{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,25]],"date-time":"2026-04-25T23:01:54Z","timestamp":1777158114666,"version":"3.51.4"},"reference-count":38,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,12,29]],"date-time":"2020-12-29T00:00:00Z","timestamp":1609200000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001602","name":"Science Foundation Ireland","doi-asserted-by":"publisher","award":["No. 16\/IA\/4470"],"award-info":[{"award-number":["No. 16\/IA\/4470"]}],"id":[{"id":"10.13039\/501100001602","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001602","name":"Science Foundation Ireland","doi-asserted-by":"publisher","award":["16\/RC\/3918"],"award-info":[{"award-number":["16\/RC\/3918"]}],"id":[{"id":"10.13039\/501100001602","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001602","name":"Science Foundation Ireland","doi-asserted-by":"publisher","award":["12\/RC\/2289 P2"],"award-info":[{"award-number":["12\/RC\/2289 P2"]}],"id":[{"id":"10.13039\/501100001602","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001602","name":"Science Foundation Ireland","doi-asserted-by":"publisher","award":["18\/CRT\/6049"],"award-info":[{"award-number":["18\/CRT\/6049"]}],"id":[{"id":"10.13039\/501100001602","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002661","name":"Fonds De La Recherche Scientifique - FNRS","doi-asserted-by":"publisher","award":["FC 33443"],"award-info":[{"award-number":["FC 33443"]}],"id":[{"id":"10.13039\/501100002661","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Synchronization is an important behavior that characterizes many natural and human made systems that are composed by several interacting units. It can be found in a broad spectrum of applications, ranging from neuroscience to power-grids, to mention a few. Such systems synchronize because of the complex set of coupling they exhibit, with the latter being modeled by complex networks. The dynamical behavior of the system and the topology of the underlying network are strongly intertwined, raising the question of the optimal architecture that makes synchronization robust. The Master Stability Function (MSF) has been proposed and extensively studied as a generic framework for tackling synchronization problems. Using this method, it has been shown that, for a class of models, synchronization in strongly directed networks is robust to external perturbations. Recent findings indicate that many real-world networks are strongly directed, being potential candidates for optimal synchronization. Moreover, many empirical networks are also strongly non-normal. Inspired by this latter fact in this work, we address the role of the non-normality in the synchronization dynamics by pointing out that standard techniques, such as the MSF, may fail to predict the stability of synchronized states. We demonstrate that, due to a transient growth that is induced by the structure\u2019s non-normality, the system might lose synchronization, contrary to the spectral prediction. These results lead to a trade-off between non-normality and directedness that should be properly considered when designing an optimal network, enhancing the robustness of synchronization.<\/jats:p>","DOI":"10.3390\/e23010036","type":"journal-article","created":{"date-parts":[[2020,12,29]],"date-time":"2020-12-29T10:04:46Z","timestamp":1609236286000},"page":"36","update-policy":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["Synchronization Dynamics in Non-Normal Networks: The Trade-Off for Optimality"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/orcid.org\/0000-0001-9093-7478","authenticated-orcid":false,"given":"Riccardo","family":"Muolo","sequence":"first","affiliation":[{"name":"Department of Mathematics and naXys, Namur Institute for Complex Systems, University of Namur, rue Graf\u00e9 2, 5000 Namur, Belgium"}]},{"ORCID":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/orcid.org\/0000-0003-2596-4503","authenticated-orcid":false,"given":"Timoteo","family":"Carletti","sequence":"additional","affiliation":[{"name":"Department of Mathematics and naXys, Namur Institute for Complex Systems, University of Namur, rue Graf\u00e9 2, 5000 Namur, Belgium"}]},{"ORCID":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/orcid.org\/0000-0003-3410-2817","authenticated-orcid":false,"given":"James P.","family":"Gleeson","sequence":"additional","affiliation":[{"name":"MACSI, Department of Mathematics and Statistics, University of Limerick, V94 T9PX Limerick, Ireland"}]},{"ORCID":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/orcid.org\/0000-0002-3335-429X","authenticated-orcid":false,"given":"Malbor","family":"Asllani","sequence":"additional","affiliation":[{"name":"Department of Mathematics and naXys, Namur Institute for Complex Systems, University of Namur, rue Graf\u00e9 2, 5000 Namur, Belgium"},{"name":"MACSI, Department of Mathematics and Statistics, University of Limerick, V94 T9PX Limerick, Ireland"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,29]]},"reference":[{"key":"ref_1","unstructured":"Newman, M.E.J. (2018). Networks: An Introduction, Oxford University Press. [2nd ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.physrep.2008.09.002","article-title":"Synchronization in complex networks","volume":"469","author":"Arenas","year":"2008","journal-title":"Phys. Rep."