Wireless May Benefit Blockchain ^†

Abstract

Wireless technologies now take every part of one’s everyday life. As such, it will be no longer a surprise if a blockchain system is composed of wirelessly connected nodes. However, wireless communication is known for its inherent unreliability caused by noise, interference, limited bandwidth, etc. Motivated by this fundamental problem, this paper investigates the impact of wireless communications on the performance of three representative consensus mechanisms, viz., proof of work (PoW), proof of stake (PoS), and proof of coverage (PoC). It features a comprehensive analytical framework that mathematically derives metrics quantifying the scalability and the level of decentralization of the three consensus mechanisms, constituting a key contribution of this work. The paper then proceeds to present extensive simulation results as a means to confirm the underpinning theoretical findings. Overall, we emphasize that the framework’s holisticity will allow it to be applied to diverse consensus mechanisms.

1. Introduction

1.1. Motivation

In the dynamic landscape of modern connectivity, the proliferation of wireless technologies has become nothing short of permeating every facet of our daily lives and redefining the way we connect, communicate, and consume information. In fact, over 55% of website traffic comes from mobile devices, and 92.3% of internet users access the internet use a mobile phone [1]. In fact, a wide variety of wireless technologies immerse our daily lives, including Wi-Fi [2], cellular technologies such as fifth-generation (5G) cellular [3] and Long-Term Evolution (LTE) [4], to vehicle-to-everything (V2X) communications [5] and even wearable technologies [6,7]. This makes a strong case where increasingly more blockchains will be established on wirelessly connected nodes.

In fact, the integration of blockchain with wireless connectivity enables transformative applications across various domains, streamlining processes and enhancing operational efficiency. In connected vehicles and intelligent transportation systems (ITSs), blockchain facilitates secure and real-time data sharing among vehicles, enabling functionalities for improved road safety [8]. Decentralized energy grids benefit from blockchain-enabled peer-to-peer (P2P) energy trading, where distributed smart meters wirelessly communicate to ensure transparent and efficient transactions [9]. Similarly, smart cities leverage blockchain to manage public resources [10]. Moreover, in the realm of unmanned aerial systems (UASs), blockchain enhances airspace management for drones, addressing the complexities of high mobility and dynamic wireless conditions [11]. Finally, in healthcare applications, blockchain enables the secure and seamless sharing of sensitive patient data from wearable devices, ensuring integrity and privacy even under varying network conditions [12]. These use cases illustrate the profound versatility of blockchain as a foundational technology that, when paired with wireless connectivity, optimizes diverse systems by ensuring data security, transparency, and decentralized coordination.

Nevertheless, one should take notice of a key challenge here: the reliability of wireless communications is generally lower in comparison to wired communications (e.g., ethernet), attributed to various random factors such as noise, interference, and limited bandwidth [13]. One should also note that this lower reliability affects the performance of consensus in blockchain [14,15].

All open public blockchains are based on the idea that they should be able to reach a consensus across a distributed network, even when there are conflicts, without putting control in one place [16]. Nonetheless, the additional dynamicity that is brought by wireless networking has not been thoroughly discussed in the literature as of yet. Therefore, it will be a valuable academic attempt to formally analyze the performance of a blockchain system established on a wireless network.

1.2. State of the Art

1.2.1. Consensus Mechanisms

PoW [17] and PoS [18] remain the two most common mechanisms, especially in the cryptocurrencies’ context [19]. PoW is a form of cryptographic proof in which one party (the prover) proves to others (the verifiers) that a certain amount of a specific computational effort has been expended [20]. The main purpose of PoW is to deter the manipulation of data by establishing large energy and hardware-control requirements to be able to do so, which inevitably leaves it to be criticized by environmentalists for their energy consumption.

Meanwhile, in an effort to avoid the high computational cost that PoW causes, PoS is a type of consensus mechanism for blockchains that are designed to elect validators in proportion to their quantity of holdings in the associated cryptocurrency.

1.2.2. Blockchain Trilemma

Moreover, there a variety of techniques that have been proposed as an effort to improve the performance of blockchains, namely, parallel structure [21], off-chain [22], reinforcement learning [23], and a combination of PoW and PoS [24]. Nonetheless, the blockchain trilemma sets a limit on this desire: no blockchain can achieve improvements on all three fronts of scalability, security, and decentralization at once. In fact, despite the large research and experimental effort, all known approaches turn out to leave tradeoffs [21].

To wit, the tradeoff stands between the security/decentralization and scalability of a blockchain network; as the number of nodes increases, security/decentralization is improved while scalability is lowered. Interestingly, the lower scalability can degrade the energy efficiency, which may be a significant issue in a blockchain network formed among wirelessly connected nodes [25].

This paper will particularly focus on the quantitative analysis of the impacts of wireless connections on scalability and decentralization, two pillars of the blockchain trilemma.

1.2.3. Blockchain over Wirelessly Connected Nodes

This makes a compelling case where we urgently need a comprehensive framework evaluating the performance of blockchains with wirelessly connected nodes with respect to the blockchain trilemma. Many of the current consensus algorithms already consider the possibilities that nodes leave, and new nodes join. However, we claim that more is needed, as due to the uncertainty in wireless connections, a blockchain system established on a wireless network will draw a completely different environment than a one with ethernet-connected nodes [26].

