Activity-Warped Power Laws for Bitcoin Price

Abstract

Abstract Bitcoin's price history follows an approximate power law in time, with \((R^2 = 0.947)\) over 2011--2026. We show that replacing uniform calendar time with activity-warped time ---where time advances faster during high-activity periods---improves both in-sample fit and out-of-sample prediction. Two warping signals are evaluated: price volatility (absolute daily log-returns) and on-chain transaction volume (daily USD value transacted). Both benefit from a power transform \((w_t^\gamma)\) that reshapes the weight distribution: \((\gamma = 2.41)\) for volatility (amplifying large-move days) and \((\gamma = 0.56)\) for transaction volume (compressing extreme spikes). Transaction volume emerges as the stronger signal, achieving \((R^2 = 0.958)\) in-sample and winning 8 of 9 walk-forward splits (mean \((\Delta R^2 = +0.414)\)). Volatility wins 5 of 9 splits but requires no external data. Transaction volume selects \((\alpha = 0)\) (pure warped time), while volatility retains a calendar component (\((\alpha \approx 0.4)\)). Neither signal benefits from smoothing. Despite being nearly uncorrelated (\((r = -0.007)\)), combining the two signals does not improve out-of-sample performance---each captures complementary but individually sufficient information about Bitcoin's growth dynamics.