I Introduction
Blockchain has been burgeoning in a plethora of domains since its inception [1, 2, 3, 4], due to the prominent traits of decentralization, transparency, traceability and immutability. Essentially, blockchain is a linear chain documenting valid transactions in a form of blocks, with each being approved by the distributed participants through consensus. The consensus mechanism functions nontrivially in preserving blockchain security since it guarantees that all the honest participants (or nodes) curate the unique shared chain, or denoted as the main chain, so as to maintain data consistency. However, there may arise disagreements among nodes over which the main chain is, and when this happens, we call the blockchain is forking.
Technically, forking refers to the phenomenon that there are a set of blocks at the same height at a time, i.e., [5], as shown in Fig. 1. This suggests divergent blockchains are maintained among nodes, leading decentralized consensus to fail. Forking can be unintentional or intentional depending on the motivations behind. By “unintentional”, we mean forking is aroused due to the inefficient propagation of the overlay network without any adversary, while the “intentional” forking involves malicious attackers who aim at inserting new features for security breaches. It is, therefore, crucial to analyze forking since it destroys the distributed trust, threatening the security and performance of blockchain.
The stateoftheart works on forking can be classified into two kinds, namely the experimentbased
[6, 7, 8] and the modelbased [9, 10, 11, 12]. As the names suggest, the former studies clarify and evaluate the insights via experimental simulations while those of the latter are through mathematical deductions. However, the experimentbased analyses are exclusive, which means the observations drawn from one study may not be feasible for another since they are scenariospecific. This makes the experimentbased analysis an illposed problem in extracting decisive forking laws from the top level, falling short in guiding countermeasures effectively.To remedy the deficiency of the experimentbased studies, the modelbased analyses heaved in sight. However, most of them are constrained to the representation of time dimension, without considering the spatial characteristics of blockchain topology. Nonetheless, forking is essentially an undesirable result of main chain propagation dynamics, which is inevitably bound to the exact positions and connectivities of nodes located in the network. Consequently, such spatialityfree forking models are incompetent to figure out the fundamental mechanism of forking. Moreover, few studies have yielded to the active defense mechanisms for thwarting forking but only recognized it passively, which caps forking prevention and cannot deter it from the root, thus motivating our work.
We face the heterogeneityindesign challenge in realizing an active defense analysis of blockchain forking through the spatialtemporal lens. That is, a finegrained quantitative analysis with spatiality requires us to embed the connection heterogeneity into the main chain propagation mechanism. To resolve this, we endow “expansivity” for connections by differentiating them as the shortrange link and the longrange link
. Specifically, shortrange links exist among direct neighbors while the longrange link can connect remote nodes probabilistically. As such, the shortrange links have no expansivity since they can only spread information among geographically adjacent nodes, while the longrange links can transmit over a long distance with various probabilities, so that we claim they have greater expansivity. Accordingly, it is the differences in expansivity that mirror the connection heterogeneity. In addition, we take advantage of the twodimensional grid network in
[13] to abstractly model the blockchain overlay network since recent theoretical developments have stated blockchain as a smallworld phenomenon [14, 15, 16]. In doing so, we come up with the definition of layer propagation conceptually which characterizes the spread of information as the ripple effect.Despite the benefits, the introduction of short/long range links increases the complexity of spatialtemporal coupling analysis
, which is referred to the second challenge in our paper. Detailedly, the remote transfer ability of the longrange links may accelerate main chain propagation with great uncertainty, complicating the propagation dynamics as a consequence. To deal with this, we thoroughly inspect the probability distributions of main chain propagation triggered by shortrange links and longrange links respectively and propose two concepts, i.e., the
activation time and activation degree, for modeling purposes, where the former denotes the time when the nodes of each layer first receive the main chain while the latter indicates the number of nodes that have received the main chain at any time. These two terms provide a direct clue for further depicting unintentional forking probability and intentional forking robust level, laying foundations for presenting an active defense mechanism to fight against forking.To conclude, the main contributions of our paper can be summarized as follows:

Spatialtemporal propagation model of the main chain. To begin with, we propose the first spatialtemporal model for the main chain propagation, by specifying the blockchain network as a twodimensional grid with short/long range links. In light of this, the forking probability and robust level in unintentional and intentional scenarios are proposed, where the former is analyzed in various time scales, facilitating countermeasures with different intensities.

