{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,9]],"date-time":"2026-06-09T14:07:52Z","timestamp":1781014072902,"version":"3.54.1"},"reference-count":87,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,9,27]],"date-time":"2025-09-27T00:00:00Z","timestamp":1758931200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012190","name":"Ministry of Science and Higher Education of the Russian Federation","doi-asserted-by":"crossref","award":["075-03-2025-448"],"award-info":[{"award-number":["075-03-2025-448"]}],"id":[{"id":"10.13039\/501100012190","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100012190","name":"Ministry of Science and Higher Education of the Russian Federation","doi-asserted-by":"crossref","award":["FSER-2025-0013"],"award-info":[{"award-number":["FSER-2025-0013"]}],"id":[{"id":"10.13039\/501100012190","id-type":"DOI","asserted-by":"crossref"}]},{"name":"EU-grant","award":["CA21169"],"award-info":[{"award-number":["CA21169"]}]},{"name":"JST, CREST","award":["JPMJCR23P4"],"award-info":[{"award-number":["JPMJCR23P4"]}]}],"content-domain":{"domain":["www.mdpi.com"],"crossmark-restriction":true},"short-container-title":["Entropy"],"abstract":"<jats:p>The increasing integration of artificial intelligence agents (AIAs) based on large language models (LLMs) is transforming many spheres of society. These agents act as human assistants, forming Distributed Intelligent Systems (DISs) and engaging in opinion formation, consensus-building, and collective decision-making. However, complex DIS network topologies introduce significant uncertainty into these processes. We propose a quantum-inspired graph signal processing framework to model collective behavior in a DIS interacting with an external environment represented by an influence matrix (IM). System topology is captured using scale-free and Watts\u2013Strogatz graphs. Two contrasting interaction regimes are considered. In the first case, the internal structure fully aligns with the external influence, as expressed by the commutativity between the adjacency matrix and the IM. Here, a renormalization-group-based scaling approach reveals minimal reservoir influence, characterized by full phase synchronization and coherent dynamics. In the second case, the IM includes heterogeneous negative (antagonistic) couplings that do not commute with the network, producing partial or complete spectral disorder. This disrupts phase coherence and may fragment opinions, except for the dominant collective (Perron) mode, which remains robust. Spectral entropy quantifies disorder and external influence. The proposed framework offers insights into designing LLM-participated DISs that can maintain coherence under environmental perturbations.<\/jats:p>","DOI":"10.3390\/e27101016","type":"journal-article","created":{"date-parts":[[2025,9,29]],"date-time":"2025-09-29T08:00:32Z","timestamp":1759132832000},"page":"1016","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Spectral Properties of Complex Distributed Intelligence Systems Coupled with an Environment"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1247-2761","authenticated-orcid":false,"given":"Alexander P.","family":"Alodjants","sequence":"first","affiliation":[{"name":"National Center for Cognitive Reaserch, ITMO University, St. Petersburg 197101, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9041-2708","authenticated-orcid":false,"given":"Dmitriy V.","family":"Tsarev","sequence":"additional","affiliation":[{"name":"National Center for Cognitive Reaserch, ITMO University, St. Petersburg 197101, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-1398-1446","authenticated-orcid":false,"given":"Petr V.","family":"Zakharenko","sequence":"additional","affiliation":[{"name":"National Center for Cognitive Reaserch, ITMO University, St. Petersburg 197101, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9857-0938","authenticated-orcid":false,"given":"Andrei Yu.","family":"Khrennikov","sequence":"additional","affiliation":[{"name":"International Center for Mathematical Modeling in Physics, Engineering, Economics, and Cognitive Science Linnaeus University, S-35195 