8)

having the following properties for all a, b, c, d ∈ [0, 1] 1. Commutativity: U (a, b) = U (b, a) 2. Associativity: U (U (a, b), c) = U (a, U (b, c)) 3. Monotonicity: U (a, b) ≤ U (c, d) if a ≤ c and b ≤ d 4. Neutral element: ∃e ∈ [0, 1] : ∀a ∈ [0, 1] : U (a, e) = a. Notice that, in Definition 4, when e = 1 appears, the specific case for the t-norm and when e = 0 becomes the t-conorm. Uninorm aggregation operators are essential in fuzzy logic systems for combining and synthesising uncertain or imprecise information from multiple sources. Moreover, they provide a flexible framework for modelling complex relationships and uncertainties. 3. Related work Consensus algorithms in the blockchain are crucial mechanisms responsible for establishing trust between the participants in the network and, at the same time, providing security services such as integrity and privacy data. For this reason, some studies have contributed to understanding the behaviour of the participants involved in the consensus algorithms. In particular, reputation-based consensus algorithms have gained attention for their ability to evaluate and integrate the behaviour of network participants into the consensus process, where the trustworthiness of validators is reflected through assigned reputation scores. These algorithms introduce an adaptive approach to validator selection, where reputation can influence which nodes participate in block validation, making the consensus process more secure and efficient. For instance, Bugday et al. [21] focused on forming a consensus group using an online learning-based reputation model, which selects nodes with high reputation values. Oliveira [22] presented the Blockchain Reputation-Based Consensus (BRBC) mechanism, which requires a node to have a reputation score above a network trust threshold to insert a new block and uses a set of judges to monitor and update node reputation. Abdo et 8

al. [23] introduced a permissionless pure reputation-based consensus algorithm. In contrast, Aluko et al. [24] proposed a Proof-of-Reputation (PoR) mechanism that uses a liquid rank algorithm to calculate node reputation. These studies collectively highlight the potential of reputation-based consensus algorithms, nevertheless, each of these works is focused on studying the problem in a specific consensus algorithm primarily focused on thresholdbased approaches to reputation management, where reputation scores are often adjusted by external monitoring mechanisms (such as judges or predefined algorithms). In this paper, the proposal focuses on a reputation-based consensus algorithm, where a validator’s reputation fluctuates according to its behaviour over time. Validators can experience increases or decreases in their reputation based on the quality of their participation in the network, allowing for a more responsive and adaptive system. This approach ensures that validators who demonstrate consistently trustworthy behaviour are rewarded, while those who act maliciously or inefficiently see their reputation decrease, eventually leading to their exclusion from the consensus process if necessary. To manage this reputation fluctuation effectively, the proposed algorithm integrates fuzzy logic principles, which are well-suited for handling uncertainty and imprecise information in dynamic environments. Prior studies on fuzzy logic in trust management provide a useful foundation. Carbó et al. [25] presented a trust management mechanism that uses fuzzy sets to handle uncertain information about others in electronic commerce. Quesada et al. [26] proposed a methodology to manage the behaviour of experts in largescale consensus reaching for group decision-making problems, using fuzzy sets and uninorm aggregation operations. However, within the blockchain area, few works integrate these tools into consensus algorithms. Recently, Ramos et al. [8] utilised fuzzy sets theory and computing with words in the PoS consensus algorithm to manage the uncertainty that exists in the stake for each validator. As a consequence, the authors designed an equitable consensus mechanism and provided diversification in the selection process to choose the validators. Following the field of fuzzy logic, the author in [27] proposed a fuzzy-based miner selection algorithm applied to the Internet of Medical Things (IoMT). The algorithm considers parameters such as node status, coin (token) stake, voting, reputation, randomness, and neighbour density. These studies collectively highlight the growing interest in modelling reputation behaviour in blockchain networks. By allowing reputation to fluctuate based on validator behaviour, the proposed approach enhances network 9

Reputation Model

Reputation Settings

Reputation Score

Selection Process Reputationbased

Figure 1: The figure illustrates three sequential stages of the structured framework for reputation models.

adaptability, fairness, and security, while ensuring and incentivising positive participation. Furthermore, integrating novel tools into blockchain systems, such as intuitionistic fuzzy sets and uninorm aggregation operations, to model reputation, may contribute to the development of more adaptive, efficient, and secure decentralised systems capable of managing uncertain and dynamic environments. 3.1. Reputation model In the literature, there are various reputation-based consensus algorithms, such as [23, 28, 29, 22, 30, 31, 32, 33, 34, 35, 36, 8, 37, 38, 39, 40, 41] that follow a general structured framework of reputation models. Such a framework consists of three sequential stages: reputation settings, reputation scoring, and reputation-based selection (see Figure. 1). In the reputation settings stage, the system parameters are configured. For instance, every node in the network is assigned a reputation value Rep ∈ [0, 1], with an initial reputation value Rep0 , that is the same for every node. The reputation scoring stage involves the algorithms used to compute each node’s reputation score. The algorithm applied to calculate the reputation score depends on the reputation-based consensus algorithm; for example, in [41] the reputation algorithm computes the score using the transaction data in the transaction block and the behaviour of nodes. Finally, the reputation-based selection process leverages these scores to prioritise or select nodes for the validation and verification process. In the following section, we present our methodology for managing reputation behaviour in reputation-based consensus algorithms for blockchain systems.

10

Reputation Model

Reputation Settings

Reputation Score

Reputationbased Selection Process

Proposal Reputation Aware Uninorm-driven

Figure 2: Figure illustrates the framework for the integration of our proposal reputation aware uninorm driven into the reputation model in reputation-based consensus algorithms.

4. Uninorm-based methodology to manage the reputation behaviour in blockchain This section presents our novel proposal, a reputation aware uninormdriven approach for managing the reputation of each node in the blockchain network. Figure 2 illustrates how our novel proposal, the reputation aware uninorm-driven approach, is integrated into the reputation model for reputationbased consensus algorithms. In the classic reputation model, the reputation selection process typically prioritises the node with the highest reputation in the current round (where a round refers to each time a block is validated). However, our proposed method extends this approach by incorporating not only the reputation from the current round but also the reputation from the previous round. By doing so, our proposal enhances the selection process, enabling more informed and effective decision-making. Therefore, our proposal is structured into two phases, which are depicted in Figure 3: • Reputation degree. In this phase, the validation process is analysed, and each validator vi ∈ V obtains a Successful Validation Rate SuccV R(vi , t) for the current round t. This rate is then used in the Intuitionistic Fuzzy Set (IFS) to compute the Reputation Degree RepD(vi , t). The RepD(vi , t) shows short-term behaviour, which can be used in the algorithm, nevertheless, calculating only the reputation degree can lead to errors. • Reputation weight. For the next phase, the UAO utilises the RepD(vi , t) to calculate the Reputation Weight w(vi , t), taking into account the rep11

Reputation Aware Uninorm-drive

Reputation degree

Reputation weight

Validation

UAO w(𝑣𝑖 , 𝑡 − 1)

𝑆𝑢𝑐𝑐𝑉𝑅(𝑣𝑖 , 𝑡)

IFS

𝑅𝑒𝑝𝐷(𝑣_𝑖, 𝑡)

𝑤(𝑣𝑖 , 𝑡)

Figure 3: Figure illustrates the methodology phases based on IFS and UAO for managing reputation behaviour in the consensus algorithms to the blockchain networks.

utation weight from the previous round w(vi , t − 1). The reputation weight w(vi , t) shows long-term behaviour, providing greater security. Therefore, the w(vi , t) is analysed and monitored to study the validator’s behaviour during the validation and verification process. Once the reputation weight w(vi , t) has been computed for all validators, the selection process is performed. First, a predefined threshold τ ∈ [0, 1] is established to filter eligible validators. Only those validators whose reputation weight satisfies w(vi , t) ≥ τ are included in the candidate set Vτ = {vi | w(vi , t) ≥ τ }. This threshold mechanism ensures that only validators demonstrating sufficient long-term trustworthy behaviour are considered for participation in the current validation round. Subsequently, rather than deterministically selecting the validator with the highest reputation weight, a randomised selection algorithm is applied over the candidate set Vτ . The random selection may follow a uniform or weight-proportional distribution based on w(vi , t), depending on the desired balance between fairness and performance. This two-stage mechanism enhances security and decentralisation by ensuring that only validators with acceptable reputation levels participate in the block validation and transaction verification process. Proposition 1 (Liveness and Safety). If at least one validator satisfies w(vi , t) ≥ τ , the selection step guarantees liveness by selecting a validator 12