Active defense mechanism proposal. Our work concludes that dwindling the longrange link factor to change the connectivity of blockchain topology serves to cut down forking completely from the root. To the best of our knowledge, our work proposes the first active defense mechanism operatively to resist forking instead of following current recognitions passively.

Rigorous verifications. Solid theoretical analyses and extensive simulations are conducted, which justify the validity of our forking models as well as the effectiveness of our active defense mechanism.
The rest of the paper is organized as follows. We first review the related work in Section II and characterize the network topology of blockchain with defined expansivity of links in Section III. Based on this, the activation time and activation degree are proposed in Sections IV and V. The modelings and analyses on the unintentional forking probability and intentional forking robust level are carried out in Sections VI and VII. Our experimental results are shown in Section VIII, and Section IX finally concludes our paper.
Ii Related work
At present, the growing interest in analyzing blockchain forking has produced a number of prominent investigations, which can be classified into two categories, i.e., the experimentbased and the modelbased.
For the experimentbased forking analyses, Chen et al. [6] proposed a mechanism called GVScheme, which introduces the role of guarantor to effectively reduce the block verification time. And the simulation results manifest the efficacy of its mechanism in reducing the forking rate. Besides, a proximityaware extension named Bitcoin Clustering Based Ping Time protocol (BCBPT) was presented by Fadihil et al. in [7], which alleviates the forking phenomenon via improving the transactions propagation process. BCBPT was designed to realize this through evaluating the ping latencies between nodes, which has been testified to be strongly effective in optimizing the performance through extensive simulations. In [8], Zhang et al. offered a backwardcompatible defense mechanism to deal with selfish mining attack, allowing forking to be solved as a side problem. Through evaluating the proposed methods extensively, they concluded that the honest miners need to adopt the longest chain if it is at least 3 blocks ahead of the competitors so as to fight back attackers.
Parallel to the above efforts, the modelbased analysis heaved in sight by Decker et al. in [9]
, which quantitatively investigated the impact of information propagation on Bitcoin security from the macro level. To that aim, they proposed a probabilistic model to estimate the average forking rate based on the measurements of block spread across the network, and concluded that propagation delay causes forks primarily. Besides, Xiao
et al. in [9] presented an analytical model to evaluate the effect of network connectivity on consensus security where various adversaries are involved. As such, both the forking rate and the revenue of miners are established to elicit the security properties of consensus. To take precautions against forking, Wang et al. [11] studied the vulnerability of blockchain incurred by intentional forks through a delicate mathematical model, taking advantage of the large deviation theory. In their model, the number of created blocks from the honest nodes and adversaries are regarded as dynamically growing queues, in which the difference between them can be analyzed to reflect the vulnerability of blockchain. Different from current efforts, Chen et al. in [12] explored the impact of competition among mining pools on forking from the miners’ perspective, based on which, a detailed model along with the derived closedform formula of forking probability and expected mining revenue of each pool was expressed. Moreover, an evolutionary game framework was then developed to further reveal the longterm trends in the computing power distributions over pools.However, the above two kinds of studies fall short in the following detriments. On the one hand, the experimentbased analyses are limited to specific scenarios, making the conclusions drawn from a certain case may be unreliable in another. As a result, they can not support forking prevention from the top level since universal and decisive forking laws are inaccessible. On the other hand, existing modelbased researches suffer from spatialityfree which means they are restricted to the representation of time dimension while ignoring the spatial characteristics of blockchain topology. However, forking is highly related to spatiality in the sense that the propagation of the main chain is carried out under a certain network topology. Therefore, the exact position and transmission capability of each node may exert a huge influence on forking. Additionally, the ongoing research only recognizes forking passively without proposing any active defense mechanism. Hence in this paper, we fill the gap through conducting forking analysis from the spatialtemporal dimension, based on which, a defense mechanism can be achieved proactively and operatively.
Iii System model
Iiia Network topology
We exploit the twodimensional grid network [13] to abstractly establish the underlying network of blockchain, as a response to the pioneer theoretical developments which characterized the blockchain as a smallworld phenomenon [14, 15, 16]. Specifically, we deem the network as a graph , where and respectively denote the sets of nodes and edges. In particular, we signify the nodes engaging in blockchain as a set of lattice points, forming a square as shown in Fig. 2. Each of them is identified by its location . In doing so, the lattice distance between node and is determined to the number of “lattice steps” detaching them: . Notably, this grid model can be interpreted simply from the “geographic” perspective, where nodes reside in the grid with neighbors in four directions. Hence, the location represents the geographical position of a node and the lattice distance matches the spatial distance accordingly. Subsequently, we distinguish the links between any two nodes as the shortrange link and longrange link, where the former specifies the link whose lattice distance of the two ends is 1 while the latter represents that of bigger than 1.
