Scaling theory of fractal complex networks: Bridging local self-similarity and global scale-invariance

Scaling theory of fractal complex networks: Bridging local self-similarity and global scale-invariance | 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 Article Scaling theory of fractal complex networks: Bridging local self-similarity and global scale-invariance Agata Fronczak, Piotr Fronczak, Mateusz Samsel, Kordian Makulski, and 2 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3093833/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 8 You are reading this latest preprint version Abstract We show that fractality in complex networks arises from the geometric self-similarity of their built-in hierarchical community-like structure, which is mathematically described by the scale-invariant equation for the masses of the boxes with which we cover the network when determining its box dimension. This approach - grounded in both scaling theory of phase transitions and renormalization group theory - leads to the consistent scaling theory of fractal complex networks, which reveals a collection of scaling exponents and different relationships between them. The exponents can be divided into two groups: microscopic (hitherto unknown) and macroscopic, characterizing respectively the local structure of fractal complex networks and their global properties. Interestingly, exponents from both groups are related to each other and only a few of them (three out of seven) are independent, thus bridging the gap between local self-similarity and global scale-invariance of fractal networks. We successfully verify our findings in real networks situated in various fields (information – the World Wide Web, biological – the human brain, and social – scientific collaboration networks) and in several fractal network models. Physical sciences/Physics/Statistical physics thermodynamics and nonlinear dynamics/Complex networks Physical sciences/Physics/Statistical physics thermodynamics and nonlinear dynamics/Statistical physics Full Text Additional Declarations No competing interests reported. Supplementary Files SMfractal.pdf Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 03 Nov, 2023 Reviews received at journal 15 Jul, 2023 Reviewers agreed at journal 02 Jul, 2023 Reviewers invited by journal 02 Jul, 2023 Editor assigned by journal 02 Jul, 2023 Editor invited by journal 02 Jul, 2023 Submission checks completed at journal 02 Jul, 2023 First submitted to journal 21 Jun, 2023 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. 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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-3093833","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":214965579,"identity":"1254cb36-673c-4677-b83d-cb52643dc1c1","order_by":0,"name":"Agata Fronczak","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABB0lEQVRIiWNgGAWjYPACOSh9wAZMSQCxDB7ljA0MDMYwLWlwLTzEajlMWAt//+LnD34wGOTzSzc//vDjzPk8gwPMB2/zMNzBqUXixjPDxh4GA8uZc44ZGPbcuF1scIAt2ZqH4Rluh904YNjAw/DHwOBGDkMCz4fbidsO8JhJ8zAcxqlF/sbxj41/GAzAWg7++XAOqIX/G14tBud7DJt5IFoYm3luHADZwoZXi+ENnsLZMkAdkjPSjJllziQX2x9mM7acY4DbL3Lnj2/4+KbCwIBfIvnxxzfH7PIk25sf3nhTcUcOlxYGiQSQ8xD8BAZmsIMP4NTBwI8mlwCl8WgZBaNgFIyCkQYAo19bGefzScsAAAAASUVORK5CYII=","orcid":"","institution":"Warsaw University of Technology","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Agata","middleName":"","lastName":"Fronczak","suffix":""},{"id":214965581,"identity":"cff4fa73-e768-4fed-869a-84503bdec519","order_by":1,"name":"Piotr Fronczak","email":"","orcid":"","institution":"Warsaw University of Technology","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Piotr","middleName":"","lastName":"Fronczak","suffix":""},{"id":214965583,"identity":"bd6c4a14-72f5-42b7-a155-baea7f04ef59","order_by":2,"name":"Mateusz Samsel","email":"","orcid":"","institution":"Warsaw University of Technology","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Mateusz","middleName":"","lastName":"Samsel","suffix":""},{"id":214965584,"identity":"aa14df93-a86a-45b6-b047-278b65b6812c","order_by":3,"name":"Kordian Makulski","email":"","orcid":"","institution":"Warsaw University of Technology","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Kordian","middleName":"","lastName":"Makulski","suffix":""},{"id":214965585,"identity":"c5ae4324-5417-4b3c-adae-089d46d23746","order_by":4,"name":"Michał Łepek","email":"","orcid":"","institution":"Warsaw University of Technology","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Michał","middleName":"","lastName":"Łepek","suffix":""},{"id":214965586,"identity":"8c84c6e4-67f0-4758-9dc0-615d9aa84734","order_by":5,"name":"Maciej J. 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