Algorithm 1 Reputation aware uninorm-driven Input : V = {v1 , v2 , ..., vn }, Current round t, Threshold τ Output : Selected validator vselec 1: Vτ ← ∅ 2: for each vi ∈ V do 3: Calculate SuccV R(vi , t) 4: Calculate RepD(vi , t) using IFS 5: Calculate w(vi , t) using UAO and w(vi , t − 1) 6: if w(vi , t) ≥ τ then 7: Vτ ← Vτ ∪ {vi } 8: end if 9: end for 10: if Vτ = ∅ then 11: Select vselec such that w(vselec , t) = max w(vi , t) vi ∈V

else 13: Randomly select vselec from Vτ 14: end if 15: return vselec 12:

in finite time. Furthermore, since w(vi , t) encodes long-term behaviour, the probability of selecting a faulty validator decreases exponentially with consecutive failures. It is important to note that the proposed selection step is modular and can be incorporated into various reputation-based consensus protocols. In particular, it can replace stake-based probability mechanisms similar to PoS schemes, while the reputation update procedure remains as defined in Section 4.1. Reputation degree and Section 4.2. Reputation weight. 4.1. Reputation degree The first step in computing the reputation degree is to calculate the successful validation rate. During the validation process, each participant has a successful validation rate, as presented in Definition 5. This rate is the number of successful validations divided by the total number of validation attempts. Definition 5. Let #SV be the number of successful validations. Then the successful validation rate, SuccV R(vi , t) by each validator vi in the current 13

𝜈! (𝑥)

𝜇! (𝑥)

𝜇! (𝑥" )

𝜈! (𝑥" ) 𝑥"

𝑋

Figure 4: Figure shows the intuitionistic fuzzy set A with the membership function µA (x) and the non-membership function νA (x).

round t is defined by the function: #SV (vi , t) (9) t Remark 1. The number of validation attempts corresponds to the number of rounds. In each round, the validator has only one attempt. SuccV R(vi , t) =

The successful validation rate will be used to compute the reputation degree through an intuitionistic fuzzy set A that expresses the linguistic variable “reputation". According to Definition 4, an intuitionistic fuzzy set A assigns to each element x of the universe X a membership degree µA (x) ∈ [0, 1] and a non-membership degree νA (x) ∈ [0, 1] such that: 0 ≤ µA (x) + νA (x) ≤ 1. Moreover, for all x ∈ X recall that πA (x) = 1 − µA (x) − νA (x). This paper focuses on studying the reputation of validators in the consensus algorithm in the blockchain. Therefore, as the universe of discourse represents the successful validation rate SuccV R(vi , t) defined on the real interval X = [0, 1], following the Definition 5. The membership function µA (x) for x = SuccV R(vi , t) ∈ A is defined as:   0 µA (x) = µ(x)   1

if x = 0 if 0 < x < 1 if x = 1 14

(10)

where µA (x) : X → [0, 1] defines the rate at which a node’s reputation increases based on its behaviour over time (see Figure 4). The design of µ(x) is guided by the principle of determining how quickly a validator can regain its reputation after exhibiting undesirable behaviour, such as failing to validate a block correctly. For example, a minor failure might allow for quicker recovery, while repeated or severe infractions could result in a slower pace. Different options for defining function µ(x) include: 1. Constant increase in reputation: In this case, a linear function is suitable. 2. Slow increase in reputation: Logarithmic and square root functions may be useful. 3. Rapid increase in reputation: Exponential and polynomial functions can be applied. 4. Initial rapid increase in reputation followed by gradual slowing: For instance, logistic and hyperbolic functions. 5. Slow initial increase in reputation followed by a rapid growth: In this scenario, a sigmoid function could be an option. The list provided above illustrates several examples of how the function µ(x) can be defined to reflect the enhancement of a validator’s reputation after encountering failures as shown in Figure 5. However, it is important to note that these are merely illustrative examples, and other functions can be utilised to define µ(x). The choice of function depends on specific factors, such as the desired rate of reputation increase. With regard to the non-membership function νA (x) for x ∈ A is defined as follows:   1 νA (x) = ν(x)   0

if x = 0 if 0 < x < 1 if x = 1

(11)

where νA (x) : X → [0, 1] is a function defined considering a similar philosophy that was used to define µ(x) in Eq. (10); however, in this case, the 15

Figure 5: Figure illustrates a set of functions that could be used to define the function µ(x)

Figure 6: Figure illustrates a set of functions that could be used to define the function ν(x)

desired rate of decrease is analysed. For instance, if a consistent penalisation is required, a linear function may be employed. Alternatively, opting for a negative logarithmic function results in a rapid initial decrease followed by a slower decline, which may be suitable for scenarios requiring swift but controlled reputation adjustment (see Figure 6). Reputation is intrinsically a concept filled with uncertainty. It does not constitute an absolute measure but rather emerges from varying perceptions of an entity’s behaviour. When new validators join the pool, the available information about their behaviour is incomplete, resulting in inherent indeterminacy, or epistemic uncertainty, regarding the extent to which they 16

Figure 7: Figure displays µA (x), νA (x), and πA (x).

are reputable (true) or non-reputable (false). This initial state of indeterminacy evolves dynamically as the blockchain network expands and validators process more transactions. Consequently, an intuitionistic framework for managing uncertainty, one that not only captures degrees of truth (membership) and falsity (non-membership) but also incorporates the dimension of indeterminacy, offers a particularly apt methodology for addressing this context. We employ an intuitionistic index πA (x) to indicate the degree of indeterminacy of the element x to the intuitionistic fuzzy set A. Figure 7 shows how πA (x) varies with respect to µA (x) and νA (x). Once the reputation is modelled through the IFS, as shown in Figure 4, the next step is to compute the reputation degree for each validator. A natural first candidate for a reputation degree would be the ratio r(x) =

µA (x) , µA (x) + νA (x)

which represents the balance between positive and negative evidence and can be interpreted as a basic reputation rating in the interval [0, 1]. However, this ratio alone presents a limitation, different combinations of µA (x) and νA (x) may produce the same value of r, even though the amount of available information differs. Therefore, this expression does not capture the degree of certainty associated with the evaluation. To address this issue, we incorpo17

rate the intuitionistic index πA (x), which quantifies the level of indeterminacy reflecting how much uncertainty remains in the assessment. When πA (x) is small, the evaluation is supported by more information, whereas larger values indicate greater indeterminacy. Instead of directly adopting r as the reputation degree, we scale it between a lower bound r0 and r, where 0 < r0 < r. The parameter r0 prevents unreliable extreme values when the certainty level is low. The scaling depends on a function s : [0, 1] → [0, 1], such that s(0) = 0 and s(1) = 1. Consequently, when the certainty level is minimal, the reputation degree approaches r0 , and when certainty is maximal, it converges to r. In this work, s(πA (x)) = πA (x)α with α > 0, which allows us to regulate the sensitivity of the transition. If 0 < α < 1, the model moves rapidly toward r even under moderate certainty; if α > 1, it remains close to r0 until high certainty is achieved. These observations are summarised in Definition 6, which formally introduces the reputation degree RepD(vi , t) within the proposed intuitionistic fuzzy framework. Definition 6. Let A be an intuitionistic fuzzy set, µA (x) the membership degree, νA (x) the non-membership degree, and πA (x) the intuitionistic index, then the reputation degree, RepD(vi , t), by each validator vi in the current round t is defined as: (πA (x))α  µA (x) (12) , RepD(vi , t) = r0 r0 (µA (x)+νA (x)) µA (x) and α > 0. where 0 < r0 < µA (x)+ν A (x)

Equation 12 given in Definition 6 introduces the reputation degree as a certainty-adjusted of the basic reputation rating derived from the IFS components. To validate that this formulation behaves consistently with its intended interpretation, it is necessary to examine its fundamental mathematical properties. In particular, the reputation degree should remain bounded, respond positively to increasing membership, negatively to increasing nonmembership, and exhibit regular behaviour. These properties are formalised in the following proposition. Proposition 2. Let µA (x), νA (x) and πA (x) be the membership, non-membership and intuitionistic index of the IFS, and let RepD(vi , t) be given by Equation (12). Then: 18

