Activity-Warped Power Laws for Bitcoin Price

Activity-Warped Power Laws for Bitcoin Price | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Activity-Warped Power Laws for Bitcoin Price Carlos Baquero This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8845008/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 4 You are reading this latest preprint version 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. Bitcoin power law warped time on-chain metrics transaction volume volatility Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 15 Feb, 2026 Editor assigned by journal 13 Feb, 2026 Submission checks completed at journal 11 Feb, 2026 First submitted to journal 10 Feb, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8845008","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":589564055,"identity":"6c4168c6-7082-449c-855d-5ab452f3090c","order_by":0,"name":"Carlos Baquero","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAmUlEQVRIiWNgGAWjYDACCQbGBxWkamE2OANhMjYQq4VNgjQt/LO70yoOVDAkbmfgPf6AOEvunN1248AZhsSdDXyJxNliIJG77fbHNobEDQd4DInXUnDwH6laGA42kKIF6JfNEgeOSRjvbOZLnEGUFv7ZvRs/HKixkd3O3nvgA1FaYJYxGDDzkKIBBAwYSNYyCkbBKBgFIwUAAOFsMlAWchvvAAAAAElFTkSuQmCC","orcid":"","institution":"University of Porto","correspondingAuthor":true,"prefix":"","firstName":"Carlos","middleName":"","lastName":"Baquero","suffix":""}],"badges":[],"createdAt":"2026-02-10 20:10:22","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8845008/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8845008/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":102746031,"identity":"f30cc692-68a0-4d9a-b4c5-c87f2d6ac972","added_by":"auto","created_at":"2026-02-16 08:55:15","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":729524,"visible":true,"origin":"","legend":"","description":"","filename":"SpringerSubmission.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8845008/v1_covered_3c4be8d7-3dc5-4f85-ad61-6cd6cbefb0e6.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Activity-Warped Power Laws for Bitcoin Price","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"digital-finance","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"dfin","sideBox":"Learn more about [Digital Finance](http://link.springer.com/journal/42521)","snPcode":"42521","submissionUrl":"https://submission.nature.com/new-submission/42521/3","title":"Digital Finance","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Bitcoin, power law, warped time, on-chain metrics, transaction volume, volatility","lastPublishedDoi":"10.21203/rs.3.rs-8845008/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8845008/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eBitcoin's price history follows an approximate power law in time, with \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((R^2 = 0.947)\\)\u003c/span\u003e\u003c/span\u003e over 2011--2026. We show that replacing uniform calendar time with \u003cem\u003eactivity-warped time\u003c/em\u003e---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 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((w_t^\\gamma)\\)\u003c/span\u003e\u003c/span\u003e that reshapes the weight distribution: \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((\\gamma = 2.41)\\)\u003c/span\u003e\u003c/span\u003e for volatility (amplifying large-move days) and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((\\gamma = 0.56)\\)\u003c/span\u003e\u003c/span\u003e for transaction volume (compressing extreme spikes). Transaction volume emerges as the stronger signal, achieving \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((R^2 = 0.958)\\)\u003c/span\u003e\u003c/span\u003e in-sample and winning 8 of 9 walk-forward splits (mean \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((\\Delta R^2 = +0.414)\\)\u003c/span\u003e\u003c/span\u003e). Volatility wins 5 of 9 splits but requires no external data. Transaction volume selects \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((\\alpha = 0)\\)\u003c/span\u003e\u003c/span\u003e (pure warped time), while volatility retains a calendar component (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((\\alpha \\approx 0.4)\\)\u003c/span\u003e\u003c/span\u003e). Neither signal benefits from smoothing. Despite being nearly uncorrelated (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((r = -0.007)\\)\u003c/span\u003e\u003c/span\u003e), combining the two signals does not improve out-of-sample performance---each captures complementary but individually sufficient information about Bitcoin's growth dynamics.\u003c/p\u003e","manuscriptTitle":"Activity-Warped Power Laws for Bitcoin Price","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-02-12 04:19:37","doi":"10.21203/rs.3.rs-8845008/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-02-16T00:54:07+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-02-13T09:43:24+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-02-11T11:08:22+00:00","index":"","fulltext":""},{"type":"submitted","content":"Digital Finance","date":"2026-02-10T19:57:06+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"digital-finance","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"dfin","sideBox":"Learn more about [Digital Finance](http://link.springer.com/journal/42521)","snPcode":"42521","submissionUrl":"https://submission.nature.com/new-submission/42521/3","title":"Digital Finance","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"4bc195b3-d4c7-4bbf-84b7-8611246777ec","owner":[],"postedDate":"February 12th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-04-22T14:10:16+00:00","versionOfRecord":[],"versionCreatedAt":"2026-02-12 04:19:37","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8845008","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8845008","identity":"rs-8845008","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}