{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,26]],"date-time":"2026-07-26T03:33:50Z","timestamp":1785036830943,"version":"3.55.0"},"reference-count":49,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,9,11]],"date-time":"2023-09-11T00:00:00Z","timestamp":1694390400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"University of Rzeszow","award":["RPPK.01.03.00-18-001\/10"],"award-info":[{"award-number":["RPPK.01.03.00-18-001\/10"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Over the past three decades, describing the reality surrounding us using the language of complex networks has become very useful and therefore popular. One of the most important features, especially of real networks, is their complexity, which often manifests itself in a fractal or even multifractal structure. As a generalization of fractal analysis, the multifractal analysis of complex networks is a useful tool for identifying and quantitatively describing the spatial hierarchy of both theoretical and numerical fractal patterns. Nowadays, there are many methods of multifractal analysis. However, all these methods take into account only the fact of connection between nodes (and eventually the weight of edges) and do not take into account the real positions (coordinates) of nodes in space. However, intuition suggests that the geometry of network nodes\u2019 position should have a significant impact on the true fractal structure. Many networks identified in nature (e.g., air connection networks, energy networks, social networks, mountain ridge networks, networks of neurones in the brain, and street networks) have their own often unique and characteristic geometry, which is not taken into account in the identification process of multifractality in commonly used methods. In this paper, we propose a multifractal network analysis method that takes into account both connections between nodes and the location coordinates of nodes (network geometry). We show the results for different geometrical variants of the same network and reveal that this method, contrary to the commonly used method, is sensitive to changes in network geometry. We also carry out tests for synthetic as well as real-world networks.<\/jats:p>","DOI":"10.3390\/e25091324","type":"journal-article","created":{"date-parts":[[2023,9,11]],"date-time":"2023-09-11T08:58:08Z","timestamp":1694422688000},"page":"1324","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Multifractality of Complex Networks Is Also Due to Geometry: A Geometric Sandbox Algorithm"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9780-4205","authenticated-orcid":false,"given":"Rafa\u0142","family":"Rak","sequence":"first","affiliation":[{"name":"Institute of Physics, College of Natural Sciences, University of Rzesz\u00f3w, Pigonia 1, 35-310 Rzesz\u00f3w, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8470-1730","authenticated-orcid":false,"given":"Ewa","family":"Rak","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, College of Natural Sciences, University of Rzesz\u00f3w, Pigonia 1, 35-310 Rzesz\u00f3w, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,11]]},"reference":[{"key":"ref_1","first-page":"17","article-title":"On the evolution of random graphs","volume":"5","author":"Erdos","year":"1960","journal-title":"Magy. Tud. Akad. Mat. Kutat\u00f3 Int. K\u00f6zl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1038\/335405a0","article-title":"Multifractal phenomena in physics and chemistry","volume":"335","author":"Stanley","year":"1988","journal-title":"Nature"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"A05214","DOI":"10.1029\/2008JA013854","article-title":"Multifractal analysis of geomagnetic storm and solar flare indices and their class dependence","volume":"114","author":"Yu","year":"2009","journal-title":"J. Geophys. Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"106603","DOI":"10.1103\/PhysRevLett.122.106603","article-title":"Many-body multifractality throughout bosonic superfluid and mott insulator phases","volume":"122","author":"Lindinger","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1038\/20924","article-title":"Multifractality in human heartbeat dynamics","volume":"399","author":"Ivanov","year":"1999","journal-title":"Nature"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"031903","DOI":"10.1103\/PhysRevE.64.031903","article-title":"Measure representation and multifractal analysis of complete genomes","volume":"64","author":"Yu","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"021913","DOI":"10.1103\/PhysRevE.68.021913","article-title":"Multifractal and correlation analyses of protein sequences from complete genomes","volume":"68","author":"Yu","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1038\/35036627","article-title":"The large-scale organization of metabolic networks","volume":"407","author":"Jeong","year":"2000","journal-title":"Nature"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1038\/35075138","article-title":"Lethality and centrality in protein networks","volume":"411","author":"Jeong","year":"2001","journal-title":"Nature"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1111\/j.1475-3995.2000.tb00204.x","article-title":"Cointegration of stochastic multifractals with application to foreign exchange rates","volume":"7","author":"Anh","year":"2000","journal-title":"Int. Trans. Op. Res."