1. RepD(vi , t) ∈ (0, 1). 2. RepD(vi , t) is monotonically increasing in µA (x). 3. RepD(vi , t) is monotonically decreasing in νA (x). 4. RepD(vi , t) is continuous on [0, 1]. For a better understanding of Definition 6, a numerical illustration is presented in the next paragraph. Suppose a validator has successful validation SuccV R = 0.75 and the membership and non-membership functions are defined by the following linear IFS functions: µA (x) = x − 0.05, νA (x) = 0.90 − x ⇒ πA (x) = 1 − (µA (x) + νA (x)) Let µA (0.75) = 0.70, νA (0.75) = 0.15, then πA (0.75) = 0.15. Moreover, for this example r0 = 0.30. Then 0.70 0.70 µA (x) = = ≈ 2.6923 r0 (µA (x) + νA (x)) 0.30(0.70 + 0.15) 0.26 Now compute the reputation degree for different α: • If α = 2: 2

RepD = 0.30 × (2.6923)0.15 = 0.30 × 1.0225 ≈ 0.3068 • If α = 1: 1

RepD = 0.30 × (2.6923)0.15 = 0.30 × 1.1602 ≈ 0.3480 • If α = 0.5: 0.5

RepD = 0.30 × (2.6923)0.15

= 0.30 × 1.4675 ≈ 0.4402

This example shows that when α is higher the reputation degree decrease, while lower α the reputation degree increases. This observation is formalised in Remark 2. 19

Remark 2. The function given in Definition 6 can be expressed as  s(π) r RepD = r0 , r0 µA (x) , 0 < r0 < r. Additionally, the parameter r0 is where r(x) = µA (x)+ν A (x) chosen as a function of r in order to encode how strict the system behaves under uncertainty. For instance, one may define r0 = βr with 0 < β < 1, so that the reputation degree is proportionally reduced when indeterminacy is high. This choice directly relates the baseline value r0 to the underlying membership/non-membership pair (µA (x), νA (x)), since r itself is determined by their balance. Consequently, r0 controls how severely the system penalises uncertain evaluations while preserving consistency with the original fuzzy evidence.

A particular case from Eq. 12, when µA (x) = c · νA (x) with c = 1: 

µA (x) RepD(vi , t) = r0 r0 (µA (x) + µA (x))  πA (x)α µA (x) = r0 r0 (2µA (x) πA (x)α  1 = r0 2r0

πA (x)α

That is semantically appealing, as it suggests “There is some evidence that you are honest, but also some evidence that you are dishonest, which makes me less sure of your honesty”. In the general case, if µA (x) = c · νA (x) with 0 ≤ c ≤ 1, then µA (x)

RepD(vi , t) = r0

!πA (x)α (13)

r0 (µA (x) + µAc(x) ) 

µA (x) · c = r0 r0 (µA (x) · c + µA (x)  πA (x)α c = r0 r0 (c + 1) 20

πA (x)α

(14) (15)

Observe that if πA (x) → 0, then RepD(vi , t) rapidly converges to r0 , even under high uncertainty. Conversely, when πA (x) → 1, RepD(vi , t) remains c until a high level of certainty is achieved. The parameter α allows close to c+1 the behaviour of the function to be tuned: values of α close to zero imply that RepD(vi , t) depends primarily on the relative magnitudes of µA (x) and νA (x), leading to an increase in the reputation degree, whereas larger values of α make it more dependent on the absolute magnitude of µA (x), resulting in a decrease in the reputation degree. 4.2. Reputation weight In this section, we employ the reputation degree computed in the preceding section to determine the corresponding reputation weight. This methodology proposes using an uninorm aggregation operation to model reputation behaviour; through these operators, the reputation degree is aggregated to obtain a reputation weight. The reputation degree and reputation weight are used to monitor and demonstrate the importance of reinforcement. For instance, if the validator ti exhibits both a high reputation degree and a high reputation weight, it receives positive reinforcement. Conversely, if the validator has both a low reputation degree and a low reputation weight, it receives negative reinforcement. Otherwise, the validator may receive average reinforcement, as shown in Figure 8. The following definition is provided for computing the reputation weight: Definition 7. Let U be the uninorm aggregation operator with the associated neutral element e. Then, the reputation weight w(vi , t), by each validator vi at the round t is computed as follows: ( e if t = 1 w(vi , t) = (16) U (RepD(vi , t), w(vi , t − 1)) if t > 1 Note that U in Definition 7 can be defined considering the structure of an uninorm shown in Figure 8. For instance, the operator U can be defined as a piecewise function where a t-norm operator is used if both the reputation degree and the reputation weight are less than the neutral element e, a tconorm is utilised if both the reputation degree and the reputation weight are greater than the neutral element e and less than 1. In other cases, the arithmetic mean between the reputation degree and the reputation weight can be applied. As a particular example, the following Fodor’s uninorm [42] is shown: 21

Y

Reputation weight

1 Average reinforcement

Positive reinforcement

Negative reinforcement

Average reinforcement

𝑒

0 0

𝑒

Reputation degree

1

X

Figure 8: Structure of uninorm: reputation weight for different types of reputation degree.

 2xy    2(x + y − 2xy − e) U (x, y) = x+y  2   

if 0 ≤ x, y ≤ e if e ≤ x, y ≤ 1 if min(x, y) < e < max(x, y)

(17)

Figure 9 describes the graphic behaviour of Fodor’s uninorm with a neutral element set to e = 0.5. In all experiments, we adopt the following uninorm with neutral element e:  if a, b ≤ e, T (a, b)/e, U (a, b) = 1 − S(1 − a, 1 − b) , if a, b > e, 1−e where T is the minimum t-norm and S is the maximum t-conorm. By following this methodology, blockchain networks can leverage uninorm aggregation operators to incorporate information about validator behaviour. The proposed approach can be integrated into consensus algorithms in which participant selection relies on fluctuating reputation as a key parameter. To 22

Figure 9: Fodor’s uninorm with e = 0.5

demonstrate the use of intuitionistic fuzzy sets and uninorm aggregation operators, the next section presents experimental results and discusses the performance of the proposed methodology. 5. Experiments and results This section performs and discusses a set of experiments designed to assess the performance of the proposed methodology. The experiments were developed using the following software and computer specifications. It includes a CPU and an Intel® Core™ i7-7500U processor, featuring a clock speed of 2.70GHz and four cores. The operating system used is Ubuntu 22.04.3. For compiling, A C++ compiler, GCC version 7.4.0 is utilised. The programming language employed is Python, specifically version 3.10.12. 5.1. Experimental results: General evaluation The objective of this experiment is to depict the performance of the methodology through the illustrative example of defining and applying specific functions for each phase. According to this methodology, the first phase consists of computing the reputation degree. To do so, simulations have been developed, and the successful validation rate is computed in each round using the function defined in Eq. (9). Table 2 shows the successful validation rate 23

for the validators v1 and v2 in the rows SucV R(v1 , t) and SucV R(v2 , t) over seven rounds. Notice that the validator v1 makes a mistake in rounds 2 and 5, while the validator v2 was wrong in rounds 2 and 3. Round(t) SuccV R(v1 , t) RepD(v1 , t) w(v1 , t) SuccV R(v2 , t) RepD(v2 , t) w(v2 , t)

1 1.00 0.62 0.50 1.00 0.62 0.50

2 3 0.50 0.67 0.40 0.55 0.40 0.48 0.50 0.33 0.40 0.22 0.40 0.18

4 5 0.75 0.60 0.59 0.50 0.53 0.52 0.50 0.60 0.40 0.50 0.14 0.14

6 0.67 0.55 0.56 0.67 0.55 0.35

7 0.71 0.58 0.63 0.71 0.58 0.46

Table 2: Table shows the successful validation rate SucV r(), the reputation degree RepD(), and the reputation weight w() for the validator v1 and v2 at seven rounds.

