Impact of the Hash Rateon the Theoretical Fork Rate of Blockchain
Akira Sakurai

Kazuyuki Shudo

Tokyo Institute of Technology
Tokyo, Japan

Kyoto University
Kyoto, Japan

Abstract—Forks in a blockchain make it less secure. So thefork rate can be used as a measure of security. Theoretical fork
rates have been used in a number of studies, not all of which
consider the distribution of hash rates. In this paper, we proposea theoretical fork rate that considers the distribution of hashrates. We compare this theoretical fork rate that considers the
distribution of hash rates, a theoretical fork rate that does notconsider the distribution of hash rates, and the real fork rate
obtained by simulation experiments. Using a simulator, we showthat the proposed theoretical fork rate is closer to the real value.

rate deﬁned from TW = T F compared to the theoretical fork
rate that does not consider the distribution of hash rates. In
section IV, we describe the application of the theoretical fork
rate to real blockchain networks and simulation experiments.
Section V gives a summary and describes future work. In the
appendix, we show through experiments that TW = T F holds
under various conditions, and that the theoretical fork ratedeﬁned from TW = T F is effective as an evaluation criterion
for blockchain networks.

Index Terms—blockchain, fork rate, block propagation time

II. T HEORETICAL VERIFICATION
Here we show from a theoretical point of view that TW =T F holds.
Let p be the probability of generating a valid block in one
hash calculation, V be the set of nodes participating in theblockchain network, and Mi (i ∈ V ) be the hash rate of eachnode i. At this time, the block generation success probability
does not depend on the previous hash calculation. Therefore,
the probability of successfully generating a block in N hashcalculations is (1 − p)N −1 p. Therefore, the average numberof hash calculations required to successfully generate a blockis 1/p. ∑From this, if the hash rate of the entire network is
Mall = i∈V Mi , the expected value of the block generationinterval of the entire network is 1/(p · Mall ); let this be T .
Next, we show the fork rate when node i succeeds increating a block. ui (t) is the sum of the hash rates of nodes thathave not yet received a block after t units of time since node
i successfully generated a block. This ui (t) takes a different
value for each block even if the same i is used, but for the sakeof simplicity, we assume that it does not depend on individual
blocks. Then, the probability that node i succeeds in creating
a block and the block is∫ distributed throughout the network∞
without forks is (1 − p) 0 ui (t)dt . Therefore, the probability
Fi that a fork will occur before the block created by node i
spreads throughout the network is as follows.

I. I NTRODUCTION
Blockchain is a distributed ledger that is the underlying
technology of cryptocurrencies such as Bitcoin [1].
A fork is a branching of the blockchain. The fork rate is
the percentage of forked blocks out of the total number ofblocks created. Forks are caused by malicious nodes and blockpropagation delays [2].For example, a fork occurs when a node
successfully generates a block and another node that has notreceived the block generates another block. A fork divides thecomputational resources used for proof of work (PoW) in the
network, increasing the risk of double spending attacks andselﬁsh mining [3].
In [2], a theoretical fork rate that does not consider the
distribution of hash rates is used. This is likely due to the
difﬁculty of knowing the distribution of hash rates in a real
blockchain network. However, such a theoretical fork rate may
deviate from the real fork rate when the distribution of hashrate is biased. Based on the above, in this paper, we ﬁrst showthat TW = T F holds in order to deﬁne the theoretical fork
rate considering the distribution of hash rates, where TW isthe average block propagation time weighted by the hash rate,
T is the block generation interval, and F is the fork rate. This
once again conﬁrms the strong relationship between fork rate
and block propagation time. We also deﬁne the theoretical fork
rate from this relationship. This theoretical fork rate takes into
consideration the distribution of hash rates. In this paper, we
compare this theoretical fork rate with a theoretical fork ratethat does not consider the distribution of hash rates and thereal fork rate. Using a simulator, we show that the theoreticalfork rate that takes into account the distribution of hash rates
is closer to the real fork rate.
In section II, we show that TW = T F theoretically. In
section III, we show the effectiveness of the theoretical fork∫∞