To elaborate, as a solution to practical limitations imposed by high node density and fluctuating wireless connectivity, a 6 G-empowered wireless network architecture was proposed [26], which in fact led to a significant improvement in several metrics, i.e., scalability, packet loss ratio, and latency. Moreover, there is a notion that signal ambiguity can also be a possible cause of a Byzantine fault. A recent proposal [27] mitigates the signal ambiguity by adopting the Second Order Blind Identification (SOBI) algorithm.

Moreover, diverse avenues of recent research were sparked, exploring topics such as blockchain performance in the context of IoT [28], communication resource consumption [29], efficient consensus mechanisms among wireless nodes [30], connectivity-adaptive contract mechanisms [31], and Byzantine fault-tolerant (BFT) consensus among wireless nodes [32]. Specifically, the existing analytical framework (e.g., universal scalability law [33]) overlooked the impacts of wireless connectivity, which may cause serious imprecision these days with such a high proportion of wireless technologies in any given network.

1.3. Contributions

As has been articulated in Section 1.2, most of the previous work in the literature on blockchain and distributed systems presents experimental results, which can only be applied to certain practical scenarios with a set of specific parameter settings. Moreover, no prior work directly shed light on the fundamental impacts of uncertainty induced by intermittent connections, which becomes the central challenge in a blockchain network formed among wireless nodes. Therefore, this paper attempts to provide a more general framework that can be applied to any other blockchain for the measurement of their consensus performance.

Addressing the limitations mentioned above in the existing literature, this paper is dedicated to answering the following research question: How will wireless connectivity affect the performance of blockchain consensus? It aims at accurately evaluating the performance of consensus mechanisms when the nodes are wirelessly connected. We also claim that this work is an extension of the author’s previous work [34]. The specific technical contributions are summarized as follows:

• It provides an analytical framework for calculation of scalability and decentralization of a blockchain consensus process.
• It quantifies the impacts of wireless connections on the scalability and decentralization by adopting a probabilistic analysis.
• It presents a comparative study among PoW, PoS, vs. PoC with respect to scalability and decentralization.

2. System Model

The key focus of this paper is the nodes being wirelessly connected. As such, the system model is supposed in such a way that the impacts of wireless connection on the blockchain performance can be clearly characterized.

2.1. Spatial Distribution of Wireless Nodes

A two-dimensional space ${\mathbb{R}}^2$ is defined as a 1 km by 1 km square, as illustrated in Figure 1. The nodes are assumed to be located at fixed positions. We model spatial distribution of the stationary nodes as a Poisson point process (PPP) in ${\mathbb{R}}^2$ with density $\lambda$. Figure 1 visualizes an example distribution of nodes with $\lambda =400$. Notice that the $\lambda$ is the parameter that will be varied according to the type of consensus mechanism. For instance, PoS has a smaller $\lambda$ compared to PoW since its consensus is operated only among the validators who are with more than the required amount of stake.

As shall be elaborated in Section 2.3, node failures occur in a “clustered” manner; the magenta-colored boxes indicate the areas where the failures transpire. It is assumed that the node failures take place within the clusters only; the example presented in Figure 1 assumes the probability of each node failure to be ${\mathsf{p}}_{\mathrm{fail}}=0.5$. We will vary this ${\mathsf{p}}_{\mathrm{fail}}$ as a method to evaluate the performance of the consensus mechanisms, viz., PoW, PoS, and PoC, which will be presented in Section 5.

The selection of PPP is justified as follows. Every realization of a finite PPP is a binomial point process (BPP) with the number of realized points [35]. PPP is used to model or abstract a network composed of a possibly infinite number of nodes randomly and independently coexisting in a finite or infinite service area (e.g., large-scale wireless networks such as cellular networks [36]). In contrast, BPP is a more adequate model when the total number of nodes is known, and the service area is finite. Therefore, our choice of PPP is in our pursuit of the generalization of the model. As such, owing to its general applicability in scenarios where nodes appear independently and are randomly distributed over a given area, this paper’s selection of PPP aligns well with the characteristics of decentralized blockchain networks, where nodes are typically deployed without a specific pattern or clustering.

2.2. Communication Among Wireless Nodes

The nodes are connected to each other via wireless communications. Each node holds 100 m of communication range. This paper assumes a general ad hoc network [37] in which no central mediator exists, but instead, the nodes are connected to their own neighbors, e.g., device-to-device communications [38], C-V2X mode 4 [39], etc. The rationale is that a centralized network (e.g., cellular, Wi-Fi, etc.) will likely employ a permissioned blockchain (or a private blockchain), whose performance is not exactly what we are interested in measuring. Importantly, the aforementioned assumptions are not to represent a certain setup or an arbitrary value. Instead, the objective is to establish a generalized analytical framework in which such parameters are treated as variables. This enhances the robustness of the proposed framework by explicitly demonstrating how variations in these parameters influence the consensus performance of a blockchain network formed among wireless nodes.

As a means to accomplish the finality of data among nodes, distributed systems often adopt a gossip protocol for propagation of information [40]. In a network of size N, these protocols consist of N local, pairwise, and periodical gossiping operations between neighboring nodes.

Figure 1 shows an example of the gossip protocol operated on a wireless ad hoc network. The green dot in Figure 1a indicates the node commencing the flooding, while the black dots are all the other nodes that will potentially receive the block that will be propagated. As shown in Figure 1b, each node propagates the block to all the neighboring nodes within its communication range. Each connection is supposed to cause a Byzantine fault at the rate of ${\mathsf{p}}_{\mathrm{fail}}$, which is shown as a red line. Such a failed connection yields a failed reception at the receiver node, which is drawn as a red dot.

