A Mechanistic Derivation of the Bitcoin Price Power Law: Network Adoption Dynamics and Generalised Metcalfe Scaling

Abstract

We demonstrate that the price of Bitcoin follows a robust power law in time,P(t) ∼ tβ, with exponent β = 5.69±0.05 and coefficient of determination R2 = 0.961over a dataset spanning 5,696 daily observations from July 2010 to February 2026(days 560–6,255 since the Genesis Block on 3 January 2009). Crucially, we showthat this exponent is not a free parameter but is determined by two independentlyestablished physical mechanisms acting in composition. First, the count of non-zerobalanceBitcoin addresses grows as N(t) ∼ tβA with βA = 3.046±0.012 (R2 = 0.977),a cubic scaling consistent with epidemic spreading on a heterogeneous scale-freenetwork (Colgate et al., 1989; Bacchetti & Koch, 1989). Second, price scales withaddress count as P ∼ NβM with βM = 1.838 ± 0.031 (R2 = 0.951), a generalisedMetcalfe-type network value law (Metcalfe 1983; Zhang et al., 2015). The compositionidentity β = βA×βM = 3.046×1.838 = 5.60 agrees with the directly measuredβobs = 5.69 to within 1.6%, with the small residual attributable to measurementvariance in the component exponents. This factorisation connects the global pricetrajectory to two well-studied universality classes: spreading processes on heterogeneouscontact networks, and value scaling in communication networks. The lognormalresiduals from the power law fit (σ = 0.302 dex, zero mean) are stationaryand exhibit no secular drift, confirming long-run stability of the scaling relation.We discuss the theoretical foundations of cubic adoption growth via the saturationwavemechanism, interpret the Metcalfe exponent through network economics, andexamine the falsifiability conditions under which the power law would be expected tobreak down. Three independent empirical tests of scale invariance are additionallyprovided: a multi-asset pair-ratio scaling test, a direct collapse test of the identityP(λt)/P(t) = λβ∗recovering β∗ = 5.59 from 5,298 price ratios, a rolling temporalstability analysis spanning 2011–2026 finding median β∗ = 5.73±0.58 with no seculardrift; and a sequential Bayesian analysis using 1,899 local scaling estimates thatyields a final posterior β ∼ N(5.729, 0.0132) with posterior uncertainty shrinkingas n−1/2 throughout 2010–2026 with no structural breaks. These results suggestthat Bitcoin’s price trajectory is not primarily driven by speculative dynamics butreflects the deterministic mathematical consequences of its network topology andvalue architecture.