{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:14:55Z","timestamp":1760242495502,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2017,7,26]],"date-time":"2017-07-26T00:00:00Z","timestamp":1501027200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Many out-of-equilibrium systems respond to external driving with nonlinear and self-similar dynamics. This near scale-invariant behavior of relaxation events has been modeled through sand pile cellular automata. However, a common feature of these models is the assumption of a local connectivity, while in many real systems, we have evidence for longer range connectivity and a complex topology of the interacting structures. Here, we investigate the role that longer range connectivity might play in near scale-invariant systems, by analyzing the results of a sand pile cellular automaton model on a Newman\u2013Watts network. The analysis clearly indicates the occurrence of a crossover phenomenon in the statistics of the relaxation events as a function of the percentage of longer range links and the breaking of the simple Finite Size Scaling (FSS). The more complex nature of the dynamics in the presence of long-range connectivity is investigated in terms of multi-scaling features and analyzed by the Rank-Ordered Multifractal Analysis (ROMA).<\/jats:p>","DOI":"10.3390\/e19080383","type":"journal-article","created":{"date-parts":[[2017,7,26]],"date-time":"2017-07-26T11:45:01Z","timestamp":1501069501000},"page":"383","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Avalanching Systems with Longer Range Connectivity: Occurrence of a Crossover Phenomenon and Multifractal Finite Size Scaling"],"prefix":"10.3390","volume":"19","author":[{"given":"Simone","family":"Benella","sequence":"first","affiliation":[{"name":"Dipartimento di Scienze Pure ed Applicate, Universit\u00e0 degli Studi di Urbino Carlo Bo, 61029 Urbino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3403-647X","authenticated-orcid":false,"given":"Giuseppe","family":"Consolini","sequence":"additional","affiliation":[{"name":"INAF-Istituto di Astrofisica e Planetologia Spaziali, 00133 Roma, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fabio","family":"Giannattasio","sequence":"additional","affiliation":[{"name":"Istituto Nazionale di Geofisica e Vulcanologia, 00143 Roma, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tom","family":"Chang","sequence":"additional","affiliation":[{"name":"Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology (MIT), Cambridge, MA 02139, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marius","family":"Echim","sequence":"additional","affiliation":[{"name":"Royal Belgian Institute for Space Aeronomy, B-1180 Brussels, Belgium"},{"name":"Institute of Space Science, 077125 Magurele, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,7,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1103\/PhysRevLett.59.381","article-title":"Self-organized criticality: An explanation of 1\/f noise","volume":"59","author":"Bak","year":"1987","journal-title":"Phys. Rev. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Sornette, D. (2001). Critical Phenomena in Natural Sciences, Springer-Verlag.","DOI":"10.1007\/978-3-662-04174-1"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/s11214-015-0155-x","article-title":"25 Years of Self-organized Criticality: Concepts and Cotroversies","volume":"198","author":"Watkins","year":"2016","journal-title":"Space Sci. Rev."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Jensen, H.J. (1998). Self-Organized Criticality: Emergent Complex Behavior in Physical and Biological Systems, Cambridge University Press.","DOI":"10.1017\/CBO9780511622717"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1377","DOI":"10.1088\/0034-4885\/62\/10\/201","article-title":"Self-organized criticality","volume":"62","author":"Turcotte","year":"1999","journal-title":"Rep. Prog. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Aschwanden, M.J. (2011). Self-Organized Criticality in Astrophysics, Springer-Verlag.","DOI":"10.1007\/978-3-642-15001-2"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"L89","DOI":"10.1086\/186180","article-title":"Avalanches and the distribution of solar flares","volume":"380","author":"Lu","year":"1991","journal-title":"Astrophys. J."},{"key":"ref_8","unstructured":"Consolini, G. (1997, January 8\u201310). Sandpile cellular automata and magnetospheric dynamics. Proceedings of the 8th GIFCO Conference, Cosmic Physics in the Year 2000: Scientific Perspectives and New Instrumentation, Como, Italy."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1142\/S0218348X02001397","article-title":"Self-Organized Criticality: A new paradigm for the magnetotail dynamics","volume":"10","author":"Consolini","year":"2002","journal-title":"Fractals"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1580","DOI":"10.1007\/s00585-998-1580-x","article-title":"Low frequency 1\/f-like fluctuations of the AE-index as a possible manifestation of self-organized criticality in the magnetosphere","volume":"16","author":"Uritsky","year":"1998","journal-title":"Ann. Geophys."},{"key":"ref_11","first-page":"1426","article-title":"Scale-free statistics of spatiotemporal auroral emissions as depicted by POLAR UVI images: Dynamic magnetosphere is an avalanching system","volume":"1078","author":"Uritsky","year":"2002","journal-title":"J. Geophys. Res."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1047","DOI":"10.1086\/308867","article-title":"Time Variability of the \u201cQuiet\u201d Sun Observed with TRACE. II. Physical Parameters, Temperature Evolution, and Energetics of Extreme-Ultraviolet Nanoflares","volume":"535","author":"Aschwanden","year":"2000","journal-title":"Astrophys. J."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Chang, T.T.S. (2015). An Introduction to Space Plasma Complexity, Cambridge University Press.","DOI":"10.1017\/CBO9780511980251"},{"key":"ref_14","first-page":"A04108","article-title":"Long-range magnetic couplings between solar flares and coronal mass ejections observed by SDO and STEREO","volume":"116","author":"Schrijver","year":"2011","journal-title":"J. Geophys. Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1016\/j.physrep.2013.11.002","article-title":"Power laws and self-organized criticality in theory and nature","volume":"536","author":"Gros","year":"2014","journal-title":"Phys. Rep."