The LTS [33–37] is an essential tool, which is used to provide the available linguistic variables for linguistic computational models. It is a set that consists of an odd number of linguistic terms. There are two classes of LTSs, which are categorized into asymmetric LTSs and symmetric LTSs, respectively. The definition of a common LTS is given as follows.

Definition 1 [33]. Let S={sε|ε=−θ,…,−1,0,1,…,θ} denote a symmetric LTS, in which the linguistic terms are ordered according to the ascending order of their subscripts. θ is an even number and θ+1 is the cardinality of the LTS S. s−θ and sθ are the lower bound and upper bound of S.

The definition of PHFLTSs was proposed by Chen et al. [38] to express the collective voting information from a group of voting nodes. It is designed based on LTS and probability distribution information of linguistic terms [39–41]. Its mathematical definition can be described as follows:

Definition 2 [38]. Let S={sε|ε=−θ,…,−1,0,1,…,θ} denote an LTS and P={(sε,pε)|ε=−θ,…,−1,0,1,…,θ} be the probability distribution information of linguistic terms, then a PHFLTS PS={(sε,pε)|ε=−θ,…,−1,0,1,…,θ}, where 0≤pε≤1 and ∑ε=−θθ⁡pε=1.

For each voted node, its received voting information can be modeled as a PHFLTS. To rank nodes and select delegates, the PHFLTSs should be compared.

Chen et al. [38] developed the possibility degree to compare PHFLTSs. The definition of possibility degree is given as follows:

Definition 3 [38]. Let S={sε|ε=−θ,…,−1,0,1,…,θ} represent an LTS, and PS1={(sε1,pε1)|sε1∈S} and PS2={(sε2,pε2)|sε2∈S} be any two PHFLTSs, then the possibility degree between them is defined as:

(1)ρ(PS1≥PS2)=∑ε1∈{−θ,…,−1,0,1,…,θ}⁡∑ε2∈{−θ,…,−1,0,1,…,θ}⁡R(sε1,sε2),

where R(sε1,sε2) is the relation value between sε1 and sε2.

Definition 4 [38]. Let S={sε|ε=−θ,…,−1,0,1,…,θ} represent an LTS, and PS1={(sε1,pε1)|sε1∈S} and PS2={(sε2,pε2)|sε2∈S} be any two PHFLTSs, then the relation value between sε1 and sε2 is defined as:

(2)R(sε1,sε2)={pε1pε2,sε1>sε212pε1pε2,sε1=sε20,sε1<sε2,

where pε1 and pε2 are the probability distribution information of sε1 and sε2, respectively.

Property 1 [38]. Let PS1 and PS2 be any two PHFLTSs, then we have ρ(PS1≥PS2)+ρ(PS1≤PS2)=1. Especially, if PS1=PS2, then we have ρ(PS1≥PS2)=ρ(PS1≤PS2)=0.5.

To ensure the fairness of the delegate selection results in the DPoS consensus mechanism, we propose a novel full associate voting architecture for collecting the voting information of all the blockchain nodes.

In the full associate voting architecture, each blockchain node should choose a linguistic term from the LTS S to vote on each blockchain node in the blockchain network. Thus, each blockchain node will receive the voting information from all the blockchain nodes including itself as shown in Fig. 1.

Here, a symmetric LTS S is used to provide the fine-grained voting options for blockchain nodes as follows : S={s−2=″Veryopposed″,s−1=″Opposed″,s0=″Neutral″,s1=″Supported″,s2=″Verysupported″}.

By using this symmetric LTS, each blockchain node can express its more fine-grained opinion by choosing one linguistic term from this LTS. The linguistic term s0 indicates the neutral attitude of the blockchain nodes. If the blockchain nodes abstain, then the linguistic term s0 is used to express their opinions.

Example 1. For the blockchain node 1 as depicted in Fig. 1, the collective voting information from all the blockchain nodes for the blockchain node 1 can be represented as a PHFLTS PS={(s−2,0),(s−1,0.1),(s0,0.4),(s1,0.2),(s2,0.3)}. We can find that the PHFLTS shows accurate information way without information loss.

