Power Laws of Bitcoin: Podcast on 17 Oct 2024 for The Bitcoin Layer (Nik Bhatia moderator)

An introduction to the Bitcoin Power Law long term trend. It is contrasted with exponentials that are constant compound annual growth. Bitcoin is not that, but instead a very steep power law. In fact it is multiple power laws for hashrate, adoption, and price behavior.

1. THE POWER
LAWS OF BITCOIN
The Bitcoin Layer podcast
Stephen Perrenod, Ph.D.
October 17, 20241

2. LONG-TERM BITCOIN PRICE ADHERES TO A
POWER LAW
Blockchain Calendar (time
chain)
Lindy Effect
Exponential or Power Law
Power Law model (OLS on
log-log, logarithmic spiral)
Quantile Regression
2
Standard deviations, skew, convexity
Declining returns and volatility are
natural
Power Law vs. gold
Occam’s razor
S-curve models (Weibull, not
logistic function)
It’s a supercomputer!

3. BLOCK CALENDAR / BLOCK TIME
It’s a Time Chain, after all
840,000 block height corresponded to 4th Halving in April
= 16 Block Years of 52,500 blocks
Block months have 4375 blocks
Since Halvings and Difficulty adjustments happen on block
boundaries, best fundamental time system for regressions
Just as the Gregorian calendar has earthly, lunar, solar
rhythms, so does Satoshi’s calendar
Already in Anno Satoshi 17 (was 16 at April 2024
Halving)
3

4. LINDY EFFECT
(LONG RUNNING SHOWS/TECH LAST LONGER)
Can apply to persistent technology (not to people)
E[T-t] = p*t , where T-t is time until expiry
p is Lindy proportion
If use annual % price gain as a p estimator and taking
geometric mean from 2011-2024, then
p = 2.4 roughly
So, if Bitcoin has survived 15+ years, it should have
another 36 years ahead (and once it reaches age 50 years,
another 120 etc.)
Taleb’s anti-fragility
1921-2018 4

5. MODELING BITCOIN VALUE:
THREE METHODS
PRICE - STOCK2FLOW ~ $70K IN 3/2023
PRICE - DIFFICULTY POWER LAW ~ $71K IN 3/2023
PRICE - TIME ~ $37K IN 3/2023 AND K = 5.42
(MID MARCH 2023 ACTUAL ABOUT $25K)
PLAN G HAD DISCOVERED A POWER LAW RELATION
CIRCA 2013 AND HAS DEVELOPED EXTENSIVELY
INCLUDING THEORETICAL UNDERPINNING WITH
METCALFE’S LAW OF NETWORK VALUE (USERS
SQUARED)
My December 2019 article in The Dark Side on Medium
5

6. WHY A POWER LAW MODEL?
Nature loves power laws:
Gravity, Electromagnetism: ~ 1/R
2
Strong nuclear force: ~ 1/R
2 for R < Rc
Weak force: ~ exponential x 1/R
2 and 1/R terms
6
Networks, both communication and social, obey Power Laws:
e.g. Metcalfe’s Law as square of nodes
ChatGPT 4o:
Here are 7 examples of power laws in biochemistry:
1. Allometric Scaling in Metabolism
2. Gene Regulation and Transcription Factor Binding
3. Protein-Protein Interaction Networks
4. Enzyme Kinetics and Michaelis-Menten Behavior
5. Evolutionary Genomics and Genome Size
6. Protein Length Distribution
7. Cellular and Molecular Noise
“Examples of [socio-economic] distributions exhibiting
power laws include
income, wealth, consumption, city populations,
fi
firm size,
family names, stock returns, and numerous others”
- Beare and Toda “On the emergence of a power law in
the distribution of COVID-19 cases” 2020

7. WHY TRY A POWER LAW?
We want a rapid rise with time
It’s simple, Occam’s razor
Network effects
Not everything is exponential (CAGR)
Exponential: Price ~ c * exp (t/to) timescale to
7
Kepler’s Law P
2 = a
3

8. EXPONENTIAL OR POWER LAW?
The data for average monthly return and monthly
return volatility is clear
Results at left are model independent
If exponential = constant CAGR both lines would
be straight
In reality, both the return and volatility fell
monotonically and considerably from Epoch 1 to
Epoch IV
Actual values are consistent with power law
model plus allowance for bubbles
8https://stephenperrenod.substack.com/publish/posts/detail/144303111

9. NOT AN EXPONENTIAL
Exponential: Price ~ c * exp (t/
to)
Characteristic timescale to
Same as CAGR
Semilog (log-linear) plot,
exponential and power law
Log-log, exponential and power
law
Both have the same CAGR now
and both pass through $63,000
9
Exponential
Power Law
Log - Linear Log - Log
Power Law
Exponential