The next step is to compute the reputation degree for v1 and v2 , considering the successful validation rate SuccV R, respectively. For this example, the membership function µA (x) presented in Eq. 10 is defined as:   0

µA (x) = exp(x)−1  exp(1)−1 

1

if x < 0 if 0 ≤ x ≤ 1

(18)

if x > 1

where the function µ(x) = exp(x)−1 . exp(1)−1 The non-membership function νA (x) in Eq. 11 is defined by:   1 νA (x) = (x − 1)2   0

if x < 0 if 0 ≤ x ≤ 1 if x > 1

(19)

where the function ν(x) = (x − 1)2 . Then, the functions in Eq 18 and Eq. 19 are used together with the function in Eq. 12 to compute the reputation degree for the validator v1 and µA (x) v2 where the parameter is set to α = 2 and r0 = 5/8 · µA (x)+ν for this A (x) example. Table 2 depicts the outcomes at the row named RepD(v1 , t) and RepD(v2 , t) for v1 and v2 , respectively. 24

Figure 10: Figure plots the results obtained in Table 2 for the validator v1

At this moment, the first phase is finished. To start with, the second phase is necessary to define the uninorm aggregation operation U . For this example, we will use Fodor’s uninorm defined in Eq. 17. After applying the function presented in Eq. 16, the reputation weight w(vi , t) is calculated for the validators v1 and v2 . The results are presented in the rows w(v1 , t) and w(v2 , t) in Table 2. The outcomes of Table 2 for the validator v1 are displayed in Figure 10. This figure illustrates the reputation behaviour for the validator v1 in ten rounds. The blue graph (dashed line) represents the successful validation rate SuccV R(v1 , t), the orange graph (dotted line) indicates the reputation degree RepD(v1 , t), and the green graph (continuous line) shows the reputation weight w(v1 , t). From Figure 10, it is observed that the validator v1 has a mistake in rounds 2 and 5; thus, the SuccV R(v1 , t), RepD(v1 , t), and w(v1 , t) decrease. For round 3, the validator starts to perform the process correctly, then SuccV R(v1 , t), RepD(v1 , t), and w(v1 , t) increase until round 4, where they decrease again. After round 6, SuccV R(v1 , t), RepD(v1 , t), and w(v1 , t) start to increase again. For the validator v2 , the outcomes of Table 2 are depicted in Figure 11. The purple graph (dashed line) represents the successful validation rate SuccV R(v2 , t), the blue graph (dotted line) indicates the reputation degree RepD(v2 , t), and the red graph (continuous line) shows the reputation weight w(v2 , t). These graphs illustrate the reputation behaviour for the valida25

Figure 11: Figure plots the results obtained in Table 2 for the validator v2

tor v2 , note that in rounds 2 and 3, the validator was incorrect, therefore, the SuccV R(v2 , t), the RepD(v2 , t), and the w(v2 , t) decrease. For the next rounds, the SuccV R(v2 , t) and the RepD(v2 , t) increase, nevertheless, w(v2 , t) starts to increase until round 5. This experiment shows that the validator v1 made a mistake in the previous round and has the opportunity to correct the mistake in the next round and increase its reputation, as shown in Figure 10. Nevertheless, if the validators were wrong two or more times consecutively, as depicted in Figure 11 for the validator v2 , the process of increasing and recovering the reputation is slower. This is because repeated mistakes can significantly impact the validator’s reputation, leading to a more gradual recovery process. The reputation recovery model is designed to penalise errors more severely, thereby encouraging validators to maintain consistent accuracy and reliability over time. By incorporating such mechanisms, the system ensures that validators who frequently make mistakes face a more challenging path to regain their status, thus promoting higher standards of performance across the network. In the next section, we have analysed the performance of the proposed methodology considering different values for the α parameter. 5.2. α parameter With respect to the parameter α introduced in Definition 12, we conducted experiments to analyse its behaviour. These experiments aim to evaluate the performance of the proposed methodology by examining variations 26

in α across ten different values. To this end, we computed the reputation degree RepD(v1 , t) and the reputation weight w(v1 , t) for validator v1 , as well as RepD(v2 , t) and w(v2 , t) for validator v2 . Tables 3 and 4 present the results for v1 and v2 , respectively. The first column reports the values of α, followed by the corresponding results obtained over twenty rounds. Parameter α α = 0.1

α = 0.5

α = 1.0

α = 1.5

α = 2.0

α = 2.5

α = 3.0

α = 3.5

α = 4.0

α = 4.5

Round(t) SuccV R(v1 , t) RepD(v1 , t) w(v1 , t) SuccV R(v1 , t) RepD(v1 , t) w(v1 , t) SuccV R(v1 , t) RepD(v1 , t) w(v1 , t) SuccV R(v1 , t) RepD(v1 , t) w(v1 , t) SuccV R(v1 , t) RepD(v1 , t) w(v1 , t) SuccV R(v1 , t) RepD(v1 , t) w(v1 , t) SuccV R(v1 , t) RepD(v1 , t) w(v1 , t) SuccV R(v1 , t) RepD(v1 , t) w(v1 , t) SuccV R(v1 , t) RepD(v1 , t) w(v1 , t) SuccV R(v1 , t) RepD(v1 , t) w(v1 , t)

1 1.00 0.62 0.50 1.00 0.62 0.50 1.00 0.62 0.50 1.00 0.62 0.50 1.00 0.62 0.50 1.00 0.62 0.50 1.00 0.62 0.50 1.00 0.62 0.50 1.00 0.62 0.50 1.00 0.62 0.50

2 0.50 0.58 0.58 0.50 0.50 0.50 0.50 0.45 0.45 0.50 0.42 0.42 0.50 0.40 0.40 0.50 0.39 0.39 0.50 0.39 0.39 0.50 0.38 0.38 0.50 0.38 0.38 0.50 0.38 0.38

3 0.67 0.79 0.82 0.67 0.68 0.68 0.67 0.61 0.53 0.67 0.57 0.49 0.67 0.55 0.48 0.67 0.54 0.46 0.67 0.53 0.46 0.67 0.53 0.45 0.67 0.52 0.45 0.67 0.52 0.45

4 0.75 0.86 0.95 0.75 0.73 0.83 0.75 0.65 0.67 0.75 0.61 0.55 0.75 0.59 0.53 0.75 0.58 0.52 0.75 0.58 0.52 0.75 0.57 0.51 0.75 0.57 0.51 0.75 0.57 0.51

5 0.60 0.72 0.97 0.60 0.62 0.87 0.60 0.56 0.71 0.60 0.52 0.57 0.60 0.50 0.52 0.60 0.49 0.50 0.60 0.48 0.50 0.60 0.47 0.49 0.60 0.47 0.49 0.60 0.47 0.49

6 0.67 0.79 0.99 0.67 0.68 0.92 0.67 0.61 0.77 0.67 0.57 0.63 0.67 0.55 0.56 0.67 0.54 0.54 0.67 0.53 0.51 0.67 0.53 0.51 0.67 0.52 0.51 0.67 0.52 0.51

7 0.71 0.84 1.00 0.71 0.72 0.95 0.71 0.64 0.84 0.71 0.60 0.70 0.71 0.58 0.63 0.71 0.57 0.60 0.71 0.56 0.57 0.71 0.56 0.56 0.71 0.55 0.56 0.71 0.55 0.56

8 0.75 0.86 1.00 0.75 0.73 0.98 0.75 0.65 0.89 0.75 0.61 0.77 0.75 0.59 0.70 0.75 0.58 0.67 0.75 0.58 0.64 0.75 0.57 0.63 0.75 0.57 0.62 0.75 0.57 0.62

9 0.78 0.88 1.00 0.78 0.74 0.99 0.78 0.66 0.92 0.78 0.62 0.83 0.78 0.6 0.76 0.78 0.59 0.73 0.78 0.59 0.7 0.78 0.59 0.69 0.78 0.58 0.69 0.78 0.58 0.68

10 0.80 0.89 1.00 0.80 0.75 0.99 0.80 0.66 0.95 0.80 0.63 0.87 0.80 0.61 0.81 0.80 0.60 0.78 0.80 0.60 0.76 0.80 0.59 0.75 0.80 0.59 0.75 0.80 0.59 0.74

11 0.73 0.85 1.00 0.73 0.72 1.00 0.73 0.64 0.96 0.73 0.60 0.90 0.73 0.58 0.84 0.73 0.57 0.81 0.73 0.57 0.79 0.73 0.56 0.78 0.73 0.56 0.78 0.73 0.56 0.77

12 0.67 0.79 1.00 0.67 0.68 1.00 0.67 0.61 0.97 0.67 0.57 0.91 0.67 0.55 0.86 0.67 0.54 0.83 0.67 0.53 0.8 0.67 0.53 0.79 0.67 0.52 0.79 0.67 0.52 0.78