For a blockchain network, there are n blockchain nodes and all the blockchain nodes should vote for each other using the voting options from the symmetric LTS S. Based on the above discussion, the steps of the delegate selection algorithm are developed as follows:

(1) Each blockchain node selects one linguistic term from the symmetric LTS S to vote for blockchain nodes in the blockchain network every time t. If the blockchain node abstains, the linguistic term s0 in the symmetric LTS S is used to express his/her voting information.

(2) For each blockchain node in the blockchain network, the voting information from all the blockchain nodes is collected, and the probability distribution information of linguistic terms is calculated. Then, the PHFLTS is used to model the collective voting information of each blockchain node i as PSi={(sεi,pεi)|sεi∈S},i=1,2,…,n.

For more details, please refer to Example 1.

Thus, all the collective voting information of n blockchain nodes in the blockchain network can be expressed as a set of n PHFLTSs. Then, the delegate selection can be formulated to be a blockchain node rank problem by comparing their PHFLTSs.

(3) The pairwise comparisons of all the blockchain nodes are conducted, and then a possibility degree matrix can be obtained as follows: p=[0.5ρ12⋯ρ1nρ210.5⋯ρ2n⋮⋮⋱⋮ρn1ρn2⋯0.5],

where ρij=ρ(PSi≥PSj), ρij+ρji=1, and ρii=0.5.

(4) The cumulative possibility degree ρi of each blockchain node i is calculated as (3)ρi=1n∑j=1nρij,i=1,2,…,n.

(5) The cumulative possibility degrees of blockchain nodes are ranked in descending order as ρ(1)≥ρ(2)≥⋯≥ρ(m−(m1+1))≥ρ(m−m1)≥⋯≥ρ(m−1)≥ρ(m)≥ρ(m+1)≥⋯≥ρ(m+m2)≥⋯≥ρ(n), where {ρ(1),ρ(2),…,ρ(n)} is a permutation of {ρ1,ρ2,…,ρn} that satisfies ρ(i)≥ρ(j).

If ρ(m)≠ρ(m+1), then the blockchain nodes with the largest m cumulative possibility degrees are selected as delegates.

If all the cumulative possibility degrees of blockchain nodes satisfy the following condition:ρ(m−(m1+1))>ρ(m−m1)=⋯=ρ(m−1)=ρ(m)=ρ(m+1)=⋯>ρ(m+m2)>ρ(m+m2+1), then the blockchain nodes that possess the largest m−(m1+1) cumulative possibility degrees are selected as delegates. Then, the lottery algorithm is used to select the rest m1+1 delegates from the blockchain nodes that own the cumulative possibility degrees ρ(m−m1),…,ρ(m−1),ρ(m),ρ(m+1),…,ρ(m+m2).

Example 2. Let us assume that there exist two nodes A and B, the LTS S is used to provide fine-grained voting options for blockchain nodes. The voting information received by them is listed in Table 1. Based on the voting information in Table 1, the collective voting information for nodes A and B can be modeled as the following two PHFLTSs: PSA={(s−2,110),(s−1,110),(s0,210),(s1,210),(s2,410)}={(s−2,0.1),(s−1,0.1),(s0,0.1),(s1,0.2),(s2,0.4)}, PSB={(s−2,110),(s−1,010),(s0,410),(s1,410),(s2,110)}={(s−2,0.1),(s−1,0.0),(s0,0.4),(s1,0.4),(s2,0.1)}.

Let us consider choosing one node from node A and node B as the delegate. If the voting result of one node is computed as the difference value of positive votes and negative votes, then the voting result of node A is computed as 4+2−1−1=4 and the voting result of node B is computed as 1+4−1−0=4. In this case, node A and node B are indistinguishable, and then the delegate cannot be chosen. That is because the neutral attitudes of blockchain nodes, and the intensity of support attitudes and opposition attitudes are not considered. If our delegate selection algorithm is used, then the possibility degree between nodes A and B is computed as ρ(PSA≥PSB)=0.595.

According to Property 1, ρ(PSB≥PSA)=0.405. Therefore, it is considered that the voting result of node B is superior to that of node A, and node B can be selected as the delegate.

