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Such frameworks have been widely used to explore the emergence of collective phenomena across diverse applications. Here, we develop an analytical framework for oscillator\u2010based Ising systems with Kuramoto\u2010type pairwise coupling and a second harmonic term. Spectral dimension reduction identifies a stability boundary governed by the balance between global coupling and second\u2010harmonic strength, while a mean\u2010field Fokker\u2013Planck analysis predicts the transition between single\u2010cluster and two\u2010cluster stationary states and clarifies the role of noise. Low\u2010energy Ising configurations are predominantly obtained when the second harmonic term is strong relative to noise. Partition\u2010function analysis further shows that, for the ring\u2010based topology, links to third neighbors yield more desired solutions than the other added non\u2010local connections. Applications to graph coloring and MaxCut demonstrate the relevance of these predictions to oscillator\u2010based Ising optimization. These results provide a predictive framework for understanding how dynamical parameters, noise, and network structure jointly shape collective states and low\u2010energy\u00a0configurations.<\/jats:p>","DOI":"10.1002\/advs.78134","type":"journal-article","created":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T03:41:10Z","timestamp":1791517270000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Analytical Prediction of Critical Transitions in Oscillator Networks With Non\u2010Local Links"],"prefix":"10.1002","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-3755-0297","authenticated-orcid":false,"given":"Qiang","family":"Li","sequence":"first","affiliation":[{"name":"Institute of Science and Technology for Brain\u2010Inspired Intelligence Fudan University Shanghai China"},{"name":"College of Future Information Technology Fudan University Shanghai China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0139-9055","authenticated-orcid":false,"given":"Chumin","family":"Sun","sequence":"additional","affiliation":[{"name":"Theory Lab, Central Research Institute, 2012 Labs Huawei Technologies Co., Ltd. Sha Tin Hong Kong"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9093-2108","authenticated-orcid":false,"given":"David","family":"Waxman","sequence":"additional","affiliation":[{"name":"Institute of Science and Technology for Brain\u2010Inspired Intelligence Fudan University Shanghai China"},{"name":"Key Laboratory of Computational Neuroscience and Brain\u2010Inspired Intelligence Ministry of Education Shanghai China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0583-4595","authenticated-orcid":false,"given":"Renaud","family":"Lambiotte","sequence":"additional","affiliation":[{"name":"Mathematical Institute University of Oxford Oxford UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2553-1804","authenticated-orcid":false,"given":"Jie","family":"Sun","sequence":"additional","affiliation":[{"name":"Theory Lab, Central Research Institute, 2012 Labs Huawei Technologies Co., Ltd. Sha Tin Hong Kong"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3225-8431","authenticated-orcid":false,"given":"Peng","family":"Ji","sequence":"additional","affiliation":[{"name":"Institute of Science and Technology for Brain\u2010Inspired Intelligence Fudan University Shanghai China"},{"name":"Key Laboratory of Computational Neuroscience and Brain\u2010Inspired Intelligence Ministry of Education Shanghai China"},{"name":"MOE Frontiers Center for Brain Science Fudan University Shanghai China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,10,8]]},"reference":[{"key":"e_1_2_8_2_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(00)00094-4"},{"key":"e_1_2_8_3_1","doi-asserted-by":"publisher","DOI":"10.1002\/advs.202310096"},{"key":"e_1_2_8_4_1","doi-asserted-by":"publisher","DOI":"10.1038\/s42005-025-02082-5"},{"key":"e_1_2_8_5_1","doi-asserted-by":"publisher","DOI":"10.1002\/advs.202408460"},{"key":"e_1_2_8_6_1","doi-asserted-by":"publisher","DOI":"10.1063\/5.0106906"},{"key":"e_1_2_8_7_1","doi-asserted-by":"publisher","DOI":"10.1126\/sciadv.adu4114"},{"key":"e_1_2_8_8_1","doi-asserted-by":"crossref","unstructured":"Y.Kuramoto \u201cSelf\u2010Entrainment of a Population of Coupled Non\u2010Linear Oscillators \u201d InInternational Symposium on Mathematical Problems in Theoretical Physics: January 23\u201329 1975 Kyoto University Kyoto\/Japan(Springer 1975) 420\u2013422.","DOI":"10.1007\/BFb0013365"},{"key":"e_1_2_8_9_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.physrep.2015.10.008"},{"key":"e_1_2_8_10_1","doi-asserted-by":"publisher","DOI":"10.1063\/1.2930766"},{"key":"e_1_2_8_11_1","doi-asserted-by":"publisher","DOI":"10.1137\/110851584"},{"key":"e_1_2_8_12_1","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1212134110"},{"key":"e_1_2_8_13_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.automatica.2014.04.012"},{"key":"e_1_2_8_14_1","doi-asserted-by":"publisher","DOI":"10.1103\/PRXEnergy.3.043004"},{"key":"e_1_2_8_15_1","doi-asserted-by":"publisher","DOI":"10.1038\/s41928-025-01393-3"},{"key":"e_1_2_8_16_1","doi-asserted-by":"publisher","DOI":"10.1002\/advs.202413247"},{"key":"e_1_2_8_17_1","doi-asserted-by":"crossref","unstructured":"T.WangandJ.Roychowdhury \u201cOim: Oscillator\u2010Based Ising Machines for Solving Combinatorial Optimisation Problems \u201d InInternational Conference on Unconventional Computation and Natural Computation(Springer 2019) 232\u2013256.","DOI":"10.1007\/978-3-030-19311-9_19"},{"key":"e_1_2_8_18_1","doi-asserted-by":"publisher","DOI":"10.1038\/s42254-022-00440-8"},{"key":"e_1_2_8_19_1","doi-asserted-by":"publisher","DOI":"10.1126\/sciadv.abe7953"},{"key":"e_1_2_8_20_1","doi-asserted-by":"publisher","DOI":"10.1126\/science.aat2025"},{"key":"e_1_2_8_21_1","doi-asserted-by":"publisher","DOI":"10.3389\/fphy.2019.00048"},{"key":"e_1_2_8_22_1","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-019-11484-3"},{"key":"e_1_2_8_23_1","doi-asserted-by":"publisher","DOI":"10.1126\/science.aah5178"},{"key":"e_1_2_8_24_1","doi-asserted-by":"crossref","unstructured":"S.Shirasaka \u201cInvestigation on Oscillator\u2010Based Ising Machines \u201d InPhotonic Neural Networks with Spatiotemporal Dynamics: Paradigms of Computing and Implementation(Springer 2023) 175\u2013199.","DOI":"10.1007\/978-981-99-5072-0_9"},{"key":"e_1_2_8_25_1","doi-asserted-by":"publisher","DOI":"10.1038\/s41928-021-00546-4"},{"key":"e_1_2_8_26_1","doi-asserted-by":"publisher","DOI":"10.1103\/PRXQuantum.4.030335"},{"key":"e_1_2_8_27_1","doi-asserted-by":"publisher","DOI":"10.1007\/s11047-021-09845-3"},{"key":"e_1_2_8_28_1","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-023-37695-3"},{"key":"e_1_2_8_29_1","doi-asserted-by":"crossref","unstructured":"M. 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