2.3. Byzantine Faults and Wireless Connection

Byzantine faults in safety-critical systems are real and occur with failure rates far more frequently than ${10}^{-9}$ faults per operational hour [41]. A typical example of a Byzantine fault is a digital signal that is stuck at “1/2”, e.g., a voltage that is anywhere between the voltages for a valid logical “0” and a valid logical “1”. Such Byzantine faults usually propagate through traditional fault containment zones, thereby invalidating system architectural assumptions.

The problem is that the wireless connection can increase the odds of Byzantine faults. Specifically, we identify “clustered occurrence” of faults attributed to interference among each other [35] as the key characteristic that the wireless connection adds to Byzantine faults, which yields that the nodes located in certain areas face a higher probability of disconnection. We attribute such clustered disconnections to interference among nodes. In fact, an ad hoc wireless network is more susceptible to interference [42] due to its decentralized network structure where no central mediator exists to coordinate scheduling among nodes.

3. Consensus Among Wireless Nodes

Taking over from Section 2.3, we delve into the impacts of wireless connections on Byzantine faults. As the first step, we define the consensus mechanisms that we analyze in this paper, i.e., PoW, PoS, and PoC, and identify their key characteristics that may be affected by the wireless connections.

Moreover, it is noteworthy that we pay attention to PoC and compare it to PoW and PoS, the two most predominating consensus mechanisms. The rationale is that PoC adopts a significantly different principle from that of PoW and PoS, which thus distinguishes it from the vast majority of recent consensus mechanisms built on PoW and PoS.

3.1. PoW

It is widely acknowledged that if two computers can agree upon just a single block (i.e., the head block) in a blockchain, one can then iterate through all of the blocks in the blockchain and verify that the rest of it is identical. Therefore, forming consensus is just a matter of agreeing upon a single block across “honest” nodes (whose acronym is “Byzantine” nodes [43]) and replicating across all of the honest nodes. Will this be that simple if the nodes are wirelessly connected?

Let us take a public blockchain like Bitcoin as an example and identify several key characteristics on which wireless connections may affect. Bitcoin allows its nodes to join or leave the network at any time. In other words, nodes should be able to join or leave our system at any time while (i) maintaining consensus; (ii) causing no deadlock in the system; and (iii) being network partition tolerant. This differentiates PoW from traditional consensus algorithms in the sense that it has to tolerate network partitions: while traditional consensus algorithms (e.g., Paxos [44], Raft [45] and practical Byzantine fault tolerance (PBFT) [43]) terminate when a consensus has been reached, PoW keeps iterating continuously, leaving a possibility that it may change at any time what the agreed-upon blockchain is.

The problem here is that if several nodes leave the network, the consensus may never be achieved due to the inability to have a sufficient number of votes to complete the algorithm (i.e., $n>3f+1$ for Byzantine fault tolerance (BFT)). Even worse, if several nodes join the network (and if they are Byzantine nodes), then they may be able to manipulate the vote or stop the network from achieving consensus. This is where things get more complicated with the nodes being wirelessly connected; in addition, to such Byzantine nodes, even honest nodes may not be able to correctly participate in a consensus when their connections are intermittent due to uncertainties in wireless communications.

Since the Bitcoin PoW adopts a gossip protocol [40], there is a chance that two different nodes have heard about two different blocks that point to the same previous block, which forms a “fork” in the blockchain. The way that many PoW protocols resolve such a fork is to add a new block to the longest branch. Even if there is more than one branch with the same length, a new block can choose one randomly, which will immediately form only one longest branch and will grow thereafter via the gossip protocol.

The last characteristic that wireless connections may affect is the possibility of “livelock”. Suppose two different forks grow separately in a blockchain, which currently shows no sign of going back to a single coherent consensus. The fundamental problem here is that we can add blocks to a chain much faster than we can learn about other nodes adding blocks to the chain. Individual computers are much faster compared to the communication time of our network between the computers, which leaves the only feasible solution to be slowing the system down. Therefore, PoW is designed such that every time a block is added, each node sleeps for a random period of time (also known as a “timeout”), only after which the block can be added.

3.2. PoS

Let us shed light on the key characteristics of PoS on which wireless connections may have impacts. Here, we take the Ethereum 2.0 [18] as an example.

Selectivity in consensus participation is the first characteristic that is affected by wireless communications. The reason is that such a smaller number of consensus participants can make the blockchain more vulnerable to faults attributed to uncertainties of wireless connections in addition to already existing Byzantine faults. (This usually is not the case in a PoW-based blockchain.) Let us continue to characterize the Ethereum 2.0’s PoS algorithm in further detail as an effort to investigate the applicability of wireless connections.

In Ethereum’s PoS system, validators commit their capital in the form of Ethereum token (ETH) to a smart contract on the network. These validators are then tasked with validating new blocks and, on occasion, creating and disseminating new blocks themselves [18]. To participate as a validator, an individual must deposit 32 ETHs into the contract and operate three distinct software components, viz., an execution client, a consensus client, and a validator client [46]. Every slot randomly selects one validator as the block proposer, responsible for creating a new block and transmitting it to other nodes in the network. Concurrently, in each slot, a committee of validators is randomly chosen, and their votes are instrumental in determining the validity of the proposed block. The partitioning of the validator set into committees is crucial for effectively managing network load.