We claim that each node has four shortrange links with neighbors determinately but has longrange links with some nodes probabilistically. In practice, the longer the spatial distance between two nodes, the less probability they have to possess a longrange link. Hence, we describe the probability that nodes and existing a longrange link, i.e., , retains a negative exponential relationship with distance. That is, where is the longrange factor to ensure . Remarkably, such a designed factor envisions a quantifiable mechanism to defense forking actively, which will be illustrated detailedly in Sections VI and VII.
IiiB Layer propagation
Once mining a new block, the creator will propagate it to all linked nodes, which then verify its validity and further transmit it to other connected nodes. Such a process continues until all the nodes accept this block, orchestrating the main chain propagation just like the ripple effect. Suppose the creator is node , we then introduce the concept of layer to capture the spreadout characteristics of information propagation in blockchain by presenting the following definition.
Definition III.1 (Layer)

Layer #: The set of nodes belongs to layer # if the lattice distance between node in with equals 1.

Layer #: The set of nodes belongs to layer # if the lattice distance between node in with is .
Assume the longest lattice distance between any node with is , then we have . As shown in Fig. 3, nodes with lattice distance from node compose layer #1 (i.e., the green points), nodes with distance belong to layer #2 (i.e., the yellow points), the same applies to other layers. If the farthest node receiving current main chain is , and the lattice distance between and meets , then we state that and the main chain has been transmitted to layer #.
Recall the longrange link^{1}^{1}1For simplicity, we assume the longrange links only reside in adjacent layers, which means nodes in layer # can only have longrange links with nodes in layer # or #. Other longrange links between layer # and # or longer can be easily extended through our following descriptions. probability is defined as with the longrange factor . One may conclude that a smaller induces a higher probability that the node can connect further, which in turn entitles a stronger potential to stretch more peripherally. That is to say, the longrange links also mirror various longdistance transmission capabilities, which brings in the following definition of expansivity.
Definition III.2 (Expansivity)
The expansivity of link with ends is defined as where denotes the index of the layer where locates if is regarded as the creator.
According to Definition III.2, the expansivity of a shortrange link equals 0 since belongs to layer # in this case. As for a longrange link, its expansivity varies on the difference between and the lattice distance. Notably, the difference in expansivity indeed reflects connection heterogeneity.
Now we are in the position to highlight the advantages of the short/long range links to manifest the merits of our system model. To begin with, the shortrange links exist between neighbors certainly, thus they offer great feasibility for intralayer main chain propagation. On the contrary, the longrange link can probabilistically connect nodes when the lattice distance is larger than 1, which is often the case for nodes in different layers. Hence, the longrange links stand for the main pillar for interlayer main chain propagation. To conclude, the short/long range links can cover various transmission cases of blockchain comprehensively. Additionally, the short/long range links with various expansivities are valuable for shaping the heterogeneity of links in blockchain. And it is the remote transfer ability of longrange links that breaks the regular pattern to propagate the main chain layer by layer, enabling crosslayer transmission. This facilitates a more pragmatic and realistic blockchain.
However, the combination of short/long range links challenges the rigorous inspection of main chain propagation. This is because the crosslayer transmission enabled by longrange links may accelerate information propagation with great uncertainty for interlayer cases, perplexing the transmission dynamics. To remedy this, we first quantify the time when each layer # is first noticed by the main chain, i.e., the activation time . Based on this, we can derive the activation degree , defined as the number of nodes for each layer that has perceived the main chain at time , which finally realizes our main chain propagation model.
Iv Activation time
In this section, we model the activation time of each layer to describe the absolute time when the nodes in each layer first accept the main chain. To achieve this, we carry out a relativetoabsolute method, which starts by introducing the adjacent propagation time to denote the relative time difference of layer # getting the main chain from layer #. After that, we deduce the corresponding absolute time by summing all the cases.
Definition IV.1 (Adjacent propagation time)
The adjacent propagation time (APT) refers to the arriving time that the main chain reaches layer #, if the time of layer # receives the main chain is set as 0.
