Graph Attention Network-based Block Propagation with Optimal AoB and Reputation in Web 3.0

Web 3.0, also regarded as the semantic web, represents the cutting edge of web development, driven by Artificial Intelligence (AI), Internet of Things, and blockchain technologies. Nowadays, the intertwining of blockchain technologies with Web 3.0 forms a symbiotic relationship, giving rise to blockchain-enabled Web 3.0, which is effective for the transparent, traceable, and decentralized recording of on-chain content. Therefore, research aimed at enhancing blockchain performance in Web 3.0 holds unique significance, serving as a key motivation for our work. Some efforts have been conducted to achieve blockchain-enabled Web 3.0[2, 7, 12]. For example, the authors in[2] introduced a Web 3.0 framework powered by quantum blockchain technologies and discussed potential challenges and applications of integrating quantum blockchain in Web 3.0[2]. The authors in [7] proposed a semantic exchange framework leveraging blockchain technologies and discussed the relationship between semantic communication and blockchain for Web 3.0[7]. The authors in[12] developed an innovative architecture for Web 3.0, integrating AI and blockchain to establish a metaverse-based framework that aims to tackle the existing privacy and security challenges of the current Web 2.0 while improving user experience and data control. Moreover, the authors emphasized the importance of blockchain technologies in the construction of Web 3.0[12]. However, the majority of existing works do not delve into the optimization of blockchain performance as a means to enhance Web 3.0 performance. Therefore, it is important to optimize blockchain performance for Web 3.0, especially optimizing block propagation.

In recent years, GAT as a subtype of graph neural networks, has garnered considerable attention, including molecules, social networks, product recommendations, computer programs, and more[13]. GAT is a class of deep learning models specially designed for processing graph-structured data, known for their exceptional adaptability and efficiency across various applications[14]. For example, the authors in [14] proposed a novel residual edge-graph attention network that combines edge and node information to address combinatorial optimization problems[14]. The authors in[15] employed a foundational three-layer GAT network to capture spatial structural information based on the dynamic aggregation in the road network. Furthermore, when compared with the graph convolutional network, the GAT model significantly enhances the reliability of traffic flow prediction[15]. The authors in[16] introduced a performance prediction model for multipath routing, utilizing the GAT, which models the function and structure of data to predict the network stability and throughput[16]. However, the majority of the existing works do not take into account the utilization of GAT to capture the miner network information, such as the relationship between adjacent miners, thereby optimizing block propagation.

Block propagation optimization has been a prominent research topic for improving blockchain performance. There are numerous researchers investigating this issue, and their efforts can be broadly categorized into three main directions: 1) Optimizing block verification: the authors in[17] proposed a block verification mechanism based on zero-knowledge proof and smart contracts, which improves both the speed of data block verification and privacy protection[17]. The authors in[18] introduced a secure verification scheme for real-time office documents leveraging blockchain technologies, thereby enhancing the efficiency and security in block verification[18]. 2) Optimizing blockchain network topology: the authors in[19] proposed a two-layer blockchain topology to overcome transaction time issues and scalability for market procurement of reactive power[19]. The authors in[20] employed the software-defined networking paradigm to optimize the control of the peer-to-peer network topology in public blockchains, resulting in a significant reduction in node resource consumption[20]. 3) Optimizing the propagation behavior of miners: the authors in[21] introduced a method for preventing block withholding attacks, which utilizes a credit model, a punishment mechanism, and the behavior reward to evaluate the contribution of miners[21]. However, few works have been conducted on combining block propagation efficiency and miner reliability based on the GAT when optimizing block propagation. In Web 3.0, efficient and reliable blockchain technology can significantly promote further development in this ecosystem, giving users a better experience. Therefore, it is still challenging to leverage the GAT to achieve efficient and reliable block propagation optimization.

Motivated by the aforementioned research gaps, we propose a GAT-based reliable block propagation optimization framework for the performance enhancement of blockchain-enabled Web 3.0, in which we apply the AoB to quantify block propagation efficiency and propose a miner reputation mechanism to ensure the reliability of block propagation.

For the block propagation in public blockchains, we introduce the block propagation mechanism which is similar to the traditional gossip protocol[22, 23]. However, for the optimization of specific block propagation trajectories, the introduced block propagation mechanism directs the propagation of a new block to a single miner, departing from the flooding propagation characteristic of the gossip. The detailed process is depicted in Part A of Fig. 1. To prevent redundant transmissions to miners that have already received the new block, the new block is not forwarded directly[22]. Instead, a miner would send an inv message containing the hash value of the new block to its adjacent miner before propagating the block. If the adjacent miner has never received the block, the adjacent miner will issue a getdata message to request the new block [22, 24], and the getdata message is then queued until the block has been received and the difficulty check has been done[22]. To speed up block propagation, the adjacent miner sending the getdata message also sends an inv message to its adjacent miner. Additionally, the block verification process is divided into two phases, i.e., the block difficulty check and transaction validation [22]. Once the block difficulty check is completed, the block is sent out without further transaction validation, aiming to minimize processing time [22].