Vaxjo-Kalmar, Sweden"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Lawless, W., Mittu, R., Sofge, D., and Brambilla, M. (2025). Introduction to bidirectionality in human\u2013AI collaborative systems. Bi-Directionality in Human-AI Collaborative Systems, Academic Press.","DOI":"10.1016\/B978-0-44-340553-2.00007-1"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"e2313925121","DOI":"10.1073\/pnas.2313925121","article-title":"A Turing test of whether AI chatbots are behaviorally similar to humans","volume":"121","author":"Mei","year":"2024","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1007\/s44336-024-00009-2","article-title":"A survey on LLM-based multi-agent systems: Workflow, infrastructure, and challenges","volume":"1","author":"Li","year":"2024","journal-title":"Vicinagearth"},{"key":"ref_4","unstructured":"Cheng, Y., Zhang, C., Zhang, Z., Meng, X., Hong, S., Li, W., Wang, Z., Wang, Z., Yin, F., and Zhao, J. (2024). Exploring Large Language Model based Intelligent Agents: Definitions, Methods, and Prospects. arXiv."},{"key":"ref_5","unstructured":"Zhang, G., Yue, Y., Sun, X., Wan, G., Yu, M., Fang, J., Wang, K., Chen, T., and Cheng, D. (2025). G-Designer: Architecting Multi-agent Communication Topologies via Graph Neural Networks. arXiv."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"M\u00fcller, P., Heimerl, A., Hossain, S., Siegel, L., Alexandersson, J., Gebhard, P., Andr\u00e9, E., and Schneeberger, T. (2024). Recognizing Emotion Regulation Strategies from Human Behavior with Large Language Models. arXiv.","DOI":"10.1109\/ACII63134.2024.00029"},{"key":"ref_7","unstructured":"Ku, L.W., Martins, A., and Srikumar, V. Exploring Collaboration Mechanisms for LLM Agents: A Social Psychology View. Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Guleva, V., Shikov, E., Bochenina, K., Kovalchuk, S., Alodjants, A., and Boukhanovsky, A. (2020). Emerging complexity in distributed intelligent systems. Entropy, 22.","DOI":"10.3390\/e22121437"},{"key":"ref_9","first-page":"P10014","article-title":"Can human-like Bots control collective mood: Agent-based simulations of online chats","volume":"2023","year":"2013","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_10","unstructured":"Qian, C., Xie, Z., Wang, Y., Liu, W., Dang, Y., Du, Z., Chen, W., Yang, C., Liu, Z., and Sun, M. (2024). Scaling Large-Language-Model-based Multi-Agent Collaboration. arXiv."},{"key":"ref_11","unstructured":"Chen, W., You, Z., Li, R., Guan, Y., Qian, C., Zhao, C., Yang, C., Xie, R., Liu, Z., and Sun, M. (2024). Internet of Agents: Weaving a Web of Heterogeneous Agents for Collaborative Intelligence. arXiv."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/S0378-4371(00)00282-X","article-title":"Phase transitions in social impact models of opinion formation","volume":"285","author":"Holyst","year":"2000","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Allahverdyan, A., and Galstyan, A. (2016). Emergence of Leadership in Communication. PLoS ONE, 11.","DOI":"10.1371\/journal.pone.0159301"},{"key":"ref_14","unstructured":"Kamar, E. (2016, January 9\u201315). Directions in Hybrid Intelligence: Complementing AI Systems with Human Intelligence. Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence (IJCAI-16), New York, NY, USA."},{"key":"ref_15","unstructured":"Zhang, Y., Lin, C., Kirchner, L., Gajos, K.Z., and Kamar, E. (2021). Towards a Science of Human-AI Decision Making: A Survey of Empirical Studies. arXiv."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Chuang, Y.S., Goyal, A., Harlalka, N., Suresh, S., Hawkins, R., Yang, S., Shah, D., Hu, J., and Rogers, T.T. (2023). Simulating Opinion Dynamics with Networks of LLM-based Agents. arXiv.","DOI":"10.18653\/v1\/2024.findings-naacl.211"},{"key":"ref_17","unstructured":"Song, T., Tan, Y., Zhu, Z., Feng, Y., and Lee, Y. (2025). Multi-Agents are Social Groups: Investigating Social Influence of Multiple Agents in Human-Agent Interactions. arXiv."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1080\/01621459.1974.10480137","article-title":"Reaching a consensus","volume":"69","author":"DeGroot","year":"1974","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1080\/0022250X.1990.9990069","article-title":"Social influence and opinions","volume":"15","author":"Friedkin","year":"1990","journal-title":"J. Math. Sociol."