Second, due to this unique structure, PoS has a higher level of reliance on the network layer. As a means to elaborate, let us take an in-depth look at the procedure of executing a transaction in Ethereum PoS:

• A user initiates a transaction by generating and signing it using their private key. The user specifies the gas amount they are willing to pay as a tip to incentivize a validator for including the transaction in a block.
• The transaction is then submitted to an Ethereum execution client, where its validity is verified. This verification includes checking if the sender has sufficient ETH for the transaction and if it has been correctly signed with the corresponding key.
• Upon confirming the transaction’s validity, the execution client adds it to its local mempool (a list of pending transactions) and broadcasts it to other nodes through the execution layer gossip network. Other nodes, upon receiving the transaction, also add it to their mempool.
• A node is randomly selected as the block proposer for the current slot via the pseudo-random RANDAO process. This proposer is responsible for constructing and broadcasting the next block to be appended to the chain, along with updating the global state.
• Other nodes receive the new beacon block through the consensus layer gossip network. They forward it to their execution client, where transactions are re-executed locally to verify the proposed state change’s validity. The validator client then attests to the block’s validity, confirming that it logically follows the chain with the highest weight of attestations as defined in the fork choice rules. The block is added to the local database in each attesting node.
• The transaction achieves a state of being “finalized” when it becomes part of a chain with a supermajority link between two checkpoints, indicating agreement from 66% of the total staked ETH on the network regarding two specific checkpoints.

As shown from the above protocol, Ethereum’s PoS uses two P2P networks: one for transaction communication among execution clients and another for block information exchange among consensus clients [46]. Transactions are transmitted through the execution layer’s P2P network via encrypted communication between authenticated peers. When a validator proposes a block, the transactions are sent to consensus clients, encapsulated into beacon blocks, and disseminated across the P2P network. Such a high reliance on networking may lead to performance degradation when the quality of communications fluctuates to a lower level.

Lastly, wireless communication may also affect the level of “honesty”. There is a protocol that governs how honest validators are selected to propose or validate blocks, process transactions, and vote for their view of the head of the chain [47]. In scenarios where several blocks occupy a similar position near the chain’s head, a fork-choice mechanism is employed to pick blocks constituting the “heaviest” chain, determined by the count of validators endorsing the blocks, weighted in accordance with their staked ether balance. Elevated uncertainty in wireless communication can lead to inaccuracies in this honesty-assessing protocol, subsequently impacting the overall Byzantine fault tolerance of the network.

3.3. PoC

The consensus mechanism employed by Helium, known as PoC [48], is a focal point of analysis in this paper. It stands out significantly from the dominant mechanisms, namely PoW and PoS. As elaborated in Section 5, PoC is found to amalgamate the benefits of PoW, such as enhanced decentralization due to fluctuations in wireless connectivity, and PoS, offering greater scalability as only specific nodes partake in the consensus process.

Within the PoC framework, miners establish physical hotspots, typically small computers with internet connectivity and antennas for wireless communication with nearby hotspots. These hotspots receive wireless data. During each “challenge”, 10 witnesses are randomly selected from all the hotspots within the beacon’s transmission range. Notably, this random selection of witnesses can change with every challenge, occurring at 30 min intervals; shorter intervals enhance PoC’s decentralization [49], aiming for a level akin to PoW.

The fundamental principle of PoC is to validate that a given hotspot is genuinely operational and serving a real area. The term “consensus” in this context pertains to whether other hotspots can substantiate that the challenged hotspot is genuinely providing service. This verified decision by witnesses constitutes a single block in the Helium network.

This paper focuses on quantifying the impact of uncertainties arising from wireless communications on the performance of PoC consensus. For example, PoC miners facing significant uncertainties may fail to provide accurate input to the consensus, thereby compromising the blockchain’s scalability. Conversely, highly variable communication channels are likely to result in the replacement of current miners with others, effectively increasing decentralization [50].

4. Analysis on Consensus Performance

This section presents the theoretical analysis framework that this paper proposes. The framework features two metrics measuring the performance of a consensus mechanism: normalized number of transactions and Gini coefficient [51] for scalability and decentralization, respectively. Refer to

Table 1. Table of acronyms and mathematical symbols.

Acronym/Symbol	Description
$\mathsf{G}$	Gini coefficient as the decentralization measure
${\mathsf{p}}_{\mathrm{fail}}$	Probability of failure at each node
PoC	Proof of coverage
PoS	Proof of stake
PoW	Proof of work
$\mathsf{R}$	Scalability
${\mathsf{r}}_{\mathrm{cls}}$	Rate of clustered area
${\mathsf{r}}_{\mathrm{v}}$	Rate of validators
$\mathsf{S}$	Universal scalability [52,53]
${\mathbb{S}}_{\left(·\right)}$	Set of nodes belonging to consensus type $\left(·\right)$
${\mathsf{T}}_c$	Length of time taken to complete a consensus
USL	Universal scalability law [52,53]

for acronms and mathematical symbols that will be used throughout the remainder of this paper.

4.1. Scalability

The literature of distributed systems often relies on the universal scalability law [52,53] as the key scalability measure, which is given by

$$\begin{array}{c}\hfill {\displaystyle \mathsf{S}=\frac{N}{1+\alpha \left(N-1\right)+\beta N\left(N-1\right)}}\end{array}$$

(1)

where N denotes the number of nodes in the network; $\alpha$ gives the level of contention; and $\beta$ indicates the delay for achieving coherency across the network. Nonetheless, we acknowledged that this quantity is usually obtained empirically, which makes it almost impossible to analytically quantify $\alpha$ and $\beta$. For this reason, we decided to move on to find another metric that can accommodate both distributed systems’ and wireless communications’ characteristics.

