{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T21:52:35Z","timestamp":1778622755808,"version":"3.51.4"},"reference-count":43,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,2,8]],"date-time":"2025-02-08T00:00:00Z","timestamp":1738972800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Secretar\u00eda de Investigaci\u00f3n y Posgrado (SIP) of the Insituto Polit\u00e9cnico Nacional\u2014M\u00e9xico"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The use of the Visibility Graph Algorithm (VGA) has proven to be a valuable tool for analyzing both real and synthetic seismicity series. Specifically, VGA transforms time series into a network representation in which structural properties such as node connectivity, clustering, and community structure can be quantitatively measured, thereby revealing underlying correlations and dynamics that may remain hidden in traditional linear or spectral analyses. The time series transformation into complex networks with VGA provides a new approach to analyze seismic dynamics, allowing scientists to extract trends and behaviors that may not be possible by classical time-series analysis. On the other hand, many studies attempt to find viable trends in order to identify preparation mechanisms prior to a strong earthquake or to analyze the aftershocks. In this work, the seismic activity of Southern California Earthquake was analyzed focusing only on the significant earthquakes. For this purpose, seismic series preceding and following each earthquake were constructed using a windowing method with different overlaps and the slope of the connectivity (k) versus magnitude (M) graph (k-M slope) and the average degree were computed from the mapped complex networks. The results revealed a significant decrease in these parameters after the earthquake, due to the contribution of the aftershocks from the main event. Interestingly, the study was extended to synthetic seismicity series and the same behavior was observed for both k-M slope and average degree. This finding suggests that the spring-block model reproduces a relaxation mechanism following a large-magnitude event like those of real seismic aftershocks. However, this conclusion contrasts with conclusions drawn by other researchers. These results highlight the utility of VGA in studying events that precede and follow major earthquakes. This technique may be used to extract some useful trends in seismicity, which could eventually be employed for a deeper understanding and possible forecasting of seismic behavior.<\/jats:p>","DOI":"10.3390\/e27020178","type":"journal-article","created":{"date-parts":[[2025,2,10]],"date-time":"2025-02-10T05:53:22Z","timestamp":1739166802000},"page":"178","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Analysis of Aftershocks from California and Synthetic Series by Using Visibility Graph Algorithm"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9872-8304","authenticated-orcid":false,"given":"Alejandro","family":"Mu\u00f1oz-Diosdado","sequence":"first","affiliation":[{"name":"Unidad Profesional Interdisciplinaria de Biotecnolog\u00eda, Instituto Polit\u00e9cnico Nacional, Mexico City 07340, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0367-2366","authenticated-orcid":false,"given":"Ana Mar\u00eda","family":"Aguilar-Molina","sequence":"additional","affiliation":[{"name":"Unidad Profesional Interdisciplinaria de Biotecnolog\u00eda, Instituto Polit\u00e9cnico Nacional, Mexico City 07340, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3903-5724","authenticated-orcid":false,"given":"Eric Eduardo","family":"Solis-Montufar","sequence":"additional","affiliation":[{"name":"Unidad Profesional Interdisciplinaria de Biotecnolog\u00eda, Instituto Polit\u00e9cnico Nacional, Mexico City 07340, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2946-3899","authenticated-orcid":false,"given":"Jos\u00e9 Alberto","family":"Zamora-Justo","sequence":"additional","affiliation":[{"name":"Unidad Profesional Interdisciplinaria de Biotecnolog\u00eda, Instituto Polit\u00e9cnico Nacional, Mexico City 07340, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.physrep.2012.01.007","article-title":"Physical approach to complex systems","volume":"515","year":"2012","journal-title":"Phys. Rep."},{"key":"ref_2","first-page":"031105","article-title":"Nonconservative earthquake model of self-organized criticality on a random graph","volume":"102","author":"Smith","year":"2020","journal-title":"Phys. Rev. E"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1016\/0040-1951(90)90298-M","article-title":"Self-organized criticality and earthquakes","volume":"179","author":"Sornette","year":"1990","journal-title":"Tectonophysics"},{"key":"ref_4","unstructured":"Kanamori, H., and Boschi, E. (1983). Earthquakes: Observation, Theory and Interpretation, Elsevier."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1103\/PhysRevLett.59.381","article-title":"Self-organized criticality: An explanation of the 1\/f noise","volume":"59","author":"Bak","year":"1987","journal-title":"Phys. Rev. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"4006","DOI":"10.1103\/PhysRevLett.84.4006","article-title":"Self-Organized Criticality in the Olami-Feder-Christensen Model","volume":"84","author":"Prado","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Bak, P. (1996). How Nature Works: The Science of Self-Organized Criticality, Copernicus.","DOI":"10.1007\/978-1-4757-5426-1"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"8729","DOI":"10.1029\/92JB00427","article-title":"Variation of the Gutenberg-Richter b values and nontrivial temporal correlations in a Spring-Block Model for earthquakes","volume":"97","author":"Christensen","year":"1992","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"15635","DOI":"10.1029\/JB094iB11p15635","article-title":"Earthquakes as a self-organized critical phenomenon","volume":"94","author":"Bak","year":"1989","journal-title":"J. Geophys. Res."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1103\/RevModPhys.74.47","article-title":"Statistical mechanics of complex networks","volume":"74","year":"2002","journal-title":"Rev. Mod. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Newman, M.E.J. (2010). Networks: An Introduction, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780199206650.003.0001"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Telesca, L., Lovallo, M., Ram\u00edrez-Rojas, A., and Flores-Marquez, L. (2014). Relationship between the Frequency Magnitude Distribution and the Visibility Graph in the Synthetic Seismicity Generated by a Simple Stick-Slip System with Asperities. PLoS ONE, 9.","DOI":"10.1371\/journal.pone.0106233"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"454","DOI":"10.1140\/epjb\/e2013-40762-2","article-title":"Earthquake magnitude time series: Scaling behavior of visibility networks","volume":"86","year":"2013","journal-title":"Eur. Phys. J. B"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"238701","DOI":"10.1103\/PhysRevLett.96.238701","article-title":"Complex network from pseudoperiodic time series: Topology versus dynamics","volume":"96","author":"Zhang","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1016\/j.physrep.2006.11.001","article-title":"Recurrence plots for the analysis of complex systems","volume":"438","author":"Marwan","year":"2007","journal-title":"Phys. Rep."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"50004","DOI":"10.1209\/0295-5075\/103\/50004","article-title":"Recurrence networks from multivariate signals for uncovering dynamic transitions of horizontal oil-water stratified flows","volume":"103","author":"Gao","year":"2013","journal-title":"Europhys. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"4972","DOI":"10.1073\/pnas.0709247105","article-title":"From time series to complex networks: The visibility graph","volume":"105","author":"Lacasa","year":"2008","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Mu\u00f1oz-Diosdado, A., Sol\u00eds-Montufar, E.E., and Zamora-Justo, J.A. (2023). Visibility Graph Analysis of Heartbeat Time Series: Comparison of Young vs. Old, Healthy vs. Diseased, Rest vs. Exercise, and Sedentary vs. Active. Entropy, 25.","DOI":"10.3390\/e25040677"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Zhu, G., Li, Y., and Wen, P.P. (2012, January 4\u20137). An efficient visibility graph similarity algorithm and its application on sleep stages classification. Proceedings of the International Conference on Brain Informatics, Macau, China.","DOI":"10.1007\/978-3-642-35139-6_18"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Nilanjana, P., Anirban, B., Susmita, B., and Dipak, G. (2016). Noninvasive alarm generation for sudden cardiac arrest: A pilot study with visibility graph technique. Transl. Biomed., 7.","DOI":"10.21767\/2172-0479.100079"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1016\/j.neuroimage.2017.05.047","article-title":"Horizontal visibility graph transfer entropy (HVG-TE): A novel metric to characterize directed connectivity in large-scale brain networks","volume":"156","author":"Yu","year":"2017","journal-title":"NeuroImage"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Bhaduri, A., and Ghosh, D. (2016). Quantitative Assessment of Heart Rate Dynamics during Meditation: An ECG Based Study with Multi-Fractality and Visibility Graph. Front. Physiol., 7.","DOI":"10.3389\/fphys.2016.00044"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"50002","DOI":"10.1209\/0295-5075\/97\/50002","article-title":"Analysis of seismic sequences by using the method of visibility graph","volume":"97","author":"Telesca","year":"2012","journal-title":"Europhys. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"6571","DOI":"10.1016\/j.physa.2013.08.078","article-title":"Investigating the time dynamics of seismicity by using the visibility graph approach: Application to seismicity of Mexican subduction zone","volume":"392","author":"Telesca","year":"2013","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/j.physa.2014.08.048","article-title":"Visibility graph analysis of 2002\u20132011 Pannonian seismicity","volume":"416","author":"Telesca","year":"2014","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"4003","DOI":"10.1007\/s00024-017-1617-8","article-title":"Analysis of the 2005\u20132016 Earthquake Sequence in Northern Iran Using the Visibility Graph Method","volume":"174","author":"Khoshnevis","year":"2017","journal-title":"Pure Appl. Geophys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"4241","DOI":"10.1007\/s00024-018-1947-1","article-title":"Visibility Graph Analysis of Alaska Crustal and Aleutian Subduction Zone Seismicity: An Investigation of the Correlation between b Value and k\u2013M Slope","volume":"175","author":"Cramer","year":"2018","journal-title":"Pure Appl. Geophys."},{"key":"ref_28","first-page":"889","article-title":"Some Common Features Between a Spring-Block Self-Organized Critical Model, Stick\u2013Slip Experiments with Sandpapers and Actual Seismicity","volume":"177","year":"2019","journal-title":"Pure Appl. Geophys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"178501","DOI":"10.1103\/PhysRevLett.88.178501","article-title":"Unified Scaling Law for Earthquakes","volume":"88","author":"Bak","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1029\/2004GL020892","article-title":"Are seismic waiting time distributions universal?","volume":"31","author":"Davidsen","year":"2004","journal-title":"Geophys. Res. Lett."},{"key":"ref_31","unstructured":"Azizzadeh-Roodpish, S., Khoshnevis, N., and Cramer, C.H. (2017, January 18\u201320). Visibility graph analysis of southern California. Proceedings of the Proceedings Annual Meeting of the Seismological Society of America, Denver, CO, USA."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"5041","DOI":"10.1016\/j.physa.2012.05.049","article-title":"Visibility graph analysis of wind speed records measured in central Argentina","volume":"391","author":"Pierini","year":"2012","journal-title":"Physica A"},{"key":"ref_33","first-page":"210","article-title":"California Earthquake History","volume":"48","author":"Toppozada","year":"2024","journal-title":"Calif. Geol. Surv."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"B03403","DOI":"10.1029\/2004JB003209","article-title":"Block models of crustal motion in southern California constrained by GPS measurements","volume":"110","author":"Meade","year":"2005","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_35","first-page":"1234","article-title":"Past and Future Earthquakes on the San Andreas Fault","volume":"129","author":"Weldon","year":"2024","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_36","first-page":"1234","article-title":"Major Earthquakes on the Southern San Andreas Fault Modulated by Lake Filling Events","volume":"51","author":"Hill","year":"2024","journal-title":"Geophys. Res. Lett."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"e2023JB026346","DOI":"10.1029\/2023JB026811","article-title":"Active Dipping Interface of the Southern San Andreas Fault Revealed by Space Geodetic and Seismic Imaging","volume":"128","author":"Vavra","year":"2023","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1244","DOI":"10.1103\/PhysRevLett.68.1244","article-title":"Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes","volume":"68","author":"Olami","year":"1992","journal-title":"Phys. Rev. Lett."},{"key":"ref_39","first-page":"8049","article-title":"The moment tensor of earthquakes: Theory and applications to the 1985 Michoac\u00e1n, Mexico, earthquake","volume":"93","year":"1988","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1038\/355071a0","article-title":"Changes in frequency\u2013size relationship from small to large earthquakes","volume":"355","author":"Pacheco","year":"1992","journal-title":"Nature"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Aguilar-Molina, A.M., Mu\u00f1oz-Diosdado, A., Salinas Mart\u00ednez, A., and Angulo-Brown, F. (2023). Multifractal Properties of Time Series of Synthetic Earthquakes Obtained from a Spring-Block Model. Entropy, 25.","DOI":"10.3390\/e25050773"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"093111","DOI":"10.1063\/5.0007480","article-title":"Visibility graph analysis of synthetic earthquakes generated by the Olami\u2013Feder\u2013Christensen spring-block model","volume":"30","author":"Lovallo","year":"2020","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1038\/30918","article-title":"Collective dynamics of \u2019small-world\u2019 networks","volume":"393","author":"Watts","year":"1998","journal-title":"Nature"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/2\/178\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:29:13Z","timestamp":1760027353000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/2\/178"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,8]]},"references-count":43,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,2]]}},"alternative-id":["e27020178"],"URL":"https:\/\/doi.org\/10.3390\/e27020178","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,2,8]]}}}