As shown in Fig. 1, the proposed GAT-based reliable block propagation optimization framework for blockchain-enabled Web 3.0 is structured into three layers, i.e., a physical layer, a data layer, and a network layer. Specifically, the first layer, functioning as the foundation of the proposed framework, is the physical layer, which exists in various applications in Web 3.0, such as decentralized autonomous organizations and decentralized marketplaces for digital products[1]. The second layer is the data layer, where blockchains provide secure and reliable data management for Web 3.0 applications[2]. The final layer is the network layer, which is exacted from the blockchains of the data layer. Especially, the network layer introduces the proposed framework design for achieving block propagation optimization, including four parts. Part A presents a block propagation protocol that shows how a new block propagates between miners in the miner network[22]. Part B introduces the AoB utilized to measure the block propagation efficiency in public blockchains. Part C focuses on the miner reputation mechanism, which utilizes the subjective logic model to calculate miner reputations for ensuring reliable block propagation[10]. Part D shows the GAT, which works directly on graph-structured data and possesses the excellent ability to capture the relationships between miners in the miner network[25]. The proposed framework aims to achieve efficient and reliable block propagation optimization, thereby enhancing the overall performance of Web 3.0 systems.

Since AoI as an effective data-freshness metric, has been widely utilized in remote system status, which is denoted as the elapsed time from the generation of the latest received status update[26]. In this paper, we utilize the AoB to quantify block propagation efficiency motivated by [26, 11], which is denoted as the time elapsed from a getdata message sent by a miner to its successful receipt of the new block, consisting of procedures of getdata message arrival time interval, block waiting time, and block propagation time[11]. We denote $t_{i}$ as the time that miner $i$ sends a getdata message and $t_{i}^{{}^{\prime}}$ as the time that miner $i$ successfully receives the block. Thus, the instantaneous AoB of block propagation between adjacent miners at any time $t$ is expressed as [26]

where $I(t)=\max\{i|t_{i}^{{}^{\prime}}\leq t,i\in\mathcal{M}\}$ represents the index of the miner that recently received the block. Based on (1), the overall AoB over the time range $(0,\delta)$ is given by

and the average AoB over the time range $(0,{\delta})$ is expressed as

Now, we show the characterization of the average AoB of block propagation between adjacent miners in public blockchains. We define the $i$-th time interval, denoted by $X_{i}$, as the time elapsed from miner $(i-1)$ sending the getdata message to miner $i$ sending the getdata message, which is given by

We consider $X_{i}$ as an exponentially distributed random variable[26], i.e., $X_{i}\sim Exp(\mu)$, where $\mathbb{E}[X_{i}]=1/\mu$ and $\mu$ represents the average getdata delivery ratio of miners. Then, we denote the system time of miner $i$ as $D_{i}$, equaling the elapsed time from miner $i$ sending a getdata message at time $t_{i}$ to successfully receive the block at $t_{i}^{{}^{\prime}}$, which is given by

where system time $D_{i}$ consists of block waiting time $W_{i}$ of miner $i$ and block propagation time $P_{i}$ from miner $(i-1)$ to miner $i$. As shown in Fig. 2, $Q_{i}$ signifies the geometric area of a trapezoidal configuration, arising from the difference between two isosceles right-angled triangles. Based on (4) and (5), $Q_{i}$ can be expressed as[26]

Therefore, the AoB of block propagation between miner $(i-1)$ and miner $i$ in public blockchains is given by[26]

Considering the system time $D_{i}$ of miner $i$, we divide the system time $D_{i}$ into block waiting time $W_{i}$ and block propagation time $P_{i}$, which is denoted as

Since $P_{i}$ is independent of time interval $X_{i}$, we can obtain

According to [26], we can express the block waiting time of miner $i$ as $W_{i}=(D_{i-1}-X_{i})^{+}$. Given specific interarrival time $X_{i}=x$, the expected block waiting time is [26]

where $f_{D}(t)$ is the Probability Density Function (PDF) of system time $D$. It is shown that the exponential distribution is a reasonable model for block propagation time[27], i.e., $P_{i}\sim Exp(1/\Gamma_{i})$, where $\Gamma_{i}$ is the block propagation time between miner $(i-1)$ and miner $i$, which is calculated using the Shannon formula as follows [10]:

where $B_{block}$, $b$, $\rho_{s}$, ${d_{i-1,i}}$, $c^{0}$, $\varepsilon$, and $N_{0}$ represent the block size, the channel bandwidth between adjacent miners, the transmit power of miners, the distance between miner $(i-1)$ and miner $i$, the unit channel power gain, the path-loss coefficient, and the noise power density, respectively. For the system time $D_{i}$ of miner $i$, $f_{D_{i}}(t)$ is given by[26]

