{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T15:46:17Z","timestamp":1768319177935,"version":"3.49.0"},"reference-count":57,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,12,4]],"date-time":"2024-12-04T00:00:00Z","timestamp":1733270400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Dutch National Police","award":["Hyperion 2454972"],"award-info":[{"award-number":["Hyperion 2454972"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Complex networks, from neuronal assemblies to social systems, can exhibit abrupt, system-wide transitions without external forcing. These endogenously generated \u201cnoise-induced transitions\u201d emerge from the intricate interplay between network structure and local dynamics, yet their underlying mechanisms remain elusive. Our study unveils two critical roles that nodes play in catalyzing these transitions within dynamical networks governed by the Boltzmann\u2013Gibbs distribution. We introduce the concept of \u201cinitiator nodes\u201d, which absorb and propagate short-lived fluctuations, temporarily destabilizing their neighbors. This process initiates a domino effect, where the stability of a node inversely correlates with the number of destabilized neighbors required to tip it. As the system approaches a tipping point, we identify \u201cstabilizer nodes\u201d that encode the system\u2019s long-term memory, ultimately reversing the domino effect and settling the network into a new stable attractor. Through targeted interventions, we demonstrate how these roles can be manipulated to either promote or inhibit systemic transitions. Our findings provide a novel framework for understanding and potentially controlling endogenously generated metastable behavior in complex networks. This approach opens new avenues for predicting and managing critical transitions in diverse fields, from neuroscience to social dynamics and beyond.<\/jats:p>","DOI":"10.3390\/e26121050","type":"journal-article","created":{"date-parts":[[2024,12,4]],"date-time":"2024-12-04T04:47:22Z","timestamp":1733287642000},"page":"1050","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Cascades Towards Noise-Induced Transitions on Networks Revealed Using Information Flows"],"prefix":"10.3390","volume":"26","author":[{"given":"Casper","family":"van Elteren","sequence":"first","affiliation":[{"name":"Institute of Informatics, University of Amsterdam, 1098 XH Amsterdam, The Netherlands"},{"name":"Institute for Advanced Study, 1012 GC Amsterdam, The Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0299-0074","authenticated-orcid":false,"given":"Rick","family":"Quax","sequence":"additional","affiliation":[{"name":"Institute of Informatics, University of Amsterdam, 1098 XH Amsterdam, The Netherlands"},{"name":"Institute for Advanced Study, 1012 GC Amsterdam, The Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3848-5395","authenticated-orcid":false,"given":"Peter M. A.","family":"Sloot","sequence":"additional","affiliation":[{"name":"Institute of Informatics, University of Amsterdam, 1098 XH Amsterdam, The Netherlands"},{"name":"Institute for Advanced Study, 1012 GC Amsterdam, The Netherlands"},{"name":"Complexity Science Hub Viennna, 1080 Vienna, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/s13194-012-0056-8","article-title":"What Is a Complex System?","volume":"3","author":"Ladyman","year":"2013","journal-title":"Eur. J. Philos. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"902","DOI":"10.1016\/j.tree.2016.09.011","article-title":"What Do You Mean, \u2018Tipping Point\u2019?","volume":"31","author":"Arani","year":"2016","journal-title":"Trends Ecol. Evol."},{"key":"ref_3","unstructured":"Kandel, E.R., Schwartz, J.H., and Jessell, T.M. (2000). Principles of Neural Science, McGraw-Hill Medical. [4th ed.]."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1016\/j.neuron.2015.09.034","article-title":"Rhythms for Cognition: Communication through Coherence","volume":"88","author":"Fries","year":"2015","journal-title":"Neuron"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Galam, S., and Cheon, T. (2020). Tipping Points in Opinion Dynamics: A Universal Formula in Five Dimensions. Front. Phys., 8.","DOI":"10.3389\/fphy.2020.566580"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"601","DOI":"10.5194\/esd-12-601-2021","article-title":"Interacting Tipping Elements Increase Risk of Climate Domino Effects under Global Warming","volume":"12","author":"Wunderling","year":"2021","journal-title":"Earth Syst. Dyn."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Beggs, J.M., and Timme, N. (2012). Being Critical of Criticality in the Brain. Front. Physiol., 3.","DOI":"10.3389\/fphys.2012.00163"},{"key":"ref_8","unstructured":"Mitchell, M., Hraber, P., and Crutchfield, J.P. (1993). Revisiting the Edge of Chaos: Evolving Cellular Automata to Perform Computations. arXiv."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1137\/17M1142028","article-title":"A Primer on Noise-Induced Transitions in Applied Dynamical Systems","volume":"60","author":"Forgoston","year":"2018","journal-title":"SIAM Rev."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Czaplicka, A., Holyst, J.A., and Sloot, P.M.A. (2013). Noise Enhances Information Transfer in Hierarchical Networks. Sci. Rep., 3.","DOI":"10.1038\/srep01223"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Nicolis, G., and Nicolis, C. (2016). Stochastic Resonance, Self-Organization and Information Dynamics in Multistable Systems. Entropy, 18.","DOI":"10.3390\/e18050172"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"4854","DOI":"10.1103\/PhysRevA.39.4854","article-title":"Theory of Stochastic Resonance","volume":"39","author":"McNamara","year":"1989","journal-title":"Phys. Rev. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1016\/S0031-8914(40)90098-2","article-title":"Brownian Motion in a Field of Force and the Diffusion Model of Chemical Reactions","volume":"7","author":"Kramers","year":"1940","journal-title":"Physica"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2181","DOI":"10.1038\/s41467-017-01916-3","article-title":"Dynamic Patterns of Information Flow in Complex Networks","volume":"8","author":"Harush","year":"2017","journal-title":"Nat. Commun."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1038\/nature18019","article-title":"Universal Resilience Patterns in Complex Networks","volume":"536","author":"Gao","year":"2016","journal-title":"Nature"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"e1922831118","DOI":"10.1073\/pnas.1922831118","article-title":"Optimal Resilience of Modular Interacting Networks","volume":"118","author":"Dong","year":"2021","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_17","first-page":"nwaa229","article-title":"Efficient Network Immunization under Limited Knowledge","volume":"8","author":"Liu","year":"2021","journal-title":"Natl. Sci. Rev."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1160","DOI":"10.1016\/j.isci.2019.07.043","article-title":"An Algorithmic Information Calculus for Causal Discovery and Reprogramming Systems","volume":"19","author":"Zenil","year":"2019","journal-title":"iScience"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1038\/s42256-018-0005-0","article-title":"Causal Deconvolution by Algorithmic Generative Models","volume":"1","author":"Zenil","year":"2019","journal-title":"Nat. Mach. Intell."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Guo, C., Yang, L., Chen, X., Chen, D., Gao, H., and Ma, J. (2020). Influential Nodes Identification in Complex Networks via Information Entropy. Entropy, 22.","DOI":"10.3390\/e22020242"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2554","DOI":"10.1073\/pnas.79.8.2554","article-title":"Neural Networks and Physical Systems with Emergent Collective Computational Abilities","volume":"79","author":"Hopfield","year":"1982","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"294","DOI":"10.1063\/1.1703954","article-title":"Time-Dependent Statistics of the Ising Model","volume":"4","author":"Glauber","year":"1963","journal-title":"J. Math. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1038\/s41467-023-44609-w","article-title":"Remotely Sensing Potential Climate Change Tipping Points across Scales","volume":"15","author":"Lenton","year":"2024","journal-title":"Nat. Commun."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1930022","DOI":"10.1142\/S0218127419300222","article-title":"Detecting and Predicting Tipping Points","volume":"29","author":"Peng","year":"2019","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"e2106140118","DOI":"10.1073\/pnas.2106140118","article-title":"Deep Learning for Early Warning Signals of Tipping Points","volume":"118","author":"Bury","year":"2021","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.plrev.2014.11.001","article-title":"Statistical Physics of Crime: A Review","volume":"12","author":"Perc","year":"2015","journal-title":"Phys. Life Rev."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"043129","DOI":"10.1063\/1.5142827","article-title":"How Motifs Condition Critical Thresholds for Tipping Cascades in Complex Networks: Linking Micro- to Macro-Scales","volume":"30","author":"Wunderling","year":"2020","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"248302","DOI":"10.1103\/PhysRevLett.119.248302","article-title":"Cascading Failures as Continuous Phase-Space Transitions","volume":"119","author":"Yang","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"eaan3184","DOI":"10.1126\/science.aan3184","article-title":"Small Vulnerable Sets Determine Large Network Cascades in Power Grids","volume":"358","author":"Yang","year":"2017","journal-title":"Science"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Cover, T.M., and Thomas, J.A. (2005). Elements of Information Theory, Wiley-Interscience.","DOI":"10.1002\/047174882X"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"126889","DOI":"10.1016\/j.physa.2022.126889","article-title":"Dynamic Importance of Network Nodes Is Poorly Predicted by Static Structural Features","volume":"593","author":"Quax","year":"2022","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"20130568","DOI":"10.1098\/rsif.2013.0568","article-title":"The Diminishing Role of Hubs in Dynamical Processes on Complex Networks","volume":"10Q","author":"Quax","year":"2013","journal-title":"J. R. Soc. Interface R. Soc."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"238701","DOI":"10.1103\/PhysRevLett.116.238701","article-title":"Information Flows? A Critique of Transfer Entropies","volume":"116","author":"James","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/cogs.12142","article-title":"Information Processing and Dynamics in Minimally Cognitive Agents","volume":"39","author":"Beer","year":"2015","journal-title":"Cogn. Sci."