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1140\/epjb\/e2012-20570-0","article-title":"A multifractal analysis of Asian foreign exchange markets","volume":"85","author":"Oh","year":"2012","journal-title":"Eur. Phys. J. B"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1145\/316194.316229","article-title":"On power-law relationships of the Internet topology","volume":"29","author":"Faloutsos","year":"2008","journal-title":"Comput. Commun. Rev."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"770","DOI":"10.1109\/90.650138","article-title":"A quantitative comparison of graph-based models for Internet topology","volume":"5","author":"Zegura","year":"1997","journal-title":"IEEE\/ACM Trans. Netw."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"258701","DOI":"10.1103\/PhysRevLett.87.258701","article-title":"Dynamical and correlation properties of the Internet","volume":"87","author":"Vazquez","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1038\/43601","article-title":"Diameter of the world-wide web","volume":"401","author":"Albert","year":"1999","journal-title":"Nature"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1038\/35012148","article-title":"Souped-up search engines","volume":"405","author":"Butler","year":"2000","journal-title":"Nature"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1016\/S1389-1286(00)00083-9","article-title":"Graph structure in the web","volume":"33","author":"Broder","year":"2000","journal-title":"Comput. Netw."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"907","DOI":"10.1038\/35082140","article-title":"The web of human sexual contacts","volume":"411","author":"Liljeros","year":"2001","journal-title":"Nature"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"5616","DOI":"10.1073\/pnas.1410931111","article-title":"Links that speak: The global language network and its association with global fame","volume":"111","author":"Ronen","year":"2014","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1073\/pnas.98.2.404","article-title":"The structure of scientific collaboration networks","volume":"98","author":"Newman","year":"2001","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"590","DOI":"10.1016\/S0378-4371(02)00736-7","article-title":"Evolution of the social network of scientific collaborations","volume":"311","author":"Barabasi","year":"2002","journal-title":"Phys. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1114","DOI":"10.1016\/j.joi.2017.09.009","article-title":"Hierarchical organization of H. Eugene Stanley scientific collaboration community in weighted network representation","volume":"11","author":"Kulig","year":"2017","journal-title":"J. Informetr."},{"key":"ref_23","first-page":"cnz017","article-title":"Universal features of mountain ridge networks on Earth","volume":"8","author":"Rak","year":"2020","journal-title":"J. Complex Netw."},{"key":"ref_24","unstructured":"Mandelbrot, B.B. (1982). The Fractal Geometry of Nature, W.H. Freeman."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"392","DOI":"10.1038\/nature03248","article-title":"Self-similarity of complex networks","volume":"433","author":"Song","year":"2005","journal-title":"Nature"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1038\/nphys266","article-title":"Origins of fractality in the growth of complex networks","volume":"2","author":"Song","year":"2006","journal-title":"Nat. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2825","DOI":"10.1073\/pnas.1106612109","article-title":"A small world of weak ties provides optimal global integration of self-similar modules in functional brain networks","volume":"109","author":"Gallos","year":"2012","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"114003","DOI":"10.7566\/JPSJ.84.114003","article-title":"Fractal and small-world networks formed by self-organized critical dynamics","volume":"84","author":"Watanabe","year":"2015","journal-title":"J. Phys. Soc. Jpn."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1126\/science.286.5439.509","article-title":"Emergence of scaling in random networks","volume":"286","author":"Albert","year":"1999","journal-title":"Science"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Rak, R., and Rak, E. (2020). The Fractional Preferential Attachment Scale-Free Network Model. Entropy, 22.","DOI":"10.3390\/e22050509"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"27602","DOI":"10.1038\/srep27602","article-title":"A statistical physics characterization of the complex systems dynamics: Quantifying complexity from spatiotemporal interactions","volume":"6","author":"Koorehdavoudi","year":"2016","journal-title":"Sci. Rep."