Furthermore, using iterated expectation and the exponential $(\mu)$ PDF of $X_{i}$[26], $\mathbb{E}[W_{i}X_{i}]$ can be denoted as

Therefore, based on (7), the AoB between miner $(i-1)$ and miner $i$ can be given by

(15) demonstrates that low overall AoB means the comprehensive reduction of the time of block propagation process, including the getdata message arrival time interval, the block waiting time, and the block propagation time, which indicates the enhanced block propagation efficiency. Furthermore, the PDF of a block to be mined can be given by[28]

Based on (16), we can obtain the block fork probability, which is a function of block propagation time and the average getdata delivery ratio of miners, given by [28]

where $\Gamma$ is the overall block propagation time. Thus, combining (17) and (26), it can be inferred that a low overall AoB can reduce the block fork probability when $\mu$ is a fixed value.

The subjective logic model, which utilizes logic and recommended opinions to express subjective beliefs, is a comprehensive model of subjective beliefs evaluation. Moreover, the miner interactions in the blockchain can be considered as a form of group behavior, which can be calculated by direct and indirect opinions[10]. Therefore, for a better fusion of probabilistic information, we utilize the subjective logic model as an effective tool to quantify the reliability of miners in the blockchain network.

Based on the subjective logic model, the subjective logic divides miner reputations into three levels, i.e., trust, distrust, and uncertainty[29, 10]. For the miner $j$, its compositive reputation opinions consist of the indirect reputation opinions of other adjacent miners and the direct reputation opinions of miner $i$ that directly interact with the miner $j$. Especially, the direct reputation opinions of miner $i$ are considered to be local reputation opinions, and the indirect reputation opinions of other adjacent miners are considered to be recommended reputation opinions[29].

In this paper, we focus on achieving reliable block propagation optimization in public blockchains by obtaining the solution path of optimal block propagation $\bm{\pi}=\left(\pi_{1},\ldots,\pi_{m}\right)$, where $\pi_{t}\in\{1,\ldots,m\}\subseteq\mathcal{M}$ and $\pi_{t}\neq\pi_{t^{\prime}},\forall t\neq t^{\prime}$. Note that $\bm{\pi}$ is the miner set of the optimal block propagation trajectory and $\pi_{1}$ represents the miner that obtains the bookkeeping right of this block consensus.

Based on the consensus mechanism of public blockchains (i.e., Proof of Work), we define $m=\lfloor{M\times r^{ratio}}\rfloor$ as the number of miners that participate in block validation, where $\lfloor\cdot\rfloor$ denotes the floor function and $r^{ratio}$ represents the proportion of the overall number of miners. Therefore, the problem of block propagation optimization is to minimize the overall AoB of block propagation with the given reputation constraint, which can be formulated as

where $\sigma\in(0,1)$ is a miner reputation threshold for ensuring block propagation. Considering $\Gamma_{i}<(1/\mu)$[26], it is easy to prove that the AoB $\Delta_{i}$ will increase as block propagation time $\Gamma_{i}$ increases. Moreover, it is obvious that $\Gamma_{i}$ is a monotonic function concerning the distance $d_{i}$ between miner $(i-1)$ and miner $i$ according to (12). Motivated by the above analysis, we propose a GAT-based block propagation optimization model to solve the optimization problem, described in Section V.

In this section, we propose a GAT-based block propagation optimization model for blockchain-enabled Web 3.0, which minimizes the average AoB of block propagation between adjacent miners with the specific reputation constraint, thereby obtaining the optimal block propagation trajectory.

In public blockchains, the miner network can be regarded as a graph network[3]. As a powerful tool that is specifically designed for processing graph-structured data[25], we apply the GAT to find the optimal block propagation trajectory, where GAT utilizes attention mechanisms to capture the relationships between miners, enhancing the extraction of relational representations among each miner. As shown in Fig. 2, the GAT architecture of the proposed framework consists of encoder and decoder components. More precisely, the encoder is used to extract the structural characteristics of the miner network, and the decoder systematically generates the miner sequence of the optimal block propagation trajectory.