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0167-2789(00)00094-4","article-title":"From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators","volume":"143","author":"Strogatz","year":"2000","journal-title":"Physica D"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Pikovsky, A., Rosenblum, M., and Kurths, J. (2001). Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press.","DOI":"10.1017\/CBO9780511755743"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/j.physd.2006.09.007","article-title":"Maximum performance at minimum cost in network synchronization","volume":"224","author":"Nishikawa","year":"2006","journal-title":"Physica D"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"065106","DOI":"10.1103\/PhysRevE.73.065106","article-title":"Synchronization is optimal in nondiagonalizable networks","volume":"73","author":"Nishikawa","year":"2006","journal-title":"Phys. Rev. E"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"094807","DOI":"10.1063\/1.4954221","article-title":"Optimal synchronization of directed complex networks","volume":"26","author":"Skardal","year":"2016","journal-title":"Chaos"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"013111","DOI":"10.1063\/1.4974029","article-title":"Optimal phase synchronization in networks of phase-coherent chaotic oscillators","volume":"27","author":"Skardal","year":"2017","journal-title":"Chaos"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2109","DOI":"10.1103\/PhysRevLett.80.2109","article-title":"Master Stability Functions for Synchronized Coupled Systems","volume":"80","author":"Pecora","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"054101","DOI":"10.1103\/PhysRevLett.89.054101","article-title":"Synchronization in Small-World Systems","volume":"89","author":"Barahona","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"48001","DOI":"10.1209\/0295-5075\/117\/48001","article-title":"Analysing degeneracies in networks spectra","volume":"117","author":"Marrec","year":"2017","journal-title":"Europhys. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Trefethen, L.N., and Embree, M. (2005). Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton University Press.","DOI":"10.1515\/9780691213101"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"Eaau9403","DOI":"10.1126\/sciadv.aau9403","article-title":"Structure and dynamics of non-normal networks","volume":"4","author":"Asllani","year":"2018","journal-title":"Sci. Adv."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"042302","DOI":"10.1103\/PhysRevE.97.042302","article-title":"Topological resilience in non-normal networked systems","volume":"97","author":"Asllani","year":"2018","journal-title":"Phys. Rev. E"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/j.jtbi.2019.07.004","article-title":"Patterns of non-normality in networked systems","volume":"480","author":"Muolo","year":"2019","journal-title":"J. Theor. Biol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"eaba2282","DOI":"10.1126\/sciadv.aba2282","article-title":"Efficient communication over complex dynamical networks: The role of matrix non-normality","volume":"6","author":"Baggio","year":"2020","journal-title":"Sci. Adv."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"032214","DOI":"10.1103\/PhysRevE.98.032214","article-title":"Non-normal amplification of stochastic quasicycles","volume":"98","author":"Nicoletti","year":"2018","journal-title":"Phys. Rev. E"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"083123","DOI":"10.1063\/1.5099538","article-title":"Resilience for stochastic systems interacting via a quasi-degenerate network","volume":"29","author":"Nicoletti","year":"2019","journal-title":"Chaos"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"578","DOI":"10.1126\/science.261.5121.578","article-title":"Hydrodynamic Stability Without Eigenvalues","volume":"261","author":"Trefethen","year":"1993","journal-title":"Science"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"653","DOI":"10.1890\/0012-9658(1997)078[0653:ATRFMT]2.0.CO;2","article-title":"Alternatives to resilience for measuring the responses of ecological systems to