‏First research paper that systematically shows that Bitcoin’s power law is a real phenomenon.

Bitcoin Power Law

Paper abstract:
The research paper is extremely long (about 45,000 characters)

- Introduction:
The emergence of stable power laws in complex systems is one of the central topics in nonlinear dynamics and statistical physics. Power laws appear in diverse contexts such as earthquake intensity, neural cell breakdowns, degree distributions in scale-free networks, biological growth, and the spread of epidemics over heterogeneous contact networks. The common thread is that the behavior of a power law indicates the absence of a preferred scale, which is a defining feature of systems operating near a critical point or evolving on a scale-free substrate.
Bitcoin, the decentralized monetary network with the first proof-of-work system that first went live in practice on January 3, 2009, provides an extraordinary opportunity: a complex socio-economic system whose complete transaction ledger is publicly recorded, along with its construction data and price. This record covers more than 15 years of continuous observation. Several scientists have noted that the price of Bitcoin shows strong power-law growth over time, but these observations have remained largely empirical, with the growth coefficient being a fitted parameter rather than a quantity derived from first principles.

- Main Empirical Results:
- Adoption power law: $$ N_t \propto t^{3.05} $$ ($$R^2 = 0.977$$), reflecting a saturation-wave mechanism on heterogeneous networks.
- General Metcalfe’s law: $$ P \propto N^{1.84} $$ ($$R^2 = 0.951$$), less than 2 due to decreasing marginal connectivity value.
- Direct price law: $$ P_t \propto t^{5.69} $$ ($$R^2 = 0.961$$), with fixed residuals and periodic market cycles lasting 4 years.

- Theoretical Analysis:
The coefficient $$\alpha \approx 3$$ comes from an epidemic spread model on scale-free networks (Colgate et al., 1989), where adoption spreads from highly connected nodes to less connected ones via a saturation wave. Meanwhile $$\mu \approx 1.84$$ reflects the general Metcalfe’s law, in which the value of connectivity decreases as the network grows. The composition $$\beta = \alpha \times \mu$$ confirms that the growth is not speculative but is the result of network architecture.

- Tests and Stability:
- No-scale tests: Bitcoin alone shows a straight line across time ratios, unlike traditional assets.
- Sequential Bayesian stability: $$\beta$$ converges to 5.73 as uncertainty shrinks as $$1/\sqrt{n}$$, without structural breaks.
- Residuals: The stable log-normal distribution remains, with 4-year cycles and no increase in variance spread.

- Conditions for Bitcoin power-law failure:
1. Collapse below the lower bound by more than 3 standard deviations for a year.
2. Address growth falls below 3.
3. Persistent bias in $$\beta$$ outside [5.0, 7.0].
4. Price decouples from addresses ($$R^2 < 0.7$$).
5. $$R^2$$ collapses below 0.80 for two years.

- Conclusion:
We demonstrated that Bitcoin’s price strongly follows a power law $$P_t \propto t^{5.69}$$ over 15 years, with a composite coefficient $$\alpha \times \mu = 5.60$$. This links price dynamics to universal expansion diffusion mechanisms and network value quantities, indicating that long-term growth is a mathematical inevitability of network engineering rather than speculation. The framework predicts changes in $$\beta$$ with network evolution and provides testable benchmarks.

$BTC $BTC ‌
#GateSquareAprilPostingChallenge #MarchNonfarmPayrollsIncoming