10. STOCK-TO-FLOW IS AN EXPONENTIAL
(MASQUERADES AS A POWER LAW)
Original S2F model = .4 (S2F)
3
S2F converges to exponential
(doubles each 4 years: 8, 24,
56, 120, 248..
A power law of exponential
Doubles each epoch, thus
Price 8 x each epoch
Thus converges to fixed
CAGR 68%
10
Exponential S2F
Exponential Price
Log - Linear
Exponential

11. WHY TRY A POWER LAW? — PRICE ~ T
K
11
Square law: y = c * x
2
Cube law: y = c * x
3
Bitcoin Power law: Price = c * t
k
Two parameters (only): c, k
Scaling property: Double t to 2t, and get c * 2
k * t
k

12. LINDY POWER LAW MODEL
SEMI-LOG AND LOG-LOG CHARTS
Model is the same for both P ~ B
k, where B is age in block years and k is power law index = 5.40; R
2 = 0.94
US$0
US$1
US$10
US$100
US$1,000
US$10,000
US$100,000
US$1,000,000
1 10 100
R² = 0.943
R² = 1
Model Price Actual Price
Log-log chart of Bitcoin price vs. Block Years elapsed, monthly data. k = 5.4 power law index is best fit (green).
Currently ~8 years (block chain age increase 53%) for a factor of 10 increase of price. $1 million around year 27 (11 plus years away)
12

13. LOGARITHMIC SPIRAL, POWER LAW MODEL
Same OLS regression with a 5.4
index power law: Red
Counterclockwise spiral of 4 block
year periodicity (90 degrees / year)
Monthly price data: Orange
Levels of log10 Price are numbered
Green dashed line vector points to
two prior peaks
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14. QUANTILE REGRESSION POWER LAW MODEL
Quantile puts regression lines
through different percentage
levels of price history
Here 0.5 corresponds to
median (of log price) rather
than mean
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$10K
$1M
202420282032203620202016

15. QUANTILE REGRESSION FORECASTS
Quantile regression yields
probability of reaching
objective by certain date
Here we plot probability of
reaching $250K, $500K, $1
million by year
Conservative, as does not
account for bubble timing
15
$1M$250K $500K

16. SOBERING PART, NOT AN EXPONENTIAL
BUT WE SHOWED, MODEL-FREE, RETURNS SLOW
Power law index varied fair amount, but ‘locked in’ by year 8
Expected gain declines with time, gradual flattening (even
before any S-curve considered)
Expected gain fair value [(B+1)/B]^k, or 39% for this block
year and 37% year after
Standard deviation is, in log terms, .315 (.253) or a
multiplicative factor of 2.07 (1.79) for all data (last 4 years)
Power law slope
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
Block years
33.544.555.566.577.588.599.51010.51111.51212.51313.51414.51515.5
k slope
Evolution of the slope parameter, oscillated in early years but has settled down to somewhat over k =
5.
16

17. PRICE DOUBLING TIME
WITH POWER LAW K = 5.4
17
0
1
2
3
4
5
Block Years
456789101112131415161718192021222324252627282930
​​
Doubling time, block years
Doubling time, for Block Year to the 5.4 Power Law model
Block year Calendar Year Doubling time, block years CAGR
4 2012 0.55 234%
5 2013 0.68 168%
6 2014 0.82 130%
7 2015 0.96 106%
8 2016 1.10 89%
9 2017 1.23 77%
10 2018 1.37 67%
11 2019 1.51 60%
12 2020 1.64 54%
13 2021 1.78 49%
14 2022 1.92 45%
15 2023 2.05 42%
16 2024 2.19 39%
17 2025 2.33 36%
18 2026 2.47 34%
19 2027 2.60 32%
20 2028 2.74 30%
21 2029 2.88 29%
22 2030 3.01 27%
23 2031 3.15 26%
24 2032 3.29 25%
25 2033 3.42 24%
26 2034 3.56 23%
27 2035 3.70 22%
28 2036 3.83 21%
29 2037 3.97 20%
30 2038 4.11 19%
Double ((B+x)/B)^5.4 = 2 x = B* (2^(1/5.4)-1)