13 0.69 0.82 1.00 0.69 0.70 1.00 0.69 0.63 0.98 0.69 0.59 0.93 0.69 0.56 0.88 0.69 0.55 0.85 0.69 0.55 0.82 0.69 0.54 0.81 0.69 0.54 0.80 0.69 0.54 0.8

14 0.71 0.84 1.00 0.71 0.72 1.00 0.71 0.64 0.98 0.71 0.60 0.94 0.71 0.58 0.90 0.71 0.57 0.87 0.71 0.56 0.84 0.71 0.56 0.83 0.71 0.55 0.82 0.71 0.55 0.82

15 0.73 0.85 1.00 0.73 0.73 1.00 0.73 0.65 0.99 0.73 0.61 0.95 0.73 0.59 0.91 0.73 0.57 0.89 0.73 0.57 0.86 0.73 0.57 0.85 0.73 0.56 0.85 0.73 0.56 0.84

16 0.75 0.86 1.00 0.75 0.73 1.00 0.75 0.65 0.99 0.75 0.61 0.96 0.75 0.59 0.93 0.75 0.58 0.90 0.75 0.58 0.88 0.75 0.57 0.87 0.75 0.57 0.87 0.75 0.57 0.87

17 0.71 0.83 1.00 0.71 0.71 1.00 0.71 0.63 0.99 0.71 0.59 0.97 0.71 0.57 0.94 0.71 0.56 0.92 0.71 0.55 0.90 0.71 0.55 0.89 0.71 0.55 0.88 0.71 0.55 0.88

18 0.67 0.79 1.00 0.67 0.68 1.00 0.67 0.61 1.00 0.67 0.57 0.97 0.67 0.55 0.95 0.67 0.54 0.92 0.67 0.53 0.90 0.67 0.53 0.89 0.67 0.52 0.89 0.67 0.52 0.88

19 0.63 0.75 1.00 0.63 0.65 1.00 0.63 0.58 1.00 0.63 0.55 0.98 0.63 0.52 0.95 0.63 0.51 0.92 0.63 0.5 0.90 0.63 0.5 0.89 0.63 0.50 0.69 0.63 0.5 0.69

20 0.65 0.78 1.00 0.65 0.67 1.00 0.65 0.60 1.00 0.65 0.56 0.98 0.65 0.54 0.95 0.65 0.53 0.93 0.65 0.52 0.91 0.65 0.51 0.90 0.65 0.50 0.70 0.65 0.51 0.70

Table 3: Table shows the successful validation rate SucV r(v1 , t), the reputation degree RepD(v1 , t), and the reputation weight w(v1 , t) for the validator v1 at 20 rounds and multiples α values.

The results presented in Table 3 are illustrated in Figures 12 and 13. Figure 12 displays multiple graphs corresponding to the reputation degree RepD(v1 , t) for different values of α. It can be observed that when α is close to zero, the reputation degree assumes higher values. As α increases, the reputation degree decreases accordingly. Figure 13 presents the corresponding plots of the reputation weight w(v1 , t), computed using the same values of α. These results indicate that the reputation weight is more lenient for small values of α, whereas larger values of α lead to more stringent weighting. Similarly, the results in Table 4 are illustrated in Figures 14 and 15. Figure 14 shows the reputation degree RepD(v2 , t) for various α values. When α is close to zero, the reputation degree is higher, and it decreases as α increases. Figure 15 displays the reputation weight w(v2 , t) using the same α values. The plots indicate that the reputation weight is more lenient for low 27

Figure 12: Figure shows the plots related to the reputation degree RepD(v1 , t) for the validator v1 as the parameter α varies.

Figure 13: Figure shows the plots related to the reputation weight w(v1 , t) for the validator v1 as the parameter α varies. This reputation weight corresponds to the reputation degree illustrated in Figure 12.

α values and becomes more stringent as α increases. The results of this experiment reveal a consistent relationship between 28

Parameter α α = 0.1

α = 0.5

α = 1.0

α = 1.5

α = 2.0

α = 2.5

α = 3.0

α = 3.5

α = 4.0

α = 4.5

Round(t) SuccV R(v2 , t) RepD(v2 , t) w(v2 , t) SuccV R(v2 , t) RepD(v2 , t) w(v2 , t) SuccV R(v1 , t) RepD(v1 , t) w(v1 , t) SuccV R(v1 , t) RepD(v1 , t) w(v1 , t) SuccV R(v1 , t) RepD(v1 , t) w(v1 , t) SuccV R(v1 , t) RepD(v1 , t) w(v1 , t) SuccV R(v1 , t) RepD(v1 , t) w(v1 , t) SuccV R(v1 , t) RepD(v1 , t) w(v1 , t) SuccV R(v1 , t) RepD(v1 , t) w(v1 , t) SuccV R(v1 , t) RepD(v1 , t) w(v1 , t)

1 1.00 0.62 0.50 1.00 0.62 0.50 1.00 0.62 0.50 1.00 0.62 0.50 1.00 0.62 0.50 1.00 0.62 0.50 1.00 0.62 0.50 1.00 0.62 0.50 1.00 0.62 0.50 1.00 0.62 0.50

2 0.50 0.58 0.58 0.50 0.50 0.50 0.50 0.45 0.45 0.50 0.42 0.42 0.50 0.40 0.40 0.50 0.39 0.39 0.50 0.39 0.39 0.50 0.38 0.38 0.50 0.38 0.38 0.50 0.38 0.38

3 0.33 0.32 0.45 0.33 0.28 0.39 0.33 0.25 0.22 0.33 0.23 0.19 0.33 0.22 0.18 0.33 0.22 0.17 0.33 0.22 0.17 0.33 0.22 0.16 0.33 0.21 0.16 0.33 0.21 0.16

4 0.50 0.58 0.51 0.50 0.50 0.45 0.50 0.45 0.20 0.50 0.42 0.16 0.50 0.40 0.14 0.50 0.39 0.13 0.50 0.39 0.13 0.50 0.38 0.13 0.50 0.38 0.12 0.50 0.38 0.12

5 0.60 0.72 0.72 0.60 0.62 0.53 0.60 0.56 0.38 0.60 0.52 0.34 0.60 0.50 0.14 0.60 0.49 0.13 0.60 0.48 0.12 0.60 0.47 0.12 0.60 0.47 0.12 0.60 0.47 0.12

6 0.50 0.58 0.77 0.50 0.50 0.53 0.50 0.45 0.34 0.50 0.42 0.29 0.50 0.40 0.12 0.50 0.39 0.10 0.50 0.39 0.09 0.50 0.38 0.09 0.50 0.38 0.09 0.50 0.38 0.09

7 0.57 0.68 0.85 0.57 0.59 0.62 0.57 0.53 0.43 0.57 0.49 0.28 0.57 0.47 0.11 0.57 0.46 0.09 0.57 0.45 0.09 0.57 0.45 0.08 0.57 0.45 0.08 0.57 0.45 0.08

8 0.62 0.75 0.92 0.62 0.65 0.73 0.62 0.58 0.51 0.62 0.54 0.41 0.62 0.52 0.31 0.62 0.51 0.30 0.62 0.5 0.09 0.62 0.50 0.08 0.62 0.49 0.08 0.62 0.49 0.08

9 0.67 0.79 0.97 0.67 0.68 0.83 0.67 0.61 0.61 0.67 0.57 0.49 0.67 0.55 0.43 0.67 0.54 0.42 0.67 0.53 0.31 0.67 0.53 0.3 0.67 0.52 0.30 0.67 0.52 0.30

10 0.70 0.82 0.99 0.70 0.71 0.90 0.70 0.63 0.71 0.70 0.59 0.54 0.70 0.57 0.50 0.70 0.56 0.49 0.70 0.55 0.43 0.70 0.55 0.43 0.70 0.54 0.42 0.70 0.54 0.42

11 0.73 0.85 1.00 0.73 0.72 0.94 0.73 0.64 0.80 0.73 0.60 0.64 0.73 0.58 0.58 0.73 0.57 0.53 0.73 0.57 0.5 0.73 0.56 0.49 0.73 0.56 0.49 0.73 0.56 0.49