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"935","DOI":"10.1109\/TAC.2012.2224251","article-title":"Consensus problems on networks with antagonistic interactions","volume":"58","author":"Altafini","year":"2013","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Alodjants, A., Bazhenov, A., Khrennikov, A., and Boukhanovsky, A. (2022). Mean-field theory of social laser. Sci. Rep., 12.","DOI":"10.1038\/s41598-022-12327-w"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Alodjants, A., Tsarev, D., Avdyushina, A., Khrennikov, A., and Boukhanovsky, A. (2024). Quantum-inspired modeling of distributed intelligence systems with artificial intelligent agents self-organization. Sci. Rep., 14.","DOI":"10.1038\/s41598-024-65684-z"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"808","DOI":"10.1109\/JPROC.2018.2820126","article-title":"Graph Signal Processing: Overview, Challenges, and Applications","volume":"106","author":"Ortega","year":"2018","journal-title":"Proc. IEEE"},{"key":"ref_24","unstructured":"Ahn, H., Tran, Q.V., Trinh, M.H., Moore, K.L., Ye, M., and Liu, J. (2018). Cooperative Opinion Dynamics on Multiple Interdependent Topics: Modeling and Analysis. arXiv."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1109\/MSP.2023.3262906","article-title":"Graph Signal Processing: History, Development, Impact, and Outlook","volume":"40","author":"Leus","year":"2023","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1109\/MSP.2020.3014591","article-title":"Graph Signal Processing for Machine Learning: A Review and New Perspectives","volume":"37","author":"Dong","year":"2020","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1109\/MSP.2020.3016143","article-title":"Graphs, Convolutions, and Neural Networks: From Graph Filters to Graph Neural Networks","volume":"37","author":"Gama","year":"2020","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"868","DOI":"10.1109\/JPROC.2018.2798928","article-title":"A Graph Signal Processing Perspective on Functional Brain Imaging","volume":"106","author":"Huang","year":"2018","journal-title":"Proc. IEEE"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Cheung, G., Magli, E., Tanaka, Y., and Ng, M. (2018). Graph Spectral Image Processing. arXiv.","DOI":"10.1109\/JPROC.2018.2799702"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Sabatini, A., and Vollero, L. (2022, January 20\u201323). GSP based subsampling of IoT sensor networks. Proceedings of the 2022 IEEE 12th Sensor Array and Multichannel Signal Processing Workshop (SAM), Trondheim, Norway.","DOI":"10.1109\/SAM53842.2022.9827833"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1007\/978-3-030-45982-6_8","article-title":"Random Walks on Graphs","volume":"Volume 31","year":"2020","journal-title":"Markov Chains"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1023\/A:1019609420309","article-title":"An example of the difference between quantum and classical random walks","volume":"1","author":"Childs","year":"2002","journal-title":"Quantum Inf. Process."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Chung, F., and Lu, L. (2006). Complex Graphs and Networks (Cbms Regional Conference Series in Mathematics), American Mathematical Society.","DOI":"10.1090\/cbms\/107"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"051903","DOI":"10.1103\/PhysRevE.64.051903","article-title":"Spectra and eigenvectors of scale-free networks","volume":"64","author":"Goh","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"026704","DOI":"10.1103\/PhysRevE.64.026704","article-title":"Spectra of \u201creal-world\u201d graphs: Beyond the semicircle law","volume":"64","author":"Farkas","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"066122","DOI":"10.1103\/PhysRevE.65.066122","article-title":"Pseudofractal scale-free web","volume":"65","author":"Dorogovtsev","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Volchenkov, D. (2025). Mathematical Frameworks for Network Dynamics: A Six-Pillar Survey for Analysis, Control, and Inference. Mathematics, 