= 1 − (1 − p) 0 ui (t)dt∫ ∞≈ p·ui (t)dt (∵ p ≪ 1)0
∫ ∞ ui (t)dt
0 Mall=(1)
T∑
Then, the fork rate is F = i∈V Mi · Fi /Mall .
Next, we show in more detail the relationship between
the block propagation time, speciﬁcally the average blockFi

Proc. IEEE 41st Int'l Conf. on Consumer Electronics(IEEE ICCE 2023), January 2023

0.0014TABLE I
FR , FW AND FN WITH VARIOUS DISTRIBUTIONS OF HASH RATES .
FW
0.001299
0.001277
0.001128
0.001040

FW /FR
0.993
0.969
0.98261.051

FN
0.001299
0.001280
0.001266
0.001270

FN /FR
0.993
0.971
1.103
1.283

x0.971x0.969x1.283x1.103x0.9826

x1.051

distribution (3)distribution (4)0.001

fork rate

Distribution (1)Distribution (2)Distribution (3)Distribution (4)FR
0.001308
0.001318
0.001148
0.000990

x0.993x0.9930.00120.00080.00060.00040.00020

propagation time weighted by the hash rate, and the fork rate.
TW,i , the average block propagation time weighted by the hashrate when node i successfully generates a block, is as follows.
∫ ∞
TW,i

=
0

distribution (1)distribution (2)FR : real fork rate
FW: theoretical fork rate considering the distribution of hash rates

FN : theoretical fork rate not considering the distribution of hash rates−t ·u′i (t)Malldtu′i (t) in the equation in the integral on the right side indicatesthe derivative of ui (t). Actually, ui (t) is not differentiable, buthere we treat ui (t) as differentiable for simplicity. Therefore,u′ (t)− Mi all indicates the amount of change in hash rate ratio forthe entire network that received the block after t units of time
have passed since the block was generated. We obtain thefollowing result by further integration by parts.
∫ ∞u′ (t)
TW,i =−t · i dt
Mall0∫∞{[−t · ui (t)]∞0 + 0 ui (t)dt}
=
Mall
∫ ∞
ui (t)=dt
Mall
0≈ T Fi(2)
We take the average based on the distribution of hash rates by
averaging both sides and obtain the following result.
∑i∈V Mi · TW,iTW =∑ Malli∈V Mi · T · Fi
≈
Mall
= TF
(3)
What this formula means is that the average block propagation
time weighted by the hash rate is equal to the product of theblock generation interval and the fork rate. In reality, formula(3) may not hold due to forks or other factors.
III. E XPERIMENT
We deﬁne the theoretical fork rate FW = TW /T from therelationship shown in section II between the fork rate, theaverage block propagation time weighted by the hash rate,
and the block generation interval. FW takes into account the
distribution of hash rates, as is clear from the deﬁnition. On
the other hand, we deﬁne the theoretical fork rate FN without
considering the distribution of hash rates as follows.
∫∞
1 − f (t)dt
0
FN =
T

Fig. 1. FR , FW and FN with various distributions of hash rates.

f (t) indicates the percentage of nodes that have received theblock t units of time after the block was created. Originally,
this variable could take a different value for each block. Here,the mean of the values for each block is treated as f (t). Sucha theoretical fork rate that does not consider the distributionof hash rates is used in various studies [2] [4].In this section,
we compare the theoretical fork rates FN and FW and the real
fork rate FR obtained by conducting simulation experimentswith various distributions of hash rates. We show that FW iscloser to the value of FR than FN .
We used SimBlock [5] for the simulation. The number ofnodes was 1000, the maximum number of inbound connections
was 125, and the number of outbound connections was 8,reconnecting every 10 blocks. All experiments were performed
with a maximum block height of 100000. The experiments
were conducted on the following four distributions of hashrates.
Distribution (1):
Let the hash rate of all nodes be 4 · 105 (/sec)
uniformly.
Distribution (2):
Let the distribution of hash rates be a normal distribution with an average of 4 · 105 (/sec) and a standard
deviation of 1 · 105 (/sec).
Distribution (3):Let the distribution consider the current Bitcoin
distribution of hash rates [6]. Assuming that there
are mining pools, we simulate nodes with high hashrates. Speciﬁcally, we randomly select six nodes andlet their hash rates be 1.50 · 108 , 1.33 · 108 , 9.33 · 107 ,8.06 · 107 , 5.06 · 107 , and 1.26 · 107 (/sec). Let the
distribution of the hash rates of the other nodes be anormal distribution with an average of 4 · 105 (/sec)
and a standard deviation of 1 · 105 (/sec).
Distribution (4):
Let the distribution consider the current Ethereum [7]distribution of hash rates [8]. As with distribution (3),we assume there are mining pools and simulate nodes
with higher hash rates. Speciﬁcally, we randomly