12 0.75 0.86 1.00 0.75 0.73 0.97 0.75 0.65 0.86 0.75 0.61 0.72 0.75 0.59 0.66 0.75 0.58 0.61 0.75 0.58 0.54 0.75 0.57 0.53 0.75 0.57 0.53 0.75 0.57 0.53

13 0.77 0.88 1.00 0.77 0.74 0.98 0.77 0.66 0.90 0.77 0.62 0.79 0.77 0.60 0.73 0.77 0.59 0.68 0.77 0.58 0.62 0.77 0.58 0.61 0.77 0.58 0.61 0.77 0.58 0.61

14 0.71 0.84 1.00 0.71 0.72 0.99 0.71 0.64 0.93 0.71 0.60 0.83 0.71 0.58 0.77 0.71 0.57 0.72 0.71 0.56 0.66 0.71 0.56 0.65 0.71 0.55 0.65 0.71 0.55 0.65

15 0.67 0.79 1.00 0.67 0.68 0.99 0.67 0.61 0.95 0.67 0.57 0.85 0.67 0.55 0.79 0.67 0.54 0.74 0.67 0.53 0.68 0.67 0.53 0.67 0.67 0.52 0.67 0.67 0.52 0.66

16 0.69 0.81 1.00 0.69 0.70 1.00 0.69 0.62 0.96 0.69 0.58 0.88 0.69 0.56 0.82 0.69 0.55 0.77 0.69 0.54 0.71 0.69 0.54 0.7 0.69 0.54 0.69 0.69 0.54 0.69

17 0.71 0.83 1.00 0.71 0.71 1.00 0.71 0.63 0.97 0.71 0.59 0.90 0.71 0.57 0.84 0.71 0.56 0.79 0.71 0.55 0.74 0.71 0.55 0.73 0.71 0.55 0.72 0.71 0.55 0.72

18 0.67 0.79 1.00 0.67 0.68 1.00 0.67 0.61 0.98 0.67 0.57 0.91 0.67 0.55 0.86 0.67 0.54 0.81 0.67 0.53 0.76 0.67 0.53 0.74 0.67 0.52 0.73 0.67 0.52 0.73

19 0.68 0.81 1.00 0.68 0.70 1.00 0.68 0.62 0.98 0.68 0.58 0.93 0.68 0.56 0.88 0.68 0.55 0.83 0.68 0.54 0.78 0.68 0.54 0.76 0.68 0.54 0.75 0.68 0.53 0.75

20 0.70 0.82 1.00 0.70 0.71 1.00 0.70 0.63 0.99 0.70 0.59 0.94 0.70 0.57 0.89 0.70 0.56 0.85 0.70 0.55 0.80 0.70 0.55 0.78 0.70 0.54 0.77 0.70 0.54 0.77

Table 4: Table shows the successful validation rate SucV r(v2 , t), the reputation degree RepD(v2 , t), and the reputation weight w(v2 , t) for the validator v2 at 20 rounds and multiples α values.

Figure 14: Figure shows the plots related to the reputation degree RepD(v2 , t) for the validator v2 as the parameter α varies.

the parameter α and both the reputation degree RepD(vi , t) and the repu29

Figure 15: Figure shows the plots related to the reputation weight w(v2 , t) for the validator v2 as the parameter α varies. This reputation weight corresponds to the reputation degree illustrated in Figure 14.

tation weight w(vi , t) for the validators. When α is close to zero, both the reputation degree and reputation weight are higher, indicating a more lenient evaluation of validator reputation. Conversely, as α increases, both the reputation degree and reputation weight decrease, reflecting a more stringent assessment. This trend is consistent across different validators, as shown in Tables 3 and 4. Therefore, the choice of α is important in determining the strictness of reputation evaluation, directly influencing how reputation is managed and assessed within the blockchain consensus algorithm. All source files generated during the development and implementation of the reputation methodology are available at the following GitHub repository: https://github.com/RCBruno/ReputationAwareUninormDriven. 5.3. Computational Complexity Analysis This section analyses both the computational and communication complexity associated with the computation of the reputation degree and the reputation weight. Regarding the reputation degree, for each validator vi at round t, the successful validation rate SuccV R(vi , t) requires one division operation, resulting in constant time complexity O(1). The evaluation of the membership function µA (x) and the non-membership function νA (x) in30

volves a direct function evaluation (e.g., linear, logarithmic, polynomial, or sigmoid), which also requires O(1) time. The computation of the intuitionistic index πA (x) = 1−µA (x)−νA (x) and the calculation of RepD(vi , t) involve a fixed number of arithmetic operations and one exponentiation. Therefore, the computation of the reputation degree for a single validator has constant complexity O(1). For a network with n validators, computing the reputation degree for all validators in a round requires O(n) time. The reputation weight w(vi , t) is obtained by applying the uninorm operator U (RepD(vi , t), w(vi , t − 1)). Since the uninorm (including Fodor’s uninorm) is defined as a piecewise function involving a constant number of comparisons and arithmetic operations, its evaluation also has constant time complexity O(1) per validator. Consequently, updating the reputation weight for all validators requires O(n) time per round. Combining both phases, the total computational complexity per round remains linear, O(n). The proposed framework does not require additional peer-to-peer message exchanges beyond those already inherent to the underlying consensus protocol. The successful validation rate SuccV R(vi , t) is computed locally from validation outcomes that are already broadcast as part of the standard block validation process. Specifically, during each round, validators exchange consensus messages (e.g., block proposals, votes, or confirmations) according to the base consensus algorithm. The reputation mechanism passively observes these outcomes and updates local reputation values. Therefore, the reputation degree and reputation weight calculations introduce no additional all-to-all communication. Assuming the underlying consensus protocol has communication complexity O(C(n)), the integration of the proposed reputation framework preserves this complexity: O(C(n)) + O(n), where the additional O(n) term corresponds only to local updates without extra network-wide broadcasts. Hence, the proposed intuitionistic fuzzy and uninorm-based methodology introduces negligible communication overhead and maintains scalability even in large-scale blockchain networks. 6. Discussion and Future directions This work introduces a novel methodology for managing reputation behaviour in consensus algorithms for blockchain, utilising intuitionistic fuzzy 31

Innovative integration

Reputation management

Computational overhead Strengths

Computational overhead Efficiency

Optimization

Critical analysis Applications

Limitations

Real-world scenarios Scalability

Future direction

Validations

Reliability Uncertainty

Smart contracts Governance mechanism

Handling large networks

High transactions

Enhancing performance

Comparative studies Federated learning

Technologies

Security

Machine learning

Figure 16: Figure summarises the main aspects related to the critical analysis and future directions.

sets and uninorm aggregation operations. Figure 16 depicts a summary of the main aspects discussed in the critical analysis and future directions sections. 6.1. Critical analysis The integration of intuitionistic fuzzy sets and uninorm aggregation operations offers an innovative solution for managing reputation behaviour in blockchain consensus algorithms, addressing key challenges such as uncertain reputation values and the absence of reputation recovery mechanisms. The methodological advantage is its ability to handle uncertainty and imprecision in reputation values, enabling validators to recover and improve their reputation even after making errors. This flexibility is demonstrated by the results in Section 5, which show that validators can regain their standing, promoting a resilient and adaptive network. Our work serves as a valuable reference for the scientific community, illustrating the potential of integrating IFS and UAO into consensus algorithms where reputation plays a crucial role. In addition, this approach contributes to designing more equitable consensus mechanisms, ensuring diverse and inclusive validator selection processes, thereby mitigating centralisation risks and enhancing network security and fairness. While the previous paragraphs highlight several strengths of the proposed methodology, we should also consider potential limitations or challenges that have not been addressed. For instance, scalability, further empirical studies and optimisations may be required to ensure that the methodology can handle the exponential growth in transaction volumes and network size typical of large-scale blockchain deployments. For this framework, the overall computational complexity per round is O(n) Furthermore, integrating IFS and UAO into blockchain consensus algorithms could add computational overhead, which may potentially impact 32