Accordingly, APT denotes the interval of main chain propagation from layer # to #. Suppose the average transmission delays via short/long range links are denoted as , where holds obviously. Then we can demonstrate APT for each layer in the following theorem.
Theorem IV.1
The expected adjacent propagation time of each layer #, i.e., , can be summarized as
(1) 
where denotes the time that layer # gets the main chain from # via the longrange links, and with and being the number of nodes in and .
Proof:
When , the lattice distance between nodes in to node is 1, hence the propagation interval from to is via the shortrange link; when , the lattice distance between nodes in to node is 2, indicating that of between nodes in to nodes in is 1. Hence the propagation interval from to is via direct shortrange link due to . This justifies the first line of (1).
Since the short/long range links exist between nodes when their lattice distance is 1 or in adjacent layers, the main chain can be transmitted to layer # from # via both of them. Hence, the APT of layer # can be characterized as
(2) 
where and denote the probability and time that layer # gets the main chain from # via the longrange links, and and express those of via the shortrange links. We name and respectively as the longrange and shortrange parts which will be demonstrated more detailedly in the following.
(a) The longrange part. Denote the probability of nodes and residing in layer # and # having the longrange link as , then we have because of . Hence, the probability that node does not possess longrange links with any node in layer # can be specified as . Let the number of nodes in is , then we can obtain since there are nodes whose lattice distance from node is . In light of this, we can get
(3)  
Accordingly, the probability that any node in layer # can not connect any node in layer # with longrange link can be formalized as . Based on and suppose , we can conclude that
(4) 
Hence, the probability that any node in layer # has a longrange link with any node in layer # can be denoted as , which indicates that .
Notably, layer # receives the main chain from layer # via the longrange link implies that during network transmission in layer #, there is a specific node (denoted as ) that has accepted the main chain, and also possesses a longrange link with nodes in layer # exactly. As a result, node propagates the main chain to the next layer via this valuable longrange link. Let the first node accepting the main chain in layer # be . If =, then . Otherwise, since node needs to receive the main chain from node through the shortrange links internally. In this paper, we simply utilize the term for demonstration and leave an exact analysis of for future work. Until now, we can deduce the APT of layer # via the longrange link as
(5) 
(b) The shortrange part. Since the shortrange links only possess in neighbors whose lattice distance is 1, when the nodes in layer # propagate the main chain to the next layer through the shortrange links, the distance to be passed will be no more than . Hence, we can set with as the discounting factor to get the expectation. With this in mind, we can get the APT of layer # via the shortrange link as
(6) 
In summary, we can get the APT of layer # as shown in the second line of (1) through combining the above two parts. This completes our proof.
It is worth noting that the defined APT only focuses on the propagation time of the main chain between nearby layers. To derive the whole expected propagation time of the main chain from the source node to layer #, i.e., , it is necessary to sum all the corresponding cases as presented in the following:
(7) 
Through substituting (1) into (7), the expected activation time when the nodes in each layer # receive the main chain firstly can be summarized as
(8) 
in which .
V Activation Degree
This section will illustrate the main chain propagation model by specifying the number of nodes that have received the main chain at any time, which is also denoted as the activation degree, based on the above analyses on activation time. To this end, we take advantage of the epidemic model, i.e., the susceptibleinfected (SI) model [17], to characterize the evolution of nodes in blockchain. SI models are widely exerted in topologically related information diffusion phenomenons, both in social and technology networks [18, 19, 20, 11]. We believe it is the simplest method to capture the essence of information propagation in blockchain. Technically, nodes have two states: susceptible () and infected (), which are interchangeable. A node in state indicates it is inactivated that has not received the main chain while that in state means it has been activated by accepting the main chain. A node in may turn into once it accepts the main chain and work on it. In the following, we first derive the activation degree within layer locally, after which, we can obtain the global degree in the network scale.
We set the beginning of main chain diffusion in layer # as time , and the activation and inactivation degrees (or the number of nodes that are in state and ) are denoted as and respectively in any time . Then we have and since there must be one activated node initially. Based on the SI model, can be derived as:
(9) 
where represents the probability that an activated node transfers the main chain to others within layer #. Considering nodes inside the layer transfer information via shortrange links, can also be interpreted as the probability that a node possesses the shortrange link with others, which is . In doing so, we can derive formally through the logistic model, leading to
(10) 
Notably, in (10) depicts the number of activated nodes in time of layer # locally. Subsequently, we are going to find the activation degree globally, i.e., , which expresses the number of infected nodes in the network at from the very beginning when the source node produces the main chain. We reset as the time when the main chain is originally generated, in light of the above analysis, we can acquire the following theorem.