It was a challenge to find previous work exactly quantifying $\left(\alpha ,\beta \right)$ of $\mathsf{S}$ for blockchain. However, we found an extensive empirical analysis of the parameters for Zookeeper [33], one of the popular distributed systems [54]. In fact, Zookeeper has appeared on many of the blockchain systems [55,56]. This makes a case that we adopt the result of $\mathsf{S}$ found for the Zookeeper as a scalability benchmark for this paper’s result, which shall be presented in Section 5.2.1.

Motivated from the challenge, we define a generalized metric for the scalability in this paper, which is given by

$$\begin{array}{cc}\mathsf{r}\hfill & =\frac{\#\mathrm{transactions}}{\#\mathrm{seconds}}\hfill \\ \hfill & ={\displaystyle \frac{{\mathsf{n}}_{\mathrm{tx}}}{{\mathsf{T}}_c}}\hfill \end{array}$$

(2)

where ${\mathsf{T}}_c$ denotes the length of time that has been taken to complete a single consensus, which in turn can be written as

$$\begin{array}{cc}\hfill {\mathsf{T}}_c& {\displaystyle =\sum _{x=1}^X{1}_{x=X}\mathsf{T}\left(x\right)}\hfill \\ \hfill & {\displaystyle =\sum _{x=1}^X\left(1_{x=X}\sum _{i,j\in \mathbb{S}\left(x\right)}{\mathsf{T}}_{ij}\right)}\hfill \\ \hfill & {\displaystyle =\sum _{x=1}^X\left(1_{x=X}{\mathsf{n}}_{\mathbb{N}\left[\mathbb{S}\left(x\right)\right]}{\mathsf{T}}_{\mathrm{to}}\right)}\hfill \end{array}$$

(3)

Here are details on the parameters of Equation (3): x denotes each consensus attempt (which is defined as an entire round of having collected verifications from all the participating nodes $\in \mathbb{S}\left(x\right)$); $\mathbb{S}\left(x\right)$ denotes the set of nodes participating in the current consensus attempt x; $X\sim \mathrm{geo}\left(p\right)$ is the number of consensus attempts before the first successful consensus, which is modeled as a geometric random variable; $1_{\left(·\right)}$ is an indicator function that gives a 1 when $\left(·\right)$ is true, or a 0 when false; ${\mathsf{n}}_k$ gives the number of communications among k nodes participating in the current consensus attempt; and ${\mathsf{T}}_{\mathrm{to}}$ gives a timeout (in the unit of seconds) for which every node must wait as a means to avoid a network partition among different nodes, which is a uniform random variable between $\left[0,{\overline{\mathsf{T}}}_{\mathrm{to}}\right]$ where ${\overline{\mathsf{T}}}_{\mathrm{to}}$ is the maximum value for ${\mathsf{T}}_{\mathrm{to}}$.

We elaborate on ${\mathsf{T}}_{ij}$, which is defined as the delay equals the sum of the times spent at every node participating in a single networking [57]. The delay can be broken down as follows at a single node:

$$\begin{array}{cc}\hfill {\mathsf{T}}_{ij}& ={\mathsf{T}}_{\mathrm{que}}+{\mathsf{T}}_{\mathrm{prop}}+{\mathsf{T}}_{\mathrm{to}}\hfill \\ \hfill & \approx {\mathsf{T}}_{\mathrm{to}}\hfill \end{array}$$

(4)

where the subscripts “que”, “prop”, and “to” indicate delays due to queueing, propagation, and timeout, respectively. It is noteworthy from the reference that we consider no queueing delay as it is less significant than the network delay [57]. We also approximate that ${\mathsf{T}}_{\mathrm{prop}}\approx 0$ as even wireless signals propagate at the speed of light. Nonetheless, as has been discussed in Section 3.1, ${\mathsf{T}}_{\mathrm{to}}$ is too significant to ignore. This justifies the approximation ${\mathsf{T}}_{ij}\approx {\mathsf{T}}_{\mathrm{to}}$.

In Equation (3), it is also significant to notice that ${\mathsf{n}}_{\mathbb{N}\left[\mathbb{S}\left(x\right)\right]}$ heavily depends on the type of consensus, i.e., PoW, PoS, and PoC, which is defined as the number of propagations within a consensus attempt via the gossip protocol:

$$\begin{array}{c}\hfill {\mathsf{n}}_{\mathbb{N}\left[\mathbb{S}\left(x\right)\right]}=n{|}_{\theta \left(n\right)\to 0}\end{array}$$

(5)

where n denotes the number of nodes in the blockchain network and $\theta \left(n\right)$ gives the rate of nodes who have “not yet received” a block, which is formulated as [40]

$$\begin{array}{cc}\hfill \theta \left(n\right)& =\mathbb{E}\left[s_{t+1}\left(n\right)\right]\hfill \\ \hfill & =s_t{\left(1-{\displaystyle \frac{1}{n}}\right)}^{n\left(1-s_t\right)}\hfill \end{array}$$

(6)

where t is each time slot where a single-hop propagation takes place. To further clarify, let us consider the variables $s_t$ and $r_t$, representing the proportions of nodes that have “not yet received” and “already received” the block in the tth time slot. It is evident that $s_t=1-r_t$. In the initial time slot, denoted as $s_0$, we assume the master peer has a block to propagate, leading to $r_0=1/n$ and $s_0=1-1/n$, indicating that only one node has received the update. Now, assuming the randomly selected node is chosen independently of other nodes and past decisions, we express Equation (6) as the expectation of $s_t+1$ in terms of $s_t$.