13.","DOI":"10.3390\/math13132116"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"295002","DOI":"10.1088\/1751-8113\/41\/29\/295002","article-title":"Spectra of sparse random matrices","volume":"41","year":"2008","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"031116","DOI":"10.1103\/PhysRevE.78.031116","article-title":"Cavity approach to the spectral density of sparse symmetric random matrices","volume":"78","author":"Rogers","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"023504","DOI":"10.1063\/5.0231303","article-title":"Density matrices and entropy operator for non-Hermitian quantum mechanics","volume":"66","author":"Bagarello","year":"2025","journal-title":"J. Math. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"20240209","DOI":"10.1098\/rspa.2024.0209","article-title":"Quantum-like states on complex synchronized networks","volume":"480","author":"Scholes","year":"2024","journal-title":"Proc. R. Soc. A"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"035006","DOI":"10.1088\/2058-9565\/adcae3","article-title":"Sensing multiatom networks in cavities via photon-induced excitation resonance","volume":"10","author":"Chattopadhyay","year":"2025","journal-title":"Quantum Sci. Technol."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2012.11.001","article-title":"Understanding quantum measurement from the solution of dynamical models","volume":"525","author":"Allahverdyan","year":"2013","journal-title":"Phys. Rep."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1382","DOI":"10.1038\/s41567-022-01831-5","article-title":"Bell\u2019s theorem allows local theories of quantum mechanics","volume":"18","author":"Hance","year":"2022","journal-title":"Nat. Phys."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"042307","DOI":"10.1103\/PhysRevA.81.042307","article-title":"State transfer in static and dynamic spin chains with disorder","volume":"81","author":"Petrosyan","year":"2010","journal-title":"Phys. Rev. A"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"20","DOI":"10.4018\/jswis.2012100102","article-title":"Modeling indirect influence on twitter","volume":"8","author":"Shuai","year":"2012","journal-title":"Int. J. Semant. Web Inf. Syst."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"2133","DOI":"10.1140\/epjst\/e2018-800027-6","article-title":"Concept of information laser: From quantum theory to behavioural dynamics","volume":"227","author":"Khrennikov","year":"2019","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Khrennikov, A. (2010). Ubiquitous Quantum Structure: From Psychology to Finances, Springer.","DOI":"10.1007\/978-3-642-05101-2"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Busemeyer, J.R., and Bruza, P.D. (2012). Quantum Models of Cognition and Decision, Cambridge University Press.","DOI":"10.1017\/CBO9780511997716"},{"key":"ref_50","unstructured":"Romero, D.M., Meeder, B., and Kleinberg, J. (April, January 28). Differences in the mechanics of information diffusion across topics: Idioms, political hashtags, and complex contagion on twitter. Proceedings of the 20th International Conference on World Wide Web, WWW \u201911, Hyderabad, India."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Li, M., Wang, X., Gao, K., and Zhang, S. (2017). A Survey on Information Diffusion in Online Social Networks: Models and Methods. Information, 8.","DOI":"10.3390\/info8040118"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"103621","DOI":"10.1016\/j.ipm.2023.103621","article-title":"Turing instability analysis of a reaction\u2013diffusion system for rumor propagation in continuous space and complex networks","volume":"61","author":"Li","year":"2024","journal-title":"Inf. Process. Manag."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"025106","DOI":"10.1063\/1.5115323","article-title":"A short introduction to the Lindblad master equation","volume":"10","author":"Manzano","year":"2020","journal-title":"Aip Adv."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"2250004","DOI":"10.1142\/S1230161222500044","article-title":"Non-Hermitian Physics and Master Equations","volume":"29","author":"Roccati","year":"2022","journal-title":"Open Syst. Inf. Dyn."