Theorem V.1
The activation degree of the network when , satisfies
(11) 
Proof:
We prove each case of (11) in the following.
(1) Case 1: The activation degree of the network since layer # has not received the main chain when according to (7), shown as the blue node in Fig. 3.
(2) Case 2: The activation degree of the network since layer # can receive the main chain when , leading to the four directly connect neighbors being activated, shown as the blue and red nodes in Fig. 3.
(3) Case 3: The activation degree of the network since both layer # and # can receive the main chain when , making , shown as the blue, red and green nodes in Fig. 3.
(4) Case 4: when , layers ## have all accepted the main chain. Take layer # as an example, the activation time of this layer is . Hence, the main chain has been propagated within this layer for , making the activation degree in this layer as . Other layers can be done in the same manner, leading to the fact that .
Until now, we present the activation degree at any time in the network, which can be recognized as the main chain propagation model. Such a measurement is so valuable in reflecting the possible forking occurrence since the higher the degree is, the more nodes perceive the main chain, the less probability that forking may happen. In the following two sections, we carry out forking analysis for both unintentional and intentional scenarios, based on which, an active defense mechanism can be proposed.
Vi Modeling and Analysis: Unintentional forking
Technically, unintentional forking is normally caused by the inefficient propagation of the blockchain overlay network without any attacker or artificial manipulation [5],[21]. Trace the root, it happens when a conflict block is found while the previous block with the same height is propagated in the network. Accordingly, we claim that the unintentional forking probability is equivalent to that of generating new blocks (i.e., ) during the propagation of chain containing .
Assume the source node starts to transmit the produced main chain at time , and all the nodes are supposed to accept the main chain at if no forking occurs. The whole time span of can be equally divided into intervals, with each interval being . In the following, we proceed to analyze the probability of unintentional forking both in unit time, i.e., , and over a period of time, i.e., , where the former focuses on each interval while the latter specifies any defined period with . In doing so, effective countermeasures of unintentional forking are facilitated operatively at different time scales with multiple intensities.
Via Unintentional forking probability analysis in unit time
In fact, the mainstream consensus mechanism, i.e., proofofwork
(PoW), guarantees that the average work of each node follows Bernoulli distribution, which suggests that the mining process can be deemed as a Poisson process
[11, 5]. Let be the set of nodes which have not received the main chain at time and denotes the number of conflict blocks created by the nodes in at time . Hence,follows Poisson distribution with expectation
, with denoting the computing power of node . As such, we can obtain the probability that new blocks are generated during as(12) 
Note that elaborates there is no block created by the inactivated nodes in , which in turn indicates no unintentional forking occurrence. Therefore, we can derive the probability of unintentional forking as
. If the computing power is uniformly distributed among nodes, we have
(13) 
where with and is formalized in (11). Through investigating (13) thoroughly, we claim the following corollaries to reveal the inherent mechanism of unintentional forking in unit time, thus we can present direct evidence to hinder it operatively.
Corollary VI.1
A smaller longrange factor can induce a lower .
Proof:
As described above, specifies the probability that any node pair and has a longrange link. The lower is, the more likely such link will exist. In addition, the longrange link facilitates the synchronicity of crosslayer main chain propagation, which allows more nodes to receive the main chain earlier, making the number of suspectable nodes smaller. Aware of this, we can clarify that a lower enables the main chain to be broadcasted to the whole network more efficiently through more longrange links, shrinking the unintentional forking probability as a consequence.
Corollary VI.2
The unintentional forking probability goes down when the transmission delays of the short/long range links, i.e., and , decline.
Proof:
As demonstrated in Theorem IV.1 and (8), a lower will decrease the expected APT as well as the activation time for each layer #. This will in turn influence in (11) positively while affect negatively, indicating a lower forking probability as formalized in (13). An intuition for this is that lower transmission delays make blockchain consume less time to hear the main chain for the inactivated nodes. This leaves no room for them to produce competitive blocks at the same height, decreasing forking probability consequently.