In Equation (5), it is noteworthy that $\mathbb{N}\left[{\mathbb{S}}_{\mathrm{pos}}\right]<<\mathbb{N}\left[{\mathbb{S}}_{\mathrm{pow}}\right]$ as the validators in PoS are a subset of nodes. We also emphsize that $\mathbb{N}\left[{\mathbb{S}}_{\mathrm{poc}}\right]>>\mathbb{N}\left[{\mathbb{S}}_{\mathrm{pow}}\right]$ is also true; however, the identities of witnesses $\in {\mathbb{S}}_{\mathrm{poc}}$ in a PoC network are subject to change. Although it is not captured in the scalability, this rotation in the witnesses’ identities will affect the decentralization, which will be detailed in the next subsection.

Now, let us quantify p for the geometric random variable X in Equation (3), which is defined as the probability of a successful consensus among $\mathbb{N}\left[\mathbb{S}\left(x\right)\right]$ nodes. We highlight that the probability p is a function of $\mathbb{N}\left[\mathbb{S}\left(x\right)\right]$ in every round of consensus attempt, which can be formally written as

$$\begin{array}{cc}\hfill p\left(x\right)& =\mathbb{P}\left[\mathbb{N}\left[\mathbb{S}\left(x\right)\right]>3f+1\right]\hfill \\ \hfill & =1-F_{\mathbb{S}}\left(3f+1\right)\hfill \\ \hfill & =1-\exp \left(-{\lambda }_{\mathbb{S}}\right)\sum _{j=0}^{3f+1}{\displaystyle \frac{{\lambda }_{\mathbb{S}}^j}{j!}}\hfill \end{array}$$

(7)

where f denotes the number of Byzantine nodes, which is designed to fluctuate due to wireless connections. It has been proved in the literature [58] that both $\mathbb{N}\left[\mathbb{S}\left(x\right)\right]$ and f followed Poisson distributions with their own respective densities, which we denote by ${\lambda }_{\mathbb{S}}$ and ${\lambda }_f$, respectively.

Here, we propose to model the impacts of wireless channel fluctuation on f as an increase in ${\lambda }_f$. This is a plausible assumption in the sense that the number of Byzantine nodes ${\lambda }_f$ should increase as more nodes experience communication errors due to the imperfect wireless connections.

Also, notice in Equation (7) that $F_{\mathbb{S}}$ denotes the cumulative distribution function (CDF) of random variable $\mathbb{N}\left[\mathbb{S}\left(x\right)\right]$, which can be formally written as $F_{\mathbb{S}}\left(s\right)=\mathbb{P}\left[\mathbb{S}\le s\right]$.

4.2. Decentralization

Now, we proceed to formulating the decentralization part of our proposed analysis framework. We propose to adopt the Gini coefficient (denoted by $\mathsf{G}$) to represent the level of decentralization. It is noteworthy that we modify $\mathsf{G}$ in such a way that the “inequality” indicates the level of participations being concentrated to a fewer number of nodes, which can also be understood as “centralized”.

We justify the selection of the Gini coefficient as the metric measuring the decentralization level as follows. The Gini coefficient is a widely recognized and intuitive metric for measuring inequality and has been effectively adapted to assess the degree of decentralization in blockchain networks [59]. Its primary strength lies in its ability to represent the distribution of power or resources (e.g., hash rate, stake ownership) across participating nodes, providing a single scalar value that is easy to interpret. There are other metrics that could be considered for the measurement of decentralization. Theil’s index is one of them, which is derived from information theory and usually requires logarithmic computations [60], whereas, the Gini coefficient directly captures the relative dispersion within the network; this simplicity makes it particularly well suited for comparative studies and stakeholder communication. Furthermore, the Gini coefficient is scale-invariant and robust to varying network sizes, ensuring that decentralization assessments remain meaningful even as the network grows [61]. As such, the Gini coefficient was chosen for its broad adoption in prior blockchain research and its ability to succinctly encapsulate inequality, which is central to understanding and comparing the decentralization of blockchain systems.

Let us consider an example scenario. Suppose a blockchain that has accomplished 100 consensuses. We divide them up among 10 nodes, giving the first 1, the second 3, etc., so that the kth person is assigned to the distribution function $u\left(k\right)=2k-1$. Note that as such, in this distribution we are ranking the nodes in ascending order according to the number of transactions whose consensus they have taken part in. Also, suppose that there can be inequality: one node has partaken in only a single consensus, whereas another node has in 19 consensuses.

Given this distribution function, we can define the associated Lorenz curve, $L\left(x\right)$, as the graph of the cumulative proportion function. In other words, $L\left(x\right)$ is the proportion of consensuses participated by the poorest $100x\%$ of the population. In our example, $L\left(0.1\right)=1/100$, $L\left(0.2\right)=4/100$, and in general, $L\left(x\right)=x^2\forall x=0,0.1,0.2,\cdots ,1$. So, for example, $L\left(0.5\right)=0.25$ indicates that the bottom 50% of the population has participated in 25% of all the consensuses. The top 10% of the population has taken part in $1-L\left(0.9\right)=19/100$ or $19\%$ of all the consensuses.