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"12416","DOI":"10.1038\/s41598-018-30654-9","article-title":"Quantifying emergence and self-organisation of enterobacter cloacae microbial communities","volume":"8","author":"Balaban","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"036118","DOI":"10.1103\/PhysRevE.84.036118","article-title":"Multifractality of complex networks","volume":"84","author":"Furuya","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Song, C., Gallos, L.K., Havlin, S., and Makse, H.A. (2007). How to calculate the fractal dimension of a complex network: The box covering algorithm. J. Stat. Mech., P03006.","DOI":"10.1088\/1742-5468\/2007\/03\/P03006"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"080504","DOI":"10.1088\/1674-1056\/21\/8\/080504","article-title":"Multifractal analysis of complex networks","volume":"21","author":"Wang","year":"2012","journal-title":"Chin. Phys. B"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Li, B., Yu, Z., and Zhou, Y. (2014). Fractal and multifractal properties of a family of fractal networks. J. Stat. Mech., P02020.","DOI":"10.1088\/1742-5468\/2014\/02\/P02020"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"026116","DOI":"10.1063\/1.2737827","article-title":"A box-covering algorithm for fractal scaling in scale-free networks","volume":"17","author":"Kim","year":"2007","journal-title":"Chaos"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"016110","DOI":"10.1103\/PhysRevE.75.016110","article-title":"Fractality in complex networks: Critical and supercritical skeletons","volume":"75","author":"Kim","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"L835","DOI":"10.1088\/0305-4470\/20\/13\/005","article-title":"Geometrical multifractality of growing structures","volume":"20","author":"Vicsek","year":"1987","journal-title":"J. Phys. A"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/0378-4371(89)90563-3","article-title":"Determination of fractal dimensions for geometrical multifractals","volume":"159","author":"Vicsek","year":"1989","journal-title":"Phys. A"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"023103","DOI":"10.1063\/1.4907557","article-title":"Determination of multifractal dimensions of complex net-works by means of the sandbox algorithm","volume":"25","author":"Liu","year":"2015","journal-title":"Chaos"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"17628","DOI":"10.1038\/srep17628","article-title":"Multifractal analysis of weighted networks by a modified sandbox algorithm","volume":"5","author":"Song","year":"2015","journal-title":"Sci. Rep."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"7487","DOI":"10.1038\/s41598-017-07209-5","article-title":"Reliable multi-fractal characterization of weighted complex networks: Algorithms and implications","volume":"7","author":"Xue","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"043303","DOI":"10.1103\/PhysRevE.103.043303","article-title":"Computationally efficient sandbox al- gorithm for multifractal analysis of large-scale complex networks with tens of millions of nodes","volume":"103","author":"Ding","year":"2021","journal-title":"Phys. Rev. E"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"111836","DOI":"10.1016\/j.chaos.2022.111836","article-title":"Sandbox fixed-mass algorithm for multifractal unweighted complex networks","volume":"156","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_46","unstructured":"(2022, September 10). Available online: http:\/\/szhorvat.net\/pelican\/."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1129","DOI":"10.1002\/spe.4380211102","article-title":"Graph Drawing by Force-directed Placement","volume":"21","author":"Fruchterman","year":"1991","journal-title":"Softw. Pract. Exp."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1109\/TSE.1981.234519","article-title":"Tidier drawing of trees","volume":"SE-7","author":"Reingold","year":"1981","journal-title":"IEEE Trans. Softw. Eng."},{"key":"ref_49","unstructured":"(2022, September 10). Available online: https:\/\/igraph.org\/c\/doc\/igraph-Layout.html."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/9\/1324\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:48:54Z","timestamp":1760129334000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/9\/1324"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,11]]},"references-count":49,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2023,9]]}},"alternative-id":["e25091324"],"URL":"https:\/\/doi.org\/10.3390\/e25091324","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,11]]}}}