18. CONVEX ADVANTAGE: Z-SCORE
PAST 10 YEARS, 4-YEAR VOL WINDOW
Z-score: # standard deviations above/below trend, using log10 of
prices
For latest 4-year window, one S.D. is a factor of 1.79 x
Positive swings larger, up to Z = 2.28, minus no lower than Z = -1.31
Upside 10^(2.28x.253) = 3.77 times; Downside 10^(1.31x.253) = 2.15
times
E.g. last data point, model trend $53,149
Downside price $24,720 but Upside price $200,372
Risk $28K for up to $147K gain (5:1)
Highly favorable risk / reward = Convexity
Furthermore, trend is 39% p.a. at present
18
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Block years elapsed
66.577.588.599.51010.51111.51212.51313.51414.51515.5
Z-score four year

19. POWER LAW MODEL VS. GOLD
This is BTC / gold ounces vs. calendar
years, past 13 years
Power law ~ t
5.4, R
2 = 0.94
So if time increases a factor of 2, Bitcoin
measured in gold ounces increases a
factor of 42!
Projection is a “good delivery bar”, 400
ounces, around 2032 or 2033
And 20 million x 400 ounces = 8 billion
ounces exceeds all gold on Earth
19

20. ADDRESSES: POWER
LAW THEORY
Proxy for adoption: non-zero balance
addresses have been growing as the 3rd
power of Bitcoin’s age
In calendar years Addresses ~ T^3.2
Price ~ A^1.8
Price ~ T^5.7
Credit: Giovanni Santostasi (PlanG)
20

21. DIFFICULTY POWER LAW
Difficulty (and Hashrate) has been rising as 12th (!)
power of time during the past 15 years, despite CPUs
to GPUs to ASICs and Moore’s law (exponential)
Modulated by Bitcoin network various feedback
effects to be a power law
Price ~ Difficulty ^ (1/2) (0.48; difficulty and hash
rate follow price)
Slope 11.97 (error .26, R
2 = 0.975)
Maybe some flattening in recent years
21

22. OCCAM’S RAZOR
Wikipedia:
The principle is attributed to the 14th-century logician and Franciscan
friar William of Ockham, though the idea itself predates him.
In essence, Occam’s Razor can be summarized as:
“The simplest explanation is usually the correct one.”
Simpler hypotheses or models preferred:
”because their empirical content is greater; and
because they are better testable” - Karl Popper
ChatGPT 4o
Reasons for Using Occam’s Razor:
1. Simplicity and Efficiency:

•

The principle encourages simplicity, which makes theories easier to understand, test, and work with.
2. Minimizing Assumptions:

•

By favoring the simplest explanation, Occam’s Razor helps reduce the number of assumptions or speculative
elements in a theory.
3. Predictive Power:

•

Simpler theories often have greater predictive power because they are more straightforward to apply to new
situations.
4. Parsimony in Science:

•

In scientific practice, Occam’s Razor is used to choose between competing hypotheses that explain the same
phenomenon.
5. Avoiding Overfitting:

•

In data science and machine learning, Occam’s Razor can help avoid overfitting,
6. Philosophical Grounding:

•

From a philosophical standpoint, Occam’s Razor reflects a preference for elegance and simplicity in explanations.
Ockham, Surrey, England

23. SAYLOR SCENARIOS
Saylor put forth 3 scenarios (base, bear, bull)
They decrease return in stair step fashion
each year
Dashed lines forward show the three
scenarios, solid green line shows the power
law
I have extrapolated backward using same
rules; they do not match the very early data
well, the bull scenario does best but the
base is closer to a power law going forward
23

24. TECHNOLOGY S-CURVE
WEIBULL CUMULATIVE DISTRIBUTION FUNCTION
Weibull cdf: f = 1 -exp [-(t/c)^k]
Normalized to an asymptotic value
Three parameters:
Characteristic time scale, c
Scale factor k
Asymptotic value
Regression requires a double log of both sides
24

25. WEIBULL CDF
Ln (1-f) = [-(t/c)^k]; ln (-ln (1-f)) = k ln (t/c) ; substitute for LHS: y = k ln (t) - k ln (c)
So for y = ax + b; a := k and x := ln (t) and b: - k ln (c) . And can linearly regress (StatPlus)
So we are regressing a double log function of the market cap (relative to its asymptotic value) vs. the
log of time
We run a series of models, varying asymptotic market cap when fitting to data and then determine
best fit for c, k for each model
Characteristic time = time to 1 - 1/e fraction of asymptotic MC (for any k)
Scale parameter k
25