Building on this understanding, one can find a more formal way to write this decentralization analysis. We define the Gini coefficient as half of the relative mean absolute difference, which is equivalent to the definition based on the Lorenz curve [62]. The mean absolute difference is the average absolute difference of all the pairs of items of the population, and the relative mean absolute difference is the mean absolute difference divided by the average, $\overline{x}$, to normalize for scale. If $x_i$ is the number of node i’s participations in consensus, and there are n nodes, then $\mathsf{G}$ is given by

$$\begin{array}{cc}\hfill \mathsf{G}& ={\displaystyle \frac{{\displaystyle \sum _{i=1}^n\sum _{j=1}^n\left|x_i-x_j\right|}}{{\displaystyle 2\sum _{i=1}^n\sum _{j=1}^n{x}_j}}}\hfill \\ \hfill & ={\displaystyle \frac{{\displaystyle \sum _{i=1}^n\sum _{j=1}^n\left|x_i-x_j\right|}}{2n^2\overline{x}}}\hfill \end{array}$$

(8)

where ${\displaystyle \frac{1}{2}}$ exists because the ${\sum }_i{\sum }_j$ yields two sums for each pair of i and j and thus needs to be divided by 2. Further, ${\displaystyle \frac{1}{n^2}}$ is to compensate the numerator’s scale-up by being summed among $n\times n$ instances. That means, a $\mathsf{G}$ is defined to measure the wealth in a per capita normalized manner.

It is of the highest importance in this paper to quantify how intermittence due to wireless connection affects the decentralization level of a consensus mechanism. From the discussion provided so far, one can find that if the wireless disconnection is distributed across the network equally, it should not affect $\mathsf{G}$. This is proved in our results: see in Section 5.2.1.

5. Numerical Results

In this section, we present the results of $\mathsf{R}$ and $\mathsf{G}$, the metrics that this paper proposes to measure the scalability and the level of decentralization of a consensus mechanism, respectively.

5.1. Parameters and Setup

We identify several key parameters that we set for the experiments as follows.

Number of Nodes: Recall from Section 2 that we adopt a PPP for the spatial distribution of the wireless nodes. We set the density of the PPP $\lambda =100$, meaning that the number of nodes distributed in a two-dimensional space ${\mathbb{R}}^2$ (with the dimension of 1 km by 1 km) follows a Poisson random variable, i.e., $\mathrm{Poiss}\left(100\right)$. (Revisit Figure 1 for the layout of our nodes distribution principle.)

Communication Range: Each node is assumed to have 100 m of the communication range, i.e., both transmission and reception of a digital packet.

Number of Transactions per Block: We also assume 10 transaction per block. It is significant to note that this quantity is easily configurable according to various practical scenarios, which forms the key benefit of the analysis framework that this paper proposes.

Number of Iterations: This experiment is generated with several parameters that are highly random, namely, the nodes’ positions, the probability of Byzantine failure, the probability of wireless disconnection, the rate of validators (for PoS), etc. As such, we have run 1 × ${10}^3$ iterations for every experiment as a means to average out the randomness and hence produce more statistically stable results.

5.2. Results and Discussion

5.2.1. Scalability

We start with presenting the results of $\mathsf{R}$, which has been defined as Equation (2) in Section 4.1. We recall from the section that this paper adopts $\mathsf{S}$ for a Zookeeper network as the scalability benchmark, which is presented as a black solid line in all of Figure 2, Figure 3, Figure 4 and Figure 5 and is compared to the $\mathsf{R}$’s of PoW, PoS, and PoC.

Figure 2 compares the three consensus mechanisms, i.e., PoW, PoS, and PoC–on the scalability, $\mathsf{R}$. One can easily find that PoC is the most scalable among the three since it is designed to select a fixed number of witnesses, which also explains the “flatness” of its scalability versus the number of nodes. Due to such selectivity, PoC should pose a lower level of decentralization, as shall be presented in Section 5.2.2.

In Figure 2, one can find it interesting that the scalability of PoS varies in accordance with parameter ${\mathsf{R}}_{\mathrm{v}}$, indicating the rate of validators. We set the parameter as a Poisson random variable with a justification that there must be a “certain amount” of tokens that the largest number of participants hold. Suppose an Ethereum network. The number of ETHs will vary by holders, but the validators are strictly required to hold more than 32 ETHs. It means that the rate of validators can be formally written as

$$\begin{array}{c}\hfill {\mathsf{r}}_{\mathrm{v}}=1-\mathbb{P}\left[Z\ge 32\right]\end{array}$$

(9)

where $Z\sim \left({\lambda }_{\mathrm{eth}}\right)$ denotes a Poisson random variable indicating the number of ETHs held by a holder. The Poisson random variable was chosen to represent the distribution of ETHs across the globe; this modeling refers to a recent study that proved that the distribution of wealth converges to a Poisson distribution [63].

Through Figure 3 and Figure 5, we focus on visualizing the scalability $\mathsf{R}$ for each of the three consensus mechanisms, i.e., PoW, PoS, and PoC, respectively, with the probability of Byzantine failure at each node (due to the uncertain wireless connection), ${\mathsf{p}}_{\mathrm{fail}}$, varied. One can notice a straightforward but clear inversely proportional relationship between ${\mathsf{p}}_{\mathrm{fail}}$ and $\mathsf{R}$: a higher ${\mathsf{p}}_{\mathrm{fail}}$ yields a lower $\mathsf{R}$. The rationale behind this relationship is as follows. A consensus protocol adopts a gossip protocol to propagate the given block. If achieved among only $n<3f-1$ nodes, the consensus is regarded to have a Byzantine fault, which calls for another round of consensus attempt. This is where a significant increase in the consensus delay ${\mathsf{T}}_c$ occurs, which negatively affects the scalability $\mathsf{R}$. (See Equations (2) and (3) for the formulation of the two parameters.)