the efficiency and performance of the blockchain. Specifically, the fuzzy set operations and aggregation processes require more sophisticated algorithms and data handling techniques, which might slow down transaction validation processes, particularly in large-scale blockchains. 6.2. Future directions Building on the insights from the critical analysis, we outline potential avenues for future research and development. This includes exploring the scalability of the proposed approach and enhancing its efficiency to ensure optimal performance in large-scale blockchain networks. Extending the application of this methodology beyond consensus algorithms to other areas of blockchain technology, such as smart contract execution and governance mechanisms, could further enhance the system’s robustness and adaptability. While the proposed methodology shows promise in addressing key challenges related to reputation management in blockchain consensus algorithms, further exploration and validation are necessary to fully assess its practical implications and scalability in real-world blockchain deployments. These directions connect to a broader line of fuzzy systems research. Optimising the uninorm parameters, as noted above, could itself be framed as an evolving, self-adaptive fuzzy system that relearns online from streaming validator behaviour rather than being fixed offline, in the tradition of evolving fuzzy classifiers for streaming sensor data [43, 44]. Because reputation is tracked across successive rounds, modelling how a validator’s intuitionistic fuzzy reputation value evolves over time is itself a temporal problem, for which time-dependent fuzzy membership functions [45] offer a natural formalism. Expressing reputation categories as linguistic, human-interpretable labels rather than raw uninorm outputs, following computing-with-words methodologies [46], and extracting explainable fuzzy rules that justify a given reputation adjustment or validator selection [47], would make the framework’s decisions more auditable to network participants. The large-scale empirical validation called for above would benefit from testing on more realistic, adversarial deployment conditions rather than only simulated scenarios, echoing a similar lesson from clinical brain-computer interface research, where competitions moved from healthy-subject to more challenging patient data to properly stress-test reliability [48]; the same broader concern, that models must remain reliable under noisy, real-world conditions, extends beyond blockchain consensus to deep learning systems generally [49]. Finally, packaging this reputation framework as accessible open-source tooling, in the 33

spirit of toolboxes built for other fuzzy, multivariate analysis pipelines [50], would help practitioners adopt and audit it in deployment. 7. Conclusion In this work, we have explored the areas of intuitionistic fuzzy sets and uninorm aggregation operations to introduce our novel methodology for managing reputation behaviour in consensus algorithms for blockchain. This methodology addresses the challenges associated with uncertain reputation values and a lack of mechanisms for reputation recovery. The results presented in Section 5 support the proposed methodology, showing that the validator has the opportunity to increase its reputation even when it was wrong in the previous round, which is not possible concerning other consensus algorithms. Moreover, as Section 5.3 discussed the framework preserves linear computational complexity per round and does not introduce additional communication overhead beyond that required by the underlying consensus protocol, thereby maintaining scalability for large blockchain networks. This paper provides a reference for the scientific community and could be used in different consensus algorithms where reputation is considered a key parameter. Furthermore, this methodology can be utilised to design equitable consensus algorithms and provide diversification in the selection process to choose the validator node. Integrating intuitionistic fuzzy sets and uninorm aggregation operators into blockchain offers a flexible framework for synthesising uncertain or imprecise information from multiple sources within blockchain networks. Future research directions include large-scale empirical validation, optimisation of the uninorm parameters, and integration with hybrid consensus architectures to further enhance robustness and practical deployment.

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References [1] Y. Chen, H. Chen, Y. Zhang, M. Han, M. Siddula, Z. Cai, A survey on blockchain systems: Attacks, defenses, and privacy preservation, High-Confidence Computing 2 (2) (2022) 100048. doi:https://doi.org/10.1016/j.hcc.2021.100048. URL https://www.sciencedirect.com/science/article/pii/S2667295221000386 [2] A. T. Espinoza Pérez, D. A. Rossit, F. Tohmé, Óscar C. Vásquez, Mass customized/personalized manufacturing in industry 4.0 and blockchain: Research challenges, main problems, and the design of an information architecture, Information Fusion 79 (2022) 44–57. doi:https://doi.org/10.1016/j.inffus.2021.09.021. URL https://www.sciencedirect.com/science/article/pii/S1566253521001974 [3] S. Fahim, S. Rahman, S. Mahmood, Blockchain: A comparative study of consensus algorithms pow, pos, poa, pov, Int. J. Math. Sci. Comput 3 (2023) 46–57. [4] A. Back, et al., Hashcash-a denial of service counter-measure (2002). [5] S. Nakamoto, Bitcoin: A peer-to-peer electronic cash system, Decentralized business review (2008). [6] Ethereum, Ethereum mainnet for enterprise, [Online], last accessed 2023-08-12, https://ethereum.org/en/ (2023). [7] D. Larimer, Transactions as proof-of-stake, Nov-2013 909 (2013). [8] B. Ramos-Cruz, J. Andreu-Pérez, F. J. Quesada, L. Martínez, Fuzzychain: An equitable consensus mechanism for blockchain networks (2024). arXiv:2404.13337. [9] P. Zheng, Z. Jiang, J. Wu, Z. Zheng, Blockchain-based decentralized application: A survey, IEEE Open Journal of the Computer Society 4 (2023) 121–133. doi:10.1109/OJCS.2023.3251854. [10] N. Chaudhry, M. M. Yousaf, Consensus algorithms in blockchain: Comparative analysis, challenges and opportunities, in: 2018 12th International Conference on Open Source Systems and Technologies (ICOSST), 2018, pp. 54–63. doi:10.1109/ICOSST.2018.8632190. 35

[11] A. Kumar, B. Kumar Sah, T. Mehrotra, G. K. Rajput, A review on double spending problem in blockchain, in: 2023 International Conference on Computational Intelligence and Sustainable Engineering Solutions (CISES), 2023, pp. 881–889. doi:10.1109/CISES58720.2023.10183579. [12] M. Wendl, M. H. Doan, R. Sassen, The environmental impact of cryptocurrencies using proof of work and proof of stake consensus algorithms: A systematic review, Journal of Environmental Management 326 (2023) 116530. doi:https://doi.org/10.1016/j.jenvman.2022.116530. URL https://www.sciencedirect.com/science/article/pii/S030147972202103X [13] D. Larimer, Delegated proof-of-stake (dpos), Bitshare whitepaper 81 (2014) 85. [14] M. Castro, B. Liskov, et al., Practical byzantine fault tolerance, in: OsDI, Vol. 99, 1999, pp. 173–186. [15] K. P. V. Kumar, B. Lakshmi, S. Sai Kumar, M. V, N. R. Sai, V. Nagamalleswara, Blockchain technology: A comprehensive review and future directions, in: 2023 Second International Conference on Augmented Intelligence and Sustainable Systems (ICAISS), 2023, pp. 1362–1368. doi:10.1109/ICAISS58487.2023.10250707. [16] L. Zadeh, Fuzzy sets, Information and Control 8 (3) (1965) 338–353. doi:https://doi.org/10.1016/S0019-9958(65)90241-X. URL https://www.sciencedirect.com/science/article/pii/S001999586590241X [17] K. T. Atanassov, Intuitionistic Fuzzy Sets - Theory and Applications, Vol. 35 of Studies in Fuzziness and Soft Computing, Physica-Verlag, 1999. doi:10.1007/978-3-7908-1870-3. URL https://doi.org/10.1007/978-3-7908-1870-3 [18] G. C. Yalçın, K. Kara, S. Edinsel, E. G. Kaygısız, V. Simic, D. Pamucar, Authentication system selection for performance appraisal in human resource management using an intuitionistic fuzzy cimas-arlon model, Applied Soft Computing 171 (2025) 112786. doi:https://doi.org/10.1016/j.asoc.2025.112786. URL https://www.sciencedirect.com/science/article/pii/S1568494625000973

36

[19] R. R. Yager, Mam and mom bag operators for aggregation, Information Sciences 69 (3) (1993) 259–273. doi:https://doi.org/10.1016/00200255(93)90124-5. URL https://www.sciencedirect.com/science/article/pii/0020025593901245 [20] R. R. Yager, A. Rybalov, Uninorm aggregation operators, Fuzzy Sets and Systems 80 (1) (1996) 111–120, fuzzy Modeling. doi:https://doi.org/10.1016/0165-0114(95)00133-6. URL https://www.sciencedirect.com/science/article/pii/0165011495001336 [21] A. Bugday, A. Ozsoy, S. M. Öztaner, H. Sever, Creating consensus group using online learning based reputation in blockchain networks, Pervasive Mob. Comput. 59 (2019). URL https://api.semanticscholar.org/CorpusID:199016438 [22] M. T. de Oliveira, L. H. A. Reis, D. S. V. Medeiros, R. C. Carrano, S. D. Olabarriaga, D. M. F. Mattos, Blockchain reputationbased consensus: A scalable and resilient mechanism for distributed mistrusting applications, Comput. Networks 179 (2020) 107367. doi:10.1016/J.COMNET.2020.107367. URL https://doi.org/10.1016/j.comnet.2020.107367 [23] J. B. Abdo, R. E. Sibai, K. Kambhampaty, J. Demerjian, Permissionless reputation-based consensus algorithm for blockchain, Internet Technology Letters 3 (3) (2020). doi:10.1002/ITL2.151. URL https://doi.org/10.1002/itl2.151 [24] O. Aluko, A. Kolonin, Studying the applicability of proof of reputation(por) as an alternative consensus mechanism for distributed ledger systems, Computer Science & Information Technology (CS & IT) (2021). URL https://api.semanticscholar.org/CorpusID:235431384 [25] J. I. C. Rubiera, J. M. Molina, J. D. Muro, Trust management through fuzzy reputation, Int. J. Cooperative Inf. Syst. 12 (2003) 135–155. URL https://api.semanticscholar.org/CorpusID:10852864 [26] F. J. Quesada, I. Palomares, L. Martínez-López, Managing experts behavior in large-scale consensus reaching processes with uninorm aggregation operators, Appl. Soft Comput. 35 (2015) 873–887.