In the following part, another example is given to compare the traditional DPoS consensus algorithm with our proposed DPoS consensus algorithm as follows:

Example 3. Let us assume that there exist two nodes C and D, and the LTS S is used to provide fine-grained voting options for blockchain nodes. The voting information received by them is listed in Table 2. Based on the voting information in Table 2, the collective voting information for nodes C and D can be modeled as the following two PHFLTSs: PSC={(s−2,0.2),(s−1,0.3),(s0,0.1),(s1,0.1),(s2,0.3)}, PSD={(s−2,0.0),(s−1,0.1),(s0,0.6),(s1,0.0),(s2,0.3)}.

Let us consider choosing one node from node C and node D as the delegate. If the traditional DPoS consensus algorithm is used, only the number of positive votes is considered to choose the delegate. In this case, the number of positive votes of node C is 4, while the number of positive votes of node D is 3. Then, node C is selected to be the delegate when the traditional DPoS consensus algorithm is used. However, from Table 2, it can be seen that the number of negative votes of node C is 5. It is more than that of node D. Moreover, the number of neutral votes of node C is also less than that of node D. Thus, it is unreasonable to choose node C to be the delegate. When our proposed DPoS consensus algorithm is used, the possibility degree is computed as ρ(PSD≥PSC)=0.62>0.5. Thus, node D is selected as the delegate. It demonstrates the rationality of our proposed DPoS consensus algorithm. The security of the blockchain network is improved.

Let us consider an example that there exist 21 nodes, each of which can give its voting information to all the nodes including itself by using the LTS S. The voting information received by these 21 nodes is listed in Table 3.

(1) Our DPoS consensus algorithm (DPoS-PHFLTS)

Based on Table 3, the collective voting information of these 21 nodes can be modeled as the following PHFLTSs: PS1={(s−2,121),(s−1,121),(s0,0),(s1,621),(s2,1321)}, PS2={(s−2,621),(s−1,221),(s0,221),(s1,221),(s2,921)}, PS3={(s−2,421),(s−1,221),(s0,221),(s1,221),(s2,1121)}, PS4={(s−2,221),(s−1,421),(s0,421),(s1,621),(s2,521)}, PS5={(s−2,321),(s−1,621),(s0,421),(s1,221),(s2,621)}, PS6={(s−2,1021),(s−1,0),(s0,521),(s1,0),(s2,621)}, PS7={(s−2,621),(s−1,421),(s0,621),(s1,0),(s2,521)}, PS8={(s−2,221),(s−1,421),(s0,621),(s1,621),(s2,321)}, PS9={(s−2,221),(s−1,421),(s0,0),(s1,621),(s2,921)}, PS10={(s−2,221),(s−1,421),(s0,0),(s1,921),(s2,621)}, PS11={(s−2,821),(s−1,421),(s0,621),(s1,0),(s2,321)}, PS12={(s−2,321),(s−1,521),(s0,521),(s1,521),(s2,321)}, PS13={(s−2,221),(s−1,221),(s0,0),(s1,1221),(s2,521)}, PS14={(s−2,1421),(s−1,0),(s0,421),(s1,0),(s2,321)}, PS15={(s−2,121),(s−1,221),(s0,0),(s1,0),(s2,1821)}, PS16={(s−2,121),(s−1,221),(s0,0),(s1,1021),(s2,821)}, PS17={(s−2,721),(s−1,221),(s0,421),(s1,421),(s2,421)}, PS18={(s−2,421),(s−1,421),(s0,521),(s1,421),(s2,421)}, PS19={(s−2,921),(s−1,921),(s0,121),(s1,0),(s2,221)}, PS20={(s−2,1621),(s−1,0),(s0,0),(s1,521),(s2,0)}, PS21={(s−2,1121),(s−1,0),(s0,521),(s1,521),(s2,0)}.