Figure 5 reveals a particularly interesting phenomenon in regard to $\mathsf{R}$ for PoC. As has already been discussed earlier in this subsection, $\mathsf{R}$ remains flat regardless of the number of nodes participating in consensus because PoC employes exactly 10 witnesses for each consensus. However, the same principle applies to PoC as well: a higher ${\mathsf{p}}_{\mathrm{fail}}$ degrades $\mathsf{R}$ due to the need for multiple block propagation attempts for a single consensus.

5.2.2. Decentralization

Figure 6 indicates that a higher wireless disconnection rate increases the decentralization level. For PoS, the rate of validators among all the nodes is set to ${\mathsf{R}}_{\mathrm{v}}=0.2$. For PoC, the rate of node replacement in each round of consensus is set to ${\mathsf{R}}_{\mathrm{sfl}}=0.9$, and the number of rounds between a replacement is ${\mathsf{\Delta }}_{\mathrm{sfl}}=0$. To wit, ${\mathsf{R}}_{\mathrm{sfl}}$ gives how many of the current witnesses will be replaced with other ones; and ${\mathsf{\Delta }}_{\mathrm{sfl}}$ means how frequently the replacement occurs.

Recall that $\mathsf{G}$ closer to 0 gives a higher level of decentralization. (This may be a surprise to some readers: the wireless connection actually helps improve the decentralization in blockchain.) The reason for this result is straightforward. PoS and PoC allow only “selective” nodes to participate in a consensus. The intermittent wireless connection hinders such participation. Therefore, the level of “inequality” is decreased among nodes participating in consensuses. All in all, very interestingly, the intermittence in wireless connectivity contributes to improving the decentralization in a blockchain consensus process.

Notice from Figure 6 that the rate of cluster (denoted by ${\mathsf{R}}_{\mathrm{cls}}$) was varied with PoW only. The rationale is that PoW is particularly susceptible to such clusteredness of node failure transpiration. In contrast, PoS and PoC will be less impacted since they already adopt “selective” participation of nodes in a consensus anyway; to wit, if failures occur at non-participating nodes, they have almost zero impact on consensus. We elaborate that the rate of cluster differentiating the PoW curves in Figure 6 is formally written as

$$\begin{array}{c}\hfill {\mathsf{r}}_{\mathrm{cls}}={\displaystyle \frac{{\sum }_{i=1}^{\mathbb{N}\left[{\mathbb{S}}_{\mathrm{cls}}\right]}\left|{\mathbb{R}}_{\mathrm{cls},i}\right|}{\left|{\mathbb{R}}^2\right|}}\end{array}$$

(10)

where ${\mathbb{R}}_{\mathrm{cls},i}\in {\mathbb{S}}_{\mathrm{cls}}$ denotes the ith cluster within the entire region ${\mathbb{R}}^2$; and $\left|·\right|$ gives the area of a two-dimensional space.

One should also notice that the three consensus mechanisms pose the order of PoW > PoC > PoS in terms of the decentralization level. This reflects the proportion of chance participating in a consensus: PoS is the most selective according to the number of staked tokens; PoC is less selective as the random wireless connectivity gives a chance of rotating participation in consensus; and PoW is almost not selective by its definition.

Figure 7 is focused on PoS to highlight the impacts of the validator selectivity ${\mathsf{R}}_{\mathrm{v}}$ on the decentralization level $\mathsf{G}$. The result is forthright: a lower ${\mathsf{R}}_{\mathrm{v}}$ (indicating the election of a larger number of validators) yields a lower $\mathsf{G}$ (meaning a higher level of decentralization).

Figure 8 visualizes the impact of wireless disconnection on the decentralization level in PoC. The same phenomenon holds that a higher level of disconnection improves decentralization. The particular focus in the figure is the variation of ${\mathsf{\Delta }}_{\mathrm{sfl}}$, the number of consensus rounds between participating nodes elections: the decentralization is degraded as the re-election of witnesses occurs less frequently.

6. Conclusions and Future Work

This paper has drawn a theoretical framework assessing the impact of wireless connectivity on the performance of consensus in a blockchain system. It has formulated the throughput and the Gini coefficient as the metrics for measuring scalability and decentralization of a consensus mechanism, respectively.

The simulation implemented the formulations and revealed that the imperfect connectivity due to wireless deteriorated scalability but improved decentralization. The reason for the scalability degradation is that a higher chance of Byzantine failure leads to a more frequent need for a re-attempt of consensus, which in turn causes a significant delay. Meanwhile, such a higher probability of failure increases the decentralization level because a more frequent failure gives a wider variety of nodes a chance to participate in the consensus.

As future work, we will extend the findings of this paper to analyzing how mobility affects the consensus in blockchain. Interestingly, the literature of wireless communications already acknowledged that higher mobility could increase the chance of successful connections in a wireless ad hoc network [64].

Moreover, we will investigate impacts of off-chain PoC migration (as part of the recent migration to Solana [65]) on this paper’s result. The scalability and decentralization may need to be re-assessed, depending on the characteristics of the oracle that is introduced by the migration, which may necessitate modifications in this paper’s analysis framework.

In future work, we will incorporate the impacts of mobility of wireless nodes on the performance of blockchain consensus, as an effort to improve the realism of wireless blockchain scenarios.