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1016\/S0378-4371(02)00549-6","article-title":"Self-organized criticality on small world networks","volume":"308","author":"Herrmann","year":"2002","journal-title":"Physica A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1016\/j.physa.2004.10.024","article-title":"Sandpiles on Watts-Strogatz type small-words","volume":"349","author":"Lahtinen","year":"2005","journal-title":"Physica A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1038\/30918","article-title":"Collective dynamics of \u201csmall world\u201d networks","volume":"393","author":"Watts","year":"1998","journal-title":"Nature"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"195001","DOI":"10.1088\/1751-8113\/46\/19\/195001","article-title":"Critical behavior of a small world sand pile model","volume":"46","author":"Hoore","year":"2013","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"832","DOI":"10.1103\/PhysRev.185.832","article-title":"Bounded and Inhomogeneous Ising Models. I. Specific-Heat Anomaly of a Finite Lattice","volume":"185","author":"Ferdinand","year":"1969","journal-title":"Phys. Rev."},{"key":"ref_21","unstructured":"Green, M.S. (1971). Critical Phenomena. Proc. 51st Enrico Fermi Summer School, Academic Press."},{"key":"ref_22","unstructured":"Cardy, J.L. (1988). Finite-size Scaling. Current Physics Sources and Comments, Elsevier."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Christensen, K., and Moloney, N.R. (2005). Complexity and Criticality, Imperial College Press.","DOI":"10.1142\/p365"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"L363","DOI":"10.1088\/0305-4470\/24\/7\/009","article-title":"Two-state model of self-organized criticality","volume":"24","author":"Manna","year":"1991","journal-title":"J. Phys. A"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"045401(R)","DOI":"10.1103\/PhysRevE.77.045401","article-title":"Rank-ordered multifractal spectrum for intermittent fluctuations","volume":"77","author":"Chang","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"4137","DOI":"10.1063\/1.873678","article-title":"Self-organized criticality, multi-fractal spectra, sporadic localized reconnections and multiscale intermittent turbulence in the magnetotail","volume":"6","author":"Chang","year":"1999","journal-title":"Phys. Plasmas"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1023\/A:1005252807049","article-title":"Magnetotail filed topology and criticality in geotail dynamics: Relevance to substorm phenomena","volume":"95","author":"Consolini","year":"2001","journal-title":"Space Sci. Rev."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1103\/PhysRevA.38.364","article-title":"Self-organized criticality","volume":"38","author":"Bak","year":"1988","journal-title":"Phys. Rev. A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/S0375-9601(99)00757-4","article-title":"Renormalization group analysis of the small world network model","volume":"263","author":"Newman","year":"1999","journal-title":"Phys. Lett. A"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1016\/S0010-4655(99)00338-0","article-title":"Critical exponents in stochastic sand pile models","volume":"121\u2013122","author":"Chessa","year":"1999","journal-title":"Comput. Phys. Commun."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"R2677","DOI":"10.1103\/PhysRevE.58.R2677","article-title":"Rare events and breakdown of simple scaling in the Abelian sand pile model","volume":"58","author":"Stella","year":"1998","journal-title":"Phys. Rev. E"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1140\/epjb\/e2010-00198-x","article-title":"Finite size scaling in BTW like sand pile models","volume":"76","author":"Ahmed","year":"2010","journal-title":"Eur. Phys. J. B"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"4375","DOI":"10.1103\/PhysRevE.57.4375","article-title":"Temporally disordered granular flow: A model of ladslides","volume":"57","year":"1998","journal-title":"Phys. Rev. E"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"3952","DOI":"10.1103\/PhysRevLett.83.3952","article-title":"Multifractal Scaling in the Bak-Tang-Wiesenfeld Sandpile and Edge Events","volume":"83","author":"Tebaldi","year":"1999","journal-title":"Phys. Rev. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1103\/PhysRevE.61.204","article-title":"Moment analysis of the probability distribution of different sand pile models","volume":"61","year":"2000","journal-title":"Phys. Rev. E"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1275","DOI":"10.1103\/RevModPhys.80.1275","article-title":"Critical phenomena in complex networks","volume":"80","author":"Dorogovtsev","year":"2008","journal-title":"Rev. Mod. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"566","DOI":"10.1016\/j.physa.2003.10.007","article-title":"Information super-diffusion on structured networks","volume":"332","author":"Thurner","year":"2004","journal-title":"Physica A"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1007\/BF00874478","article-title":"Multifractal measures, especially for geophysicist","volume":"131","author":"Mandelbrot","year":"1989","journal-title":"Pure Appl. Geophys."},{"key":"ref_39","first-page":"341","article-title":"On the properties of small world network models","volume":"263","author":"Barrat","year":"2000","journal-title":"Eur. Phys. J."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1399","DOI":"10.1016\/S1364-6826(00)00241-8","article-title":"Substorms as nonequilibrium transitions of the magnetosphere","volume":"63","author":"Sharma","year":"2001","journal-title":"J. Atmos. Sol. Terr. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"16116","DOI":"10.1103\/PhysRevE.65.016116","article-title":"Modeling substorm dynamics of the magnetosphere: From self-organization and self-organized criticality to nonequilibrium phase transitions","volume":"65","author":"Sitnov","year":"2001","journal-title":"Phys. Rev. E"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/8\/383\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:44:03Z","timestamp":1760208243000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/8\/383"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,7,26]]},"references-count":41,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2017,8]]}},"alternative-id":["e19080383"],"URL":"https:\/\/doi.org\/10.3390\/e19080383","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2017,7,26]]}}}