Proc. IEEE 41st Int'l Conf. on Consumer Electronics(IEEE ICCE 2023), January 2023

IV. A PPLICATION
Based on the experimental results in the appendix, we consider applying the theoretical fork rate FW to real blockchain
networks and simulation experiments.
To apply the theoretical fork rate FW to a real blockchain
network, it is necessary to know the distribution of hash rates.
Therefore, we plan to estimate the distribution of hash rates
from the block generation rate of each node.
From the experimental results in the appendix, when measuring the fork rate of a blockchain network in a simulation
experiment, it is necessary to simulate many blocks. On
the other hand, by using the theoretical fork rate FW , it ispossible to determine the fork rate of the network quickly andaccurately.
V. C ONCLUSION AND FUTURE WORKThis paper shows that the average block propagation timeweighted by the hash rate is equal to the product of the block
generation interval and fork rate. Based on this relationship,
we deﬁned a new theoretical fork rate. Furthermore, we
conducted simulations to show that our theoretical fork rate
FW is closer to the actual fork rate FR than the theoretical fork
rate FN , which does not consider the hash rate distribution,
under various distributions.
In the future, we plan to evaluate block propagation acceleration methods such as neighbor node selection [9] using thetheoretical fork rate FW presented in this work.
R EFERENCES[1] S. Nakamoto, “Bitcoin: A peer-to-peer electronic cash system,” 2008.
[2] C. Decker and R. Wattenhofer, “Information propagation in the bitcoin
network,” in IEEE P2P 2013 Proceedings, 2013, pp. 1–10.
[3] A. Gervais, G. O. Karame, K. Wüst, V. Glykantzis, H. Ritzdorf,
and S. Capkun, “On the security and performance of proof of work
blockchains,” Proceedings of the 2016 ACM SIGSAC Conference on
Computer and Communications Security, 2016.
[4] R. Kanda and K. Shudo, “Block interval adjustment toward fair proof-ofwork blockchains,” 2020 IEEE 36th International Conference on Data
Engineering Workshops (ICDEW), pp. 1–6, 2020.
[5] Y. Aoki, K. Otsuki, T. Kaneko, R. Banno, and K. Shudo, “Simblock: A
blockchain network simulator,” in Proc. IEEE INFOCOM 2019 - IEEE
Conference on Computer Communications Workshops (INFOCOM 2019
Workshops), 2019, pp. 325–329.

TABLE II
FW , FR AND TW UNDER VARIOUS CONDITIONS .

2019
2015
Region-based neighborselection
Without compact block relay

FR
0.001318
0.001916FW
0.001277
0.001845

FW /FR
0.968
0.963

TW (msec)
769
1112

0.001228

0.001254

1.021

755

0.004588

0.004695

1.023

2814

x1.0230.005
0.00450.004
0.0035fork rate

select ﬁve nodes and let their hash rates be 3.31·108 ,1.60 · 108 , 1.14 · 108 , 8.00 · 107 , and 5.71 · 107 (/sec).
Let the distribution of the hash rates of the othernodes be a normal distribution with an average of
4 · 105 (/sec) and a standard deviation of 1 · 105 (/sec).
Table I and Figure 1 show the experimental results. It canbe seen that FW is close to FR in all distributions (1) to (4).
In the case of distribution (1), FN is close to FW because
the hash rate follows a uniform distribution. In the case of
distribution (2), the hash rate follows a normal distribution,but even in this case there is no signiﬁcant difference between
FR and FN . In distributions (3) and (4), the distributions of
the hash rates are greatly biased by introducing nodes withhigh hash rates. In this case, the values of FR and FN diverge
greatly.