ViB Unintentional forking probability analysis over a period of time
In this section, we formalize the probability of unintentional forking during a period of time , i.e., . According to subsection VIA, the forking probability in is denoted as , then turns to
(14) 
When the computing power is allocated uniformly, becomes . We then give the following corollaries.
Corollary VI.3
A smaller longrange factor can bring about a lower .
Corollary VI.4
drops when and decrease.
Vii Modeling and Analysis: Intentional forking
Differently with unintentional forking, intentional forking involves malicious adversaries, who aim to surreptitiously execute the PoW for creating a chain containing fake transactions, namely the fake chain [11]. If the fake chain is longer than the honest main chain^{2}^{2}2In this section, we differentiate the chains created by the honest nodes and the adversaries as the honest main chain and the fake chain, respectively., the latter is then substituted by the malicious fake one according to the longest chain principle, incurring intentional forking attack as a consequence.
In particular, the adversaries always commence the forking attack assembly as a group since the honest majority makes it challenging for an attacker to commit attacks solely. In such an adversarial group, members share knowledge of the fake chain and information interchange is required to be realtime so as to prompt the success probability of attack. Considering this, we can assume there is no transmission delay among the group which can be realized through maintaining the participants in a local area network (LAN). Based on this, Wang et al. in [11] presented the robust level of blockchain being attacked by intentional forking through analyzing the computational confrontation between the honest and malicious nodes, that is
(15) 
where is a preset threshold indicating the vulnerability probability of blockchain should be less than. In addition, and describe the number of nodes that have received the honest main chain and fake chain, and represent the computing power of the honest and malicious node on the premise of uniform distribution. Essentially, suggests the required difference in the number of blocks completed by the adversary and the honest node, to verify a transaction. The higher is, the more effort should be put to launch attacks, the safer the blockchain being protected from intentional forking.
However, the above formulation falls short in static modeling and spatialityfree, making it insufficient in depicting a realtime and pragmatic blockchain. Aware of this, we present a dynamic robust level with spatiality, i.e., as follows:
(16) 
where is determined as (11). This gives the following corollaries.
Corollary VII.1
A smaller longrange factor can lead to a higher , thus making blockchain stronger in fighting against intentional forking.
Corollary VII.2
raises when and decrease, which indicates that dwindling the transmission delays can help to hinder intentional forking.
These two corollaries can be easily derived, therefore we omit their proofs for simplicity.
Remark. Here we are going to highlight the significant merits of all the proposed corollaries in facilitating an active defense mechanism for thwarting forking. Note that for both unintentional and intentional scenarios, setting a smaller and will lead to a lower forking probability and promote the robustness of blockchain, which means 1) increasing the longrange link between any two nodes and 2) dwindling the transmission delays of the short/long range links are the two effective countermeasures. This provides clear evidence for blockchain designers to devise and detect the topology and transmission capacity of the network so as to meet our requirements in fighting back forking. More excitingly, our paper proposes the first active forking defense mechanism, that is, positively reshaping the overlay network through decreasing the longrange factor . In doing so, we can eradicate the forking phenomenon fundamentally. %ָ ķ Ȼ ̵ֲ档
Viii Experiments
In this section, we will evaluate the validity and effectiveness of our proposed models and active defense mechanism. We have simulated the blockchain network with the PoW consensus mechanism written by Python, running on machines equipped with Intel Core i78700 GPU, 3.20 GHz CPU and 8 GB RAM. We begin by verifying our main chain propagation model in subsection VIIIA, after that, the unintentional and intentional forking models established in Sections VI and VII are analyzed respectively in subsections VIIIB and VIIIC.
Viiia Evaluation on the main chain propagation model
We first carry out experiments on the activation degree and compare it with the theoretical results, to demonstrate the validity of our proposed models. To calculate , we simulate the topology of blockchain as the aforementioned grid of and 545 nodes with short/long range links. Besides, the probability of the longrange link between any node pair and is preset as where and 10. The transmission delays are determined as and respectively. Note the tested system is driven by PoW consensus and we fix the interval as for better performance. Each experiment is repeated 50 times to get the average result for statistic confidence.