According to Definitions 3 and 4, the cumulative possibility degree ρi of each node i is computed as: ρ1=121∑j=121ρ1j=0.7333,ρ2=121∑j=121ρ2j=0.5407, ρ3=121∑j=121ρ3j=0.6104,ρ4=121∑j=121ρ4j=0.5468, ρ5=121∑j=121ρ5j=0.5059,ρ6=121∑j=121ρ6j=0.4178, ρ7=121∑j=121ρ7j=0.4367,ρ8=121∑j=121ρ8j=0.5090, ρ9=121∑j=121ρ9j=0.6223,ρ10=121∑j=121ρ10j=0.5888, ρ11=121∑j=121ρ11j=0.3671,ρ12=121∑j=121ρ12j=0.4788 ρ13=121∑j=121ρ13j=0.6060,ρ14=121∑j=121ρ14j=0.2974, ρ15=121∑j=121ρ15j=0.7750,ρ16=121∑j=121ρ16j=0.6632, ρ17=121∑j=121ρ17j=0.4457,ρ18=121∑j=121ρ18j=0.4806, ρ19=121∑j=121ρ19j=0.2999,ρ20=121∑j=121ρ20j=0.2475, ρ21=121∑j=121ρ21j=0.3271.

According to the descending order of cumulative possibility degree, these 21 blockchain nodes can be ranked as: N15>N1>N16>N9>N3>N13>N10>N4>N2>N8>N5>N18>N12>N17>N7>N6>N11>N21>N19>N14>N20.

Therefore, node 15, node 1, node 16, node 9, and node 3 are selected as the delegates for validating the transactions.

(2) Difference value method (DV)

The voting result of one node is computed as the difference value of positive votes and negative votes in DV. In this part, we calculated the number of affirmative and negative votes in Table 3, and the statistical results are shown in Table 4. The difference value method is used to calculate the voting result of each blockchain node as follows: ξ1=13+6−1−1=17,ξ2=9+2−2−6=3, ξ3=11+2−2−4=7,ξ4=5+6−4−2=5, ξ5=6+2−6−3=−1,ξ6=6+0−0−10=−4, ξ7=5+0−4−6=−5,ξ8=3+6−4−2=3, ξ9=6+9−4−2=9,ξ10=9+6−4−2=9, ξ11=3+0−4−8=−9,ξ12=3+5−5−3=0, ξ13=5+12−2−2=13,ξ14=3+0−0−14=−11, ξ15=18+0−2−1=15,ξ16=8+10−2−1=15, ξ17=4+4−2−7=−1,ξ18=4+4−4−4=0, ξ19=2+0−9−9=−16,ξ20=0+5−0−16=−11, ξ21=0+5−0−11=−6.

According to the voting result, these 21 blockchain nodes are ranked as: N1>N15=N16>N13>N9=N10>N3>N4>N2=N8>N17>N12=N18>N5>N6>N7>N21>N11>N14>N20>N19.

Thus, node 1, node 15, node 16, and node 13 are directly selected as the delegates. However, the voting results of node 9 and node 10 are equal. They cannot be indistinguishable and the remaining delegate cannot be obtained when the difference value method is used.

(3) Traditional DPoS consensus algorithm (T-DPoS)

The voting result of one node is computed as the value of positive votes in T-DPoS. In this part, according to the statistical results as shown in Table 4, the traditional DPoS consensus algorithm is used to compute the voting result of each blockchain node as: δ1=13+6=19,δ2=9+2=11, δ3=11+2=13,δ4=5+6=11, δ5=6+2=8,δ6=6+0=6, δ7=5+0=5,δ8=3+6=9, δ9=9+6=15,δ10=9+6=15, δ11=3+0=3,δ12=3+5=8, δ13=5+12=17,δ14=3+0=3, δ15=18+0=18,δ16=8+10=18, δ17=4+4=8,δ18=4+4=8, δ19=2+0=2,δ20=0+5=5, δ21=0+5=5.

According to the voting result, these 21 blockchain nodes are ranked as: N1>N15=N16>N13>N10=N9>N3>N4=N2>N8>N18=N17=N12=N5>N6>N21=N20=N7>N14=N11>N19.

Thus, node 1, node 15, node 16, and node 13 are directly selected as the delegates. However, the voting results of node 9 and node 10 are equal. They cannot be distinguishable and the remaining delegate cannot be obtained when the difference value method is used.

