An improved cellular goore game-based consensus protocol for blockchain

Abstract

A blockchain is a distributed network of nodes that records transactions in a digital ledger. To ensure trust in a network with untrusted nodes, blockchain uses consensus algorithms to record transactions in its ledger. Although various consensus protocols have been proposed and implemented for blockchain lately, they still have drawbacks. The last decade has seen a lot of attention and rapid growth in artificial intelligence and blockchain technologies. The performance of blockchain can be improved, and its problems can be solved by incorporating artificial intelligence. The concept of cognitive blockchain is related to AI functionalities into blockchain systems to enhance their utility and capabilities. Cellular Goore Game-based consensus was recently proposed as intelligence consensus in the cognitive blockchain. Although this consensus algorithm improves scalability, fault tolerance, and performance, it has problems, such as high communication complexity. In this paper, we proposed an improved Cellular Goore Game-based consensus protocol, which increases fault tolerance and decreases communication complexity. We also studied theoretically the Cellular Goore Game used as a distributed model in the proposed consensus and proved the convergence of CGG in the proposed protocol in this paper. We evaluated the performance of the proposed protocol by conducting several experiments. Empirical results show that this approach improves fault tolerance and communication complexity.

Appendix A

Proof of Lemma 3: Let $ES(k)=\{{{\underline{\alpha }|\underline{\alpha }=(\alpha }_{1}\left(k\right),{\alpha }_{2}\left(k\right),\dots \dots ,{\alpha }_{{N}_{P}}\left(k\right))}^{T}\}$ be the event set that evolves state $\underline{x}\left(k\right)$. Therefore, we can express the following equation, where the definition of ${f}_{ES\left(k\right)}$ is in accordance with Eq. 14.

$$\underline{x}\left(k+1\right)={f}_{ES\left(k\right)}(\underline{x}\left(k\right))$$

(41)

Consider $Probability\left[ES\left(k\right)=e|\underline{x}\left(k\right)=\underline{x}\right]= {\rho }_{e}(\underline{x})$, wherein ${\rho }_{e}(\underline{x})$ is a real-valued function on $ES\times \mathcal{X}$. Let us define $d(\underline{x},\underline{y}),m\left({\rho }_{e}\right)$ and $\mu \left({f}_{e}\right)$ using the equations below.

$$d\left(\underline{x},\underline{y}\right)= {\sum }_{i}\left|{x}_{i}-{y}_{i}\right|$$

(42)

$$m\left( {\rho_{e} } \right) = sup_{{\underline{{x \ne x^{\prime}}} }} \frac{{\left| {\rho_{e} \left( {\underline{x} } \right) - \rho_{e} \left( {\underline{x} } \right)} \right|}}{{d\left( {\underline{x} ,\underline{{x^{\prime}}} } \right)}}$$

(43)

$$\mu \left( {f_{e} } \right) = sup_{{\underline{x} \ne \underline{{x^{\prime}}} }} \frac{{d\left( {f_{e} \left( {\underline{x} } \right) - f_{e} \left( {\underline{{x^{\prime}}} } \right)} \right)}}{{d\left( {\underline{x} ,\underline{{x^{\prime}}} } \right)}}$$

(44)

The following statements are true:

• 1. The set ES is finite.

• 2. ($\mathcal{X},d$) is a compact metric space.

• 3. For every $e\in ES$, $m\left({\rho }_{e}\right)<\infty$.

• 4. For every $e\in ES$, $\mu \left({f}_{e}\right) <1$. To prove this statement, suppose $\underline{x}$ and $\underline{y}$ as two states of the process ${\left\{\underline{x}\left(k\right)\right\}}_{k\ge 0}.$ The equation below shows that $\mu \left({f}_{e}\right) <1$ since ${0<b}_{i}<\mathrm{1,0}<\beta (k)<1, \forall i$ from Eqs. 15 and 44.

  $$\mu \left( {f_{e} } \right) = sup_{{\underline{x} \ne \underline{{x^{\prime}}} }} \frac{{d\left( {f_{e} \left( {\underline {x} } \right) - f_{e} \left( {\underline {y} } \right)} \right)}}{{d\left( {\underline {x} ,\underline {y} } \right)}} = \frac{{\mathop \sum \nolimits_{i} \left| {f_{{e_{i} }} \left( {x_{i} } \right) - f_{{e_{i} }} \left( {y_{i} } \right)} \right|}}{{\mathop \sum \nolimits_{i} \left| {x_{i} - y_{i} } \right|}} = \frac{{\mathop \sum \nolimits_{i} \left( {1 - b_{i} \beta \left( k \right)} \right)\left| {x_{i} - y_{i} } \right|}}{{\mathop \sum \nolimits_{i} \left| {x_{i} - y_{i} } \right|}}$$

  (45)

As a result, the Markovian process provided by Eq. 14 is strictly a distance diminishing process, as defined by Norman [63] in his definition.

Proof of Corollary in lemma 3: The equation that results from Lemma 3 is as follows:

$$d\left({\underline{x}}^{c},{\underline{y}}^{c}\right)= {\sum }_{i}{(1-{b}_{i}\beta (k))}^{c}\times \left|{x}_{i}-{y}_{i}\right|$$

(46)

As ${\text{c}}\to \infty$, the right hand side of the preceding equation tends to zero. As a result, ${\underline{x}}^{c }\to {\underline{y}}^{c}$ as regardless of the initial configurations $\underline{x}$ and $\underline{y}.$