{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/multifractality-in-random-networks-with-power","title":"Multifractality in random networks with power-law decaying bond strengths","arxiv_id":"1903.11733","date":"2019-03-27","proceeding":null,"authors":["Didier A. Vega-Oliveros","J. A. Méndez-Bermúdez","Francisco A. Rodrigues"],"abstract":"In this paper we demonstrate numerically that random networks whose adjacency matrices ${\\bf A}$ are represented by a diluted version of the Power--Law Banded Random Matrix (PBRM) model have multifractal eigenfunctions. The PBRM model describes one--dimensional samples with random long--range bonds. The bond strengths of the model, which decay as a power--law, are tuned by the parameter $\\mu$ as $A_{mn}\\propto |m-n|^{-\\mu}$; while the sparsity is driven by the average network connectivity $\\alpha$: for $\\alpha=0$ the vertices in the network are isolated and for $\\alpha=1$ the network is fully connected and the PBRM model is recovered. Though it is known that the PBRM model has multifractal eigenfunctions at the critical value $\\mu=\\mu_c=1$, we clearly show [from the scaling of the relative fluctuation of the participation number $I_2$ as well as the scaling of the probability distribution functions $P(\\ln I_2)$] the existence of the critical value $\\mu_c\\equiv \\mu_c(\\alpha)$ for $\\alpha<1$. Moreover, we characterise the multifractality of the eigenfunctions of our random network model by the use of the corresponding multifractal dimensions $D_q$, that we compute from the finite network-size scaling of the typical eigenfunction participation numbers $\\exp\\left\\langle\\ln I_q \\right\\rangle$.","url_abs":"https://arxiv.org/abs/1903.11733v1","url_pdf":"https://arxiv.org/pdf/1903.11733v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"multifractality-in-random-networks-with-power","repo_url":"https://github.com/didiervega/RandomMatrix","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}