26. WEIBULL CDF REGRESSION (MARKET CAP)
BLOCK YEARS [2, 15.75]
Weibull cdf: f = 1 -exp [-(t/c)^k]
For early t << c, f = 1 - [ 1 - (t/c)^k ] = (t/c)^k and it’s just a power law of k! (Market cap ~ t^6)
So as long as a power law holds we are not close to the knee at age c; knee = 1 - 1/e = 63% of
asymptotic value
And that means the longer it holds, then the asymptotic market cap can be significantly higher
Year of Knee of S-curve and Best fit price at Knee
ASYMPTOTIC MC K SHAPE C TIMESCALE R2 F YEAR PRICE
POWER LAW 5.9862x 1.93 years0.95433742025 88K
$3 T 6.051 18.34 0.95333042026 96K
$10 T 6.004 22.67 0.95333582030 310K
$30 T 5.992 27.32 0.95433692035 950K
$100 T 5.988 33.40 0.95433732041 3.1M
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27. @OrionX_net
CryptoSuper 12th
Mining Report,
May 2024:
Comparison to Top 500
Supercomputer List
Attribute
Frontier
Supercomputer (#1)
Top 500 (all)
WhatsMiner 63S
Hydro cabinet
Bitcoin Network
Equivalent
Performance 1.2 Exaflops 8.2 Exaflops 4.68 Petahash/s 581 Exahash/s
One year
increase
9% 57% 50% (vs. 53S) 57%
Chips
37,888 AMD Instinct
GPUS 6nm; 9472
AMD Epyc CPUs
Hundreds of
millons of cores
12 multithread, 5 nm
ASIC
1,489,744 ASICs
Cabinets 74 1 124,145 cabinets
Power
consumption
23 MegaWatts 89.2 KW 11,074 MW
Weight 296 tons 0.76 metric tons94,600 metric tons
Cost $600 million $107,000 $13.28 billion
Output Science
Science &
Engineering & AI
1.32 Bitcoin per year164,250 Bitcoin per year
Value Priceless Enormous $93,400 per year$11.6 billion per year
Table 5. Comparison of a
hypothetical Bitcoin network based
on the latest high-end WhatsMiner
63S Hydro system with the
Department of Energy’s Frontier
supercomputer and the Top 500 list
as a whole. The cost to build out
the global Bitcoin network is about
22 times that of the Frontier system
while the economic output is
measurable at $11.6 billion
annually, currently. Interestingly
both the Top500 and the Bitcoin
network have recently experienced
the same Moore’s Law growth rate

28. @moneyordebt
Thank You
This is not investment advice. Bitcoin is highly volatile. Past performance of back-tested models is no assurance of future performance.
Only invest what you can afford to lose. You must decide how much of your investment capital you are willing to risk with Bitcoin.
No warranties are expressed or implied.
Money has become information.
Bitcoin is energy securely encapsulated as information.
Electrons to eternal bits.
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29. Appendix
29

30. – Chat GPT o1 preview
30
**Networks of the Bitcoin Ecosystem**
1. Bitcoin Blockchain Network
2. Mining Network
 3. Lightning Network  
4. User Network
5.  Developer Network
6. Merchant Network
7. Exchange Network
8. Full Node Network
 9. Regulatory and Legal Network
10. Service Providers Network
The Bitcoin ecosystem is a complex interplay of various networks, each contributing to its overall functionality and
growth. Feedback loops within these networks can amplify effects, leading to accelerated adoption, innovation, and, at
times, challenges. Understanding these loops is essential for stakeholders, as they can influence strategic decisions,
policy-making, and future developments in the Bitcoin landscape.
BITCOIN is a META NETWORK

31. JENSEN’S INEQUALITY RATIO*
MEASURES CONVEXITY
(*NOT JENSEN ALPHA, DIFFERENT GUY)
Jensen’s inequality compares (a) mean of model values (e.g. from
power law function) over an interval with (b) function at the mean of
the interval
If average ( φ [t1, t2] ) > φ (average [t1,t2] ) then φ is a convex
function
Simply put: average of all model prices over interval relative to
expected price at the mean time of the interval
Ratio of the two averages =* Jensen inequality ratio
Positive Convexity is good as bond traders know
If it rises with more data it means the convexity is hardening
Copilot AI: “If Jensen’s inequality keeps getting stronger for a dataset,
it could indicate increased stability and/or strong convexity.”
31
0
1
2
3
4
Block Years
2 3 4 5 6 7 8 9 101112131415.0
Jensen Ratio of Bitcoin Power Law ~ Block Year^5.4

32. POWER LAW MODEL VS. M2 SUPPLY
There is even a power law of Bitcoin price vs the M2 money supply
Quarterly data since 2011
Power law ~ (M2)
10.3 (!!), R
2 = 0.85
But a polynomial (3rd order) is a better fit, R
2 = 0.94
32

33. ABOUT ME
Early start and finish in Finance
Older than Michael Saylor
Younger than fellow ΠΛΦ brother Feynman
Astrophysics and then Supercomputing
industry for a few decades followed by
Technology consulting
OrionX.net , Analyst and Partner
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