Fig. 4 reports the theoretical and simulated results of on the difference of when and 10 and and . Conclusively, we make the following observations: 1) our experimental and analytical results match perfectly under various network sizes and topologies. This demonstrates the validity of our main chain propagation model in depicting the information diffusion of blockchain. 2) From subfigures (a) and (c), we can conclude that under the same network size with , the increase of will impose a negative effect on . More specifically, when , the activation degree of is nearly 120 in the simulation (or 110 theoretically), while that of is 80 approximately in the simulation (or 70 theoretically). One conjecture is that a higher may trigger a lower probability of possessing a longrange link between any nodes and , slowing down the propagation process crosslayer and consequently declining the number of activated nodes. The same conclusion can also be acquired from subfigures (b) and (d). 3) From subfigures (a) and (b), we can find that when is identical, the network with realizes consensus globally earlier than that of with , which reports a smaller network is much easier to reach consistency. Detailedly, all the nodes get the main chain when in the network with 145 nodes, while in the network with 545 nodes stops raising until . The same can be concluded from subfigures (c) and (d).
ViiiB Evaluation on the unintentional forking model
After testifying the validity of the main chain propagation model , we proceed to evaluate the performance of forking probabilities and in unintentional forking. Besides the above described experimental settings, we set the computing power of each node equally and bring in the transmission delay ratio in which means the initially set and , and represents and .
We plot the evolutions of unintentional forking probability in unit time in Figs. 5 and 6. Specifically, Fig. 5 displays the negative relationship between and , when and 10 and and 545. That is, the forking phenomenon can be hindered probabilistically if the main chain is spread long enough. In addition, the fact that the orange lines lie above the blue ones shows that a lower will lead to a smaller forking probability, corroborating Corollary VI.1 as a result. Fig. 6 exhibits the effect of transmission delays on , which varies with the propagation time on the difference of and . The subfigures in Fig. 6 suggest that higher delays will trigger forking with a higher probability as we can inspect that the orange lines locate above the blue ones. This proves what we state in Corollary VI.2.
The trends of unintentional forking probability over a period of time, i.e., , are presented in Figs. 7, 8, based on which, we can find 1) grows as time goes by but with a gradually decreasing rate, which agrees with the proposed analytical model as shown in (14). Theoretically, is formalized according to the summation of , that is, the number of suspectable nodes from to , which increases with a declining growth rate as more and more nodes accept the main chain. 2) A smaller can incite a lower forking probability as we derived in Corollary VI.3 according to Fig. 7. 3) Fig. 8. testifies the effectiveness of Corollary VI.4 in that the forking probability over a period of time drops as the decrease of the transmission delays. To conclude, a blockchain overlay network with a lower longrange factor , a smaller network size and less delays will endure the least risk of being attacked by unintentional forking.
ViiiC Evaluation on the intentional forking model
In this section, we evaluate the dynamic robust level with spatiality, i.e., , for the intentional forking scenario. Assume there are malicious nodes, where the threshold is preset as 0.1. The results are demonstrated in Figs. 9 and 10. Fig. 9 reports the relationship between and , from which we can conclude that as time goes by, the robust level will increase since the decrease of denotes less block difference is required for hindering forking occurrence. Besides, it is testified that a smaller can lead to higher , as the blue lines are always below the orange ones. This justifies the validity of Corollary VII.1. Additionally, Fig. 10 shows varies with when different and are adopted, where we can find lower delays can help promote the robust level as clarified in Corollary VII.2.
Ix Conclusion
In this paper, we carry out the first active defense analysis of blockchain forking from the spatialtemporal dimension. To begin with, we characterize the topology of blockchain network as a twodimensional grid with distinguished short/long range links. Based on this, the concepts of layer and expansivity are defined to respectively capture the rippleeffect information propagation process and connection heterogeneity of blockchain. In addition, we propose the term of activation time to denote the time when the nodes in each layer first accept the main chain, in light of which, we finally realize the activation degree to express the main chain propagation model. In doing so, the forking probability and robust level in unintentional and intentional forking are modeled and inspected detailedly. Through our analyses, we conclude that 1) dwindling the transmission delays of the short/long range links can hinder forking and 2) positively reshaping the overlay network through shrinking the longrange factor can resolve the forking phenomenon from the root. This observation is so valuable that it delivers the first forking defense mechanism proactively and operatively. Extensive theoretical deductions and simulations are conducted to verify the effectiveness of our analysis.
Acknowledgment
This work has been supported by National Key R&D Program of China (No. 2019YFB2102600), National Natural Science Foundation of China (No. 61772044), the International Joint Research Project of Faculty of Education, Beijing Normal University, and Engineering Research Center of Intelligent Technology and Educational Application, Ministry of Education.
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