0.003
0.0025x0.9630.002
0.0015x0.968x1.0210.001
0.0005
0
2019

2015

region-based
without compact
neighbor selection
block relay

FR: real fork rate
FW: theoretical fork rate considering the distribution of hash rates

Fig. 2. FR and FW under various conditions.

[6] “Blockchain.com,” https://www.blockchain.com/pools, 2022, accessed:
Aug. 19. 2022.
[7] “Ethereum: A secure decentralised generalised transaction ledger,”https://ethereum.github.io/yellowpaper/paper.pdf.
[8] “Blockchair,”https://blockchair.com/ja/ethereum/charts/hashratedistribution, 2022, accessed: Aug. 17. 2022.
[9] Y. Aoki and K. Shudo, “Proximity neighbor selection in blockchain networks,” in Proc. 2nd IEEE Int’l Conf. on Blockchain (IEEE Blockchain2019), 2019, pp. 52–58.
[10] R. Nagayama, R. Banno, and K. Shudo, “Identifying impacts of protocol
and internet development on the bitcoin network,” in Proc. 25th IEEE
Symposium on Computers and Communications (IEEE ISCC 2020),
2020, pp. 1–6.
[11] M.
Corallo,“Compact
block
relay,”
https://github.com/bitcoin/bips/blob/master/bip-0152.mediawiki, 2016,
accessed: Aug. 16. 2022.
[12] H. Matsuura, Y. Goto, and H. Sao, “Region-based neighbor selection inblockchain networks,” in Proc. 2021 IEEE International Conference on
Blockchain (IEEE Blockchain 2021), 2021, pp. 21–28.

A PPENDIX
Here, we show that the relationship described in II holds
even in more complex situations. We also show the effectiveness of the theoretical fork rate FW on the convergence speed.
As in the experiment above, we used SimBlock with 1000
nodes, 125 inbound connections, and 8 outbound connections,reconnecting every 10 blocks. All experiments were performed
with a maximum block height of 100000.
We investigated FW , FR and TW under various conditions.
The details of each condition are as follows.
2019:
All nodes in SimBlock belong to one of six regions.
We use the 2019 regional bandwidth and latency

Proc. IEEE 41st Int'l Conf. on Consumer Electronics(IEEE ICCE 2023), January 2023

0.0035
0.003

fork rate

0.00250.002
0.00150.001
0.0005
100
4000
7900
11800
15700
19600
23500
27500
31400
35300
39200
43100
47000
50900
54800
58700
62600
66500
70400
74300
78200
82100
86000
89900
93800
97700

0

block heght

FR : real fork rate

FW : theoretical fork rate

Fig. 3. TR and TW per block height.

available in SimBlock [10]. Let the distribution ofhash rates be a normal distribution with an average of
4 · 105 (/sec) and a standard deviation of 1 · 105 (/sec).
We use compact block relay [11] and choose neighbor nodes randomly.
2015:
We use the 2015 regional bandwidth and latency
available in SimBlock. The other conditions are thesame as those of 2019.
Region-based neighbor selection:
We apply region-based neighbor selection [12]. With
this method, the number of connections between thesame regions among the outbound connections ofeach node is set to 6. The other conditions are thesame as those of 2019.
Without compact block relay:
We do not apply a compact block relay. The otherconditions are the same as those of 2019.
Table II and Figure 2 show the experimental results. As seen,
FW matches FR with high accuracy under all conditions. Thisshows that formula 3 holds even in complicated situations.
The next experiment demonstrates the effectiveness of the
proposed theoretical fork rate FW from the viewpoint of
convergence speed. As shown in Figure 3, FW is faster andmore stable than FR . From this experimental result, we can see
the effectiveness of the theoretical fork FW when evaluating
the blockchain network.

Proc. IEEE 41st Int'l Conf. on Consumer Electronics(IEEE ICCE 2023), January 2023