For the problem of block propagation optimization, $G$ as an input instance can be considered as a fully connected graph $G=(\mathcal{M},\mathcal{E})$, representing the public blockchain network composed of miners, where $\mathcal{E}$ is the edge set of the miner network. Considering that miner $j$ is an adjacent miner of miner $i$, we define $\bm{f}_{i}$ and $\bm{f}_{j}$ as the location features of miner $i$ and miner $j$, respectively. Specifically, the location feature $\bm{f}_{i}$ of miner $i$ is a $d_{f}$-dimensional feature, where $d_{f}=2$ is composed of horizontal and vertical coordinates of miner $i$ in the input instance $G$. Thus, the distance between adjacent miners can be calculated by[14]

where $\left||\cdot|\right|_{2}$ denotes the $L2$ norm to calculate the $2D$ Euclidean distance. For network parameters $\bm{\theta}$ and the input instance $G$, the corresponding solution probability $p_{\bm{\theta}}\left(\bm{\pi}|G\right)$ is given by

Therefore, the overall route length of block propagation is given by[25]

The function $L(\bm{\pi}|G)$ will be utilized to train the network in Section VI. Next, we introduce the encoder-decoder architecture constructed for the GAT-based block propagation optimization model.

Unlike the typical transformer architecture[31], we embed miner features into the context of the graph in the encoder[25]. The miner features are fed into a $d_{h}$- dimensional hidden layer, where $d_{h}=128$. In the hidden layer, performing a learnable linear projection using parameters $\bm{W}^{h}$ and $\bm{b}^{h}$, the initial $d_{h}$-dimensional miner embeddings $\bm{h}_{i}^{(0)}$ is calculated by[25]

where $\bm{W}^{h}\in\mathbb{R}^{d_{h}\times d_{x}}$ and $\bm{b}^{h}\in\mathbb{R}^{d_{h}}$. After obtaining the embedding $\bm{h}_{i}^{(0)}$, it is sent to the graph attention layer and updated with $N$ GAT layers. We denote $\bm{h}_{i}^{(r)}$ as the miner embeddings calculated by GAT layer $r\in\{1,\ldots,N\}$. Specifically, $\bar{\bm{{h}}}^{(graph)}$ is denoted as the graph embedding, which is the aggregated embedding of the input graph as the average of final miner embeddings, given by[25]

Finally, the encoder outputs the final miner embeddings $\bm{h}_{i}^{(N)}$, as well as the graph embedding $\bar{\bm{h}}^{(graph)}$ to the decoder.

Then, we introduce the major architecture of the proposed graph attention layer in detail.

The decoding step for block propagation is sequential. For each decoder step $t\in\{1,\ldots,m\}$, the current miner that would propagate the block selects its adjacent miner to append to the end of the sequential solution. We construct a special context vector $\bm{h}^{(d)}$ to represent the decoding context[25], which is composed of the outputs of the encoder and decoder up to time $t$. According to [31], $\bm{h}^{(d)}$ is given by

Here, $[\cdot,\cdot,\cdot,]$ is the horizontal concatenation operator. For each time $t$, we utilize the graph embedding $\bar{\bm{h}}^{(graph)}$ as a part of the context vector $\bm{h}^{(d)}$, for the reason that the graph embedding is designed for taking the complete graph structure into account. For $t=1$, we introduce learnable $d_{h}$-dimensional parameters $\bm{v}^{1}$ and $\bm{v}^{2}$ as input placeholders. For $t=2$, the placeholder $\bm{v}_{1}$ would be replaced by the embedding $\bm{h}_{\pi_{1}}^{(N)}$. For $t>2$, the placeholder $\bm{v}_{2}$ and the embedding $\bm{h}_{\pi_{1}}^{(N)}$ would be replaced by the embeddings $\bm{h}_{\pi_{t-1}}^{(N)}$ and $\bm{h}_{\pi_{t-2}}^{(N)}$ of miners $\pi_{t-1}$ and $\pi_{t-2}$, to achieve the local optimum of AoB, and eventually achieve the optimization objective of minimizing overall AoB. Note that the result vector $\bm{h}^{(d)}$ is $(3\cdot d_{h})$-dimension to align with the miner embeddings $\bm{h}_{\pi_{t}}^{(N)}$.

In the decoder, we compute compatibility using an MHA layer similar to the encoder. Firstly, the aggregated query $\bm{q}^{(d)}$ is calculated by the context embedding $\bm{h}^{(d)}$ while the keys $\bm{k}_{i}$ and the values $\bm{v}_{i}$ are calculated by the miner embeddings $\bm{h}_{i}^{(N)}$[25]:

Based on (44), we can use the single query $\bm{q}^{(d)}$ to compute its compatibility with all miners. If miners have already received the block at time $t$, we will set the compatibility to $-\infty$, masking miners that have already received blocks to avoid selecting them repeatedly. Therefore, the $u_{j}^{(d)}$ is given by[25]