perturbations","volume":"78","author":"Neubert","year":"1997","journal-title":"Ecology"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0025-5564(01)00087-6","article-title":"Transient Dynamics and Pattern Formation: Reactivity is Necessary for Turing Instabilities","volume":"175","author":"Neubert","year":"2002","journal-title":"Math. Biosci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"043059","DOI":"10.1103\/PhysRevResearch.2.043059","article-title":"Non-normality and non-monotonic dynamics in complex reaction networks","volume":"2","author":"Nicolaou","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3542","DOI":"10.1063\/1.1841255","article-title":"Symmetry breaking instabilities in dissipative systems","volume":"46","author":"Prigogine","year":"1967","journal-title":"J. Chem. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"P09001","DOI":"10.1088\/1742-5468\/2008\/09\/P09001","article-title":"How limit cycles and quasi-cycles are related in a system with intrinsic noise","volume":"2008","author":"Boland","year":"2008","journal-title":"J. Stat. Mech."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1140\/epjb\/e2020-10206-3","article-title":"A universal route to pattern formation in multicellular systems","volume":"93","author":"Asllani","year":"2020","journal-title":"Eur. Phys. J. B"},{"key":"ref_26","unstructured":"Sanders, J.A., Verhulst, F., and Murdock, J. (2007). Averaging Methods in Nonlinear Dynamical Systems, Springer. [2nd ed.]."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"4517","DOI":"10.1038\/ncomms5517","article-title":"The theory of pattern formation on directed networks","volume":"5","author":"Asllani","year":"2014","journal-title":"Nat. Comm."},{"key":"ref_28","unstructured":"Davis, P.J. (1994). Circulant Matrices, AMS Chelsea Publishing. [2nd ed.]."},{"key":"ref_29","unstructured":"This follows from the fact that every tridiagonal matrix is similar to a symmetric one and consequently. However, the set of eigenvectors is not the same; they are linearly independent in both cases."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1063\/1.166213","article-title":"Synchronization of chaotic systems: Transverse stability of trajectories in invariant manifolds","volume":"7","author":"Brown","year":"1997","journal-title":"Chaos"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"022818","DOI":"10.1103\/PhysRevE.92.022818","article-title":"Turing-like instabilities from a limit cycle","volume":"92","author":"Challenger","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Kuramoto, Y. (1984). Chemical Oscillations, Waves, and Turbulence, Springer.","DOI":"10.1007\/978-3-642-69689-3"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1016\/j.cnsns.2017.08.012","article-title":"Ginzburg-Landau approximation for self-sustained oscillators weakly coupled on complex directed graphs","volume":"56","author":"Fanelli","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer Simulat."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1038\/nature16948","article-title":"Universal resilience patterns in complex networks","volume":"530","author":"Gao","year":"2016","journal-title":"Nature"},{"key":"ref_35","unstructured":"Wright, T.G. (2020, December 27). EigTool, MATLAB\u00ae ToolBox. Available online: https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/www.cs.ox.ac.uk\/pseudospectra\/eigtool\/."},{"key":"ref_36","unstructured":"For a given species, e.g., \u03c6, the standard deviation at a given time t is S(t)=1n\u22121\u2211i=1n\u03c6i(t)\u2212\u03c6^(t), where n is the number of nodes and \u03c6^(t)=1n\u2211i=1n\u03c6i(t). We can observe that such quantity is 0 if the system is synchronized, i.e., \u03c6i(t)=\u03c6^(t)\u2200t, while a value different from 0 indicates that not every node has the same concentration at any given time."},{"key":"ref_37","first-page":"290","article-title":"On Random Graphs","volume":"6","year":"1959","journal-title":"Publ. Math."},{"key":"ref_38","unstructured":"Notice that the set of parameters, in this case, is far from the region where the averaging method presented in the main text is valid."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/www.mdpi.com\/1099-4300\/23\/1\/36\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:47:20Z","timestamp":1760179640000},"score":1,"resource":{"primary":{"URL":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/www.mdpi.com\/1099-4300\/23\/1\/36"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,29]]},"references-count":38,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["e23010036"],"URL":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/doi.org\/10.3390\/e23010036","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,12,29]]}}}