(4) PBFT consensus algorithm based on vague sets (VS-PBFT)

In this part, the PBFT consensus algorithm based on vague sets (VS-PBFT) [42] is used. As listed in Table 5, the collective voting information for these 21 blockchain nodes is modeled using vague sets, and their corresponding fuzzy values are computed.

According to the fuzzy values, these 21 nodes are ranked as: N1>N15=N16>N13>N9=N10>N3>N4>N8>N2>N12=N18>N5=N17>N6>N7>N21>N11>N20>N14>N19.

Thus, node 1, node 15, node 16, and node 13 are directly selected as the delegates. However, the fuzzy values of node 9 and node 10 are equal. They cannot be indistinguishable and the remaining delegate cannot be obtained.

To show the superiority of our DPoS-PHFLTS algorithm, the ranking results obtained by four algorithms are summarized in Table 6.

As shown in Table 6, it can be seen that our DPoS-PHFLTS algorithm can effectively obtain 5 delegates from all the nodes, while the rest algorithms only select 4 delegates and the rest one delegate cannot be directly obtained from node 9 and node 10. Nevertheless, we can perceive from the voting results of node 9 and node 10 that there is a stronger intention to support node 9. From the analysis result, it can be seen that our DPoS-PHFLTS algorithm has better performance than the existing algorithms in this example.

Another voting information received by these 21 nodes is listed in Table 7.

(1) Our DPoS consensus algorithm (DPoS-PHFLTS)

Based on Table 7, according to Definitions 3 and 4, the cumulative possibility degree ρi of each node i is computed as: ρ1=121∑j=121ρ1j=0.5843,ρ2=121∑j=121ρ2j=0.3673, ρ3=121∑j=121ρ3j=0.2835,ρ4=121∑j=121ρ4j=0.5329, ρ5=121∑j=121ρ5j=0.4469,ρ6=121∑j=121ρ6j=0.4764, ρ7=121∑j=121ρ7j=0.5807,ρ8=121∑j=121ρ8j=0.3930, ρ9=121∑j=121ρ9j=0.4942,ρ10=121∑j=121ρ10j=0.5749, ρ11=121∑j=121ρ11j=0.5532,ρ12=121∑j=121ρ12j=0.5234, ρ13=121∑j=121ρ13j=0.6957,ρ14=121∑j=121ρ14j=0.4430, ρ15=121∑j=121ρ15j=0.7323,ρ16=121∑j=121ρ16j=0.7245, ρ17=121∑j=121ρ17j=0.5259,ρ18=121∑j=121ρ18j=0.3235, ρ19=121∑j=121ρ19j=0.3387,ρ20=121∑j=121ρ20j=0.4700, ρ21=121∑j=121ρ21j=0.4358.

According to the descending order of cumulative possibility degree, these 21 blockchain nodes can be ranked as: N15>N16>N13>N1>N7>N10>N11>N4>N17>N12>N9>N6>N20>N5>N14>N21>N8>N2>N19>N18>N3.

(2) Difference value method (DV)

In this part, we calculated the number of affirmative and negative votes in Table 7, and the statistical results are shown in Table 8. The difference value method is used to calculate the voting result of each blockchain node as follows: ξ1=3+8−1−1=9,ξ2=1+6−0−13=−6, ξ3=1+1−1−15=−14,ξ4=3+6−1−2=6, ξ5=8+3−4−4=3,ξ6=0+7−5−0=2, ξ7=3+9−3−0=9,ξ8=2+5−4−8=−5, ξ9=6+5−2−5=4,ξ10=2+8−1−1=8, ξ11=2+10−3−4=5,ξ12=9+2−0−2=9, ξ13=4+13−0−3=14,ξ14=2+6−4−5=−1, ξ15=7+12−0−0=19,ξ16=2+16−2−1=15, ξ17=10+4−0−7=7,ξ18=4+3−9−4=−6, ξ19=6+2−8−4=−4,ξ20=4+1−0−4=1, ξ21=4+1−2−4=−1.