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Kolchinsky, A. (2022). A Novel Approach to the Partial Information Decomposition. Entropy, 24.","DOI":"10.3390\/e24030403"},{"key":"ref_36","unstructured":"Williams, P.L., and Beer, R.D. (2010). Nonnegative Decomposition of Multivariate Information. arXiv."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1038\/nrn3061","article-title":"The Benefits of Noise in Neural Systems: Bridging Theory and Experiment","volume":"12","author":"McDonnell","year":"2011","journal-title":"Nat. Rev. Neurosci."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"V\u00e1zquez-Rodr\u00edguez, B., Avena-Koenigsberger, A., Sporns, O., Griffa, A., Hagmann, P., and Larralde, H. (2017). Stochastic Resonance at Criticality in a Network Model of the Human Cortex. Sci. Rep., 7.","DOI":"10.1038\/s41598-017-13400-5"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Roy, S., and Majumdar, S. (2022). The Role of Noise in Brain Function. Noise and Randomness in Living System, Springer.","DOI":"10.1007\/978-981-16-9583-4"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1038\/nrn2258","article-title":"Noise in the Nervous System","volume":"9","author":"Faisal","year":"2008","journal-title":"Nat. Rev. Neurosci."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"rsta.2020.0236","DOI":"10.1098\/rsta.2020.0236","article-title":"Stochastic and Vibrational Resonance in Complex Networks of Neurons","volume":"379","author":"Calim","year":"2021","journal-title":"Philos. Trans. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_42","unstructured":"Lizier, J.T., Prokopenko, M., and Zomaya, A.Y. (2008, January 5\u20138). The Information Dynamics of Phase Transitions in Random Boolean Networks. Proceedings of the Eleventh International Conference on the Simulation and Synthesis of Living Systems (ALife XI), Winchester, UK."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Lizier, J.T., Flecker, B., and Williams, P.L. (2013, January 16\u201319). Towards a Synergy-Based Approach to Measuring Information Modification. Proceedings of the IEEE Symposium on Artificial Life (ALIFE), Singapore.","DOI":"10.1109\/ALIFE.2013.6602430"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Lizier, J.T., Bertschinger, N., Jost, J., and Wibral, M. (2018). Information Decomposition of Target Effects from Multi-Source Interactions: Perspectives on Previous, Current and Future Work. Entropy, 20.","DOI":"10.3390\/e20040307"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Quax, R., Har-Shemesh, O., and Sloot, P.M. (2017). Quantifying Synergistic Information Using Intermediate Stochastic Variables. Entropy, 19.","DOI":"10.3390\/e19020085"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"037109","DOI":"10.1063\/1.3486801","article-title":"Information Modification and Particle Collisions in Distributed Computation","volume":"20","author":"Lizier","year":"2010","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1038\/nature08227","article-title":"Early-Warning Signals for Critical Transitions","volume":"461","author":"Scheffer","year":"2009","journal-title":"Nature"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"041116","DOI":"10.1103\/PhysRevE.84.041116","article-title":"Relating Fisher Information to Order Parameters","volume":"84","author":"Prokopenko","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1038\/35098000","article-title":"Catastrophic Shifts in Ecosystems","volume":"413","author":"Scheffer","year":"2001","journal-title":"Nature"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"773","DOI":"10.1007\/s10098-013-0687-2","article-title":"Managing for Resilience: Early Detection of Regime Shifts in Complex Systems","volume":"16","author":"Eason","year":"2014","journal-title":"Clean Technol. Environ. Policy"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1103\/PhysRevLett.85.461","article-title":"Measuring Information Transfer","volume":"85","author":"Schreiber","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1142\/S0219525908001465","article-title":"Information Flows in Causal Networks","volume":"11","author":"Ay","year":"2008","journal-title":"Adv. Complex Syst."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41467-019-10105-3","article-title":"Inferring Causation from Time Series in Earth System Sciences","volume":"10","author":"Runge","year":"2019","journal-title":"Nat. Commun."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"037101","DOI":"10.1103\/PhysRevE.78.037101","article-title":"Functions of Neuronal Network Motifs","volume":"78","author":"Li","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_55","unstructured":"Bialek, W., and Tishby, N. (1999). Predictive Information. arXiv."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1016\/0375-9601(95)00867-5","article-title":"A Statistical Measure of Complexity","volume":"209","author":"Mancini","year":"1995","journal-title":"Phys. Lett. A"},{"key":"ref_57","first-page":"15","article-title":"SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python","volume":"17","author":"Virtanen","year":"2020","journal-title":"Nat. Methods"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/12\/1050\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:46:33Z","timestamp":1760114793000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/12\/1050"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,4]]},"references-count":57,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["e26121050"],"URL":"https:\/\/doi.org\/10.3390\/e26121050","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,12,4]]}}}