37

doi:10.1016/J.ASOC.2015.02.040. URL https://doi.org/10.1016/j.asoc.2015.02.040 [27] N. Singh, A. Kumar Das, D. Sinha, Miner selection in an internet of medical things framework using fuzzy logic, Applied Soft Computing 161 (2024) 111768. doi:https://doi.org/10.1016/j.asoc.2024.111768. URL https://www.sciencedirect.com/science/article/pii/S1568494624005428 [28] W. Cai, W. Jiang, K. Xie, Y. Zhu, Y. Liu, T. Shen, Dynamic reputation-based consensus mechanism: Real-time transactions for energy blockchain, Int. J. Distributed Sens. Networks 16 (3) (2020). doi:10.1177/1550147720907335. URL https://doi.org/10.1177/1550147720907335 [29] P. Chen, Y. Chen, X. Wang, L. Yuan, C. Tan, Y. Yang, A high-capacity slicing pbft protocol based on reputation evaluation model, WIRELESS NETWORKS 30 (9) (2024) 7469–7482. doi:10.1007/s11276-023-03636-7. [30] Z. Deng, C. Tang, T. Li, D. He, Permissioned blockchainbased trusted and robust consensus optimization orienting intelligent transportation systems, IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS (2024 NOV 27 2024). doi:10.1109/TITS.2024.3496553. [31] C. Gan, X. Xiao, Y. Zhang, Q. Zhu, J. Bi, D. K. Jain, A. Saini, An asynchronous federated learning-assisted data sharing method for medical blockchain, APPLIED INTELLIGENCE 55 (2) (JAN 2025). doi:10.1007/s10489-024-06172-9. [32] M. Hussain, A. Mehmood, M. A. Khan, R. Khan, J. Lloret, Reputation-based leader selection consensus algorithm with rewards for blockchain technology, COMPUTERS 14 (1) (JAN 2025). doi:10.3390/computers14010020. [33] G. Jia, Z. Shen, H. Sun, J. Xin, D. Wang, Rwa-bft: Reputation-weighted asynchronous bft for large-scale iot, SENSORS 25 (2) (JAN 2025). doi:10.3390/s25020413. [34] Y. Jiang, Y. Guan, A cluster reputation-based hierarchical consensus model in blockchain, Peer Peer Netw. Appl. 16 (5) (2023) 2591–2606. 38

doi:10.1007/S12083-023-01550-5. URL https://doi.org/10.1007/s12083-023-01550-5 [35] K. Lei, Q. Zhang, L. Xu, Z. Qi, Reputation-based byzantine fault-tolerance for consortium blockchain, in: 24th IEEE International Conference on Parallel and Distributed Systems, ICPADS 2018, Singapore, December 11-13, 2018, IEEE, 2018, pp. 604–611. doi:10.1109/PADSW.2018.8644933. URL https://doi.org/10.1109/PADSW.2018.8644933 [36] C. T. Nguyen, D. T. Hoang, D. N. Nguyen, D. Niyato, H. T. Nguyen, E. Dutkiewicz, Proof-of-stake consensus mechanisms for future blockchain networks: Fundamentals, applications and opportunities, IEEE Access 7 (2019) 85727–85745. doi:10.1109/ACCESS.2019.2925010. URL https://doi.org/10.1109/ACCESS.2019.2925010 [37] A. Sarfaraz, R. K. Chakrabortty, D. Essam, Reputation based proof of cooperation: an efficient and scalable consensus algorithm for supply chain applications, J. Ambient Intell. Humaniz. Comput. 14 (6) (2023) 7795–7811. doi:10.1007/S12652-023-04592-Y. URL https://doi.org/10.1007/s12652-023-04592-y [38] X. Wu, T. Meng, H. Huang, L. Liu, J. Zhang, A hybrid consensus algorithm for collaborative protection of multi-domain education data, JOURNAL OF INTERNET TECHNOLOGY 25 (7) (2024) 963–975. doi:10.70003/160792642024122507002. [39] F. Yang, W. Zhou, Q. Wu, R. Long, N. N. Xiong, M. Zhou, Delegated proof of stake with downgrade: A secure and efficient blockchain consensus algorithm with downgrade mechanism, IEEE Access 7 (2019) 118541–118555. doi:10.1109/ACCESS.2019.2935149. URL https://doi.org/10.1109/ACCESS.2019.2935149 [40] T. Zhang, Z. Huang, Fpor: Fair proof-of-reputation consensus for blockchain, ICT Express 9 (1) (2023) 45–50. doi:10.1016/J.ICTE.2022.11.007. URL https://doi.org/10.1016/j.icte.2022.11.007 [41] Q. Zhuang, Y. Liu, L. Chen, Z. Ai, Proof of reputation: A reputationbased consensus protocol for blockchain based systems, in: Proceedings of the 2019 International Electronics Communication Conference, 39

IECC 2019, Okinawa, Japan, July 7 - 9, 2019, ACM, 2019, pp. 131–138. doi:10.1145/3343147.3343169. URL https://doi.org/10.1145/3343147.3343169 [42] J. C. Fodor, R. R. Yager, A. N. Rybalov, Structure of uninorms, Int. J. Uncertain. Fuzziness Knowl. Based Syst. 5 (4) (1997) 411–428. doi:10.1142/S0218488597000312. URL https://doi.org/10.1142/S0218488597000312 [43] J. Andreu, P. Angelov, Real-time human activity recognition from wireless sensors using evolving fuzzy systems, in: International Conference on Fuzzy Systems, IEEE, 2010, pp. 1–8. [44] J. Andreu, P. Angelov, An evolving machine learning method for human activity recognition systems, Journal of Ambient Intelligence and Humanized Computing 4 (2) (2013) 195–206. [45] M. Kiani, J. Andreu-Perez, H. Hagras, A temporal type-2 fuzzy system for time-dependent explainable artificial intelligence, IEEE Transactions on Artificial Intelligence 4 (3) (2022) 573–586. [46] P. K. Gupta, J. Andreu-Perez, A gentle introduction and survey on computing with words (cww) methodologies, Neurocomputing 500 (2022) 921–937. [47] J. Fumanal-Idocin, J. Andreu-Perez, O. Cordón, H. Hagras, H. Bustince, Artxai: Explainable artificial intelligence curates deep representation learning for artistic images using fuzzy techniques, IEEE Transactions on Fuzzy Systems 32 (4) (2023) 1915–1926. [48] A. Chowdhury, J. Andreu-Perez, Clinical brain–computer interface challenge 2020 (cbcic at wcci2020): Overview, methods and results, IEEE Transactions on Medical Robotics and Bionics 3 (3) (2021) 661–670. [49] L. Chaudhary, N. Girdhar, D. Sharma, J. Andreu-Perez, A. Doucet, M. Renz, A review of deep learning models for twitter sentiment analysis: Challenges and opportunities, IEEE Transactions on Computational Social Systems 11 (3) (2023) 3550–3579.

40

[50] J. Andreu-Perez, C. Solnais, K. Sriskandarajah, Ealab (eye activity lab): a matlab toolbox for variable extraction, multivariate analysis and classification of eye-movement data, Neuroinformatics 14 (1) (2016) 51–67.

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