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1016\/j.jtbi.2011.04.022","article-title":"Quantum-like model of brain\u2019s functioning: Decision making from decoherence","volume":"281","author":"Asano","year":"2011","journal-title":"J. Theor. Biol."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1142\/S1230161211000042","article-title":"On application of Gorini-Kossakowski-Sudarshan-Lindblad equation in cognitive psychology","volume":"18","author":"Asano","year":"2011","journal-title":"Open Syst. Inf. Dyn."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1007\/s11128-024-04351-8","article-title":"High-dimensional graphs convolution for quantum walks photonic applications","volume":"23","author":"Abramov","year":"2024","journal-title":"Quantum Inf. Process"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1038\/s42254-020-00264-4","article-title":"Network geometry","volume":"3","author":"Bonamassa","year":"2021","journal-title":"Nat. Rev. Phys."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/BF01011765","article-title":"Variational approximations for renormalization group transformations","volume":"14","author":"Kadanoff","year":"1976","journal-title":"J. Stat. Phys."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"023230","DOI":"10.1103\/PhysRevResearch.3.023230","article-title":"Neural Monte Carlo renormalization group","volume":"3","author":"Chung","year":"2021","journal-title":"Phys. Rev. Res."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"033508","DOI":"10.1103\/1mqj-61yh","article-title":"Complex networks for random lasers and photonic computing","volume":"112","author":"Tsarev","year":"2025","journal-title":"Phys. Rev. A"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"20150094","DOI":"10.1098\/rsta.2015.0094","article-title":"Social laser: Action amplification by stimulated emission of social energy","volume":"374","author":"Khrennikov","year":"2016","journal-title":"Phil. Trans. R. Soc."},{"key":"ref_63","doi-asserted-by":"crossref","unstructured":"Khrennikov, A., Alodjants, A., Trofimova, A., and Tsarev, D. (2018). On interpretational questions for quantum-Like modeling of social lasing. Entropy, 20.","DOI":"10.3390\/e20120921"},{"key":"ref_64","doi-asserted-by":"crossref","unstructured":"Khrennikov, A. (2020). Social Laser, Jenny Stanford Publishing.","DOI":"10.1201\/9780367817206"},{"key":"ref_65","doi-asserted-by":"crossref","unstructured":"Holyst, J.A. (2016). Cyberemotions: Collective Emotions in Cyberspace, Springer.","DOI":"10.1007\/978-3-319-43639-5"},{"key":"ref_66","first-page":"20120375","article-title":"Network science","volume":"371","year":"2013","journal-title":"Philos. Trans. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_67","first-page":"021040","article-title":"Influence Matrix Approach to Many-Body Floquet Dynamics","volume":"11","author":"Lerose","year":"2021","journal-title":"Phys. Rev. X"},{"key":"ref_68","first-page":"1","article-title":"Social influence networks and opinion change","volume":"16","author":"Friedkin","year":"1999","journal-title":"Adv. Group Process."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1038\/30918","article-title":"Collective dynamics of \u2018small-world\u2019 networks","volume":"393","author":"Watts","year":"1998","journal-title":"Nature"},{"key":"ref_70","first-page":"021025","article-title":"Nonreciprocal Photon Transmission and Amplification via Reservoir Engineering","volume":"5","author":"Metelmann","year":"2015","journal-title":"Phys. Rev. X"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"434003","DOI":"10.1088\/1751-8121\/ab1ce0","article-title":"Spectral theory of sparse non-Hermitian random matrices","volume":"52","author":"Metz","year":"2019","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"211732","DOI":"10.1098\/rsos.211732","article-title":"Opinion dynamics in social networks under competition: The role of influencing factors in consensus-reaching","volume":"9","author":"Lang","year":"2022","journal-title":"R. Soc. Open Sci."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"89","DOI":"10.2307\/3213262","article-title":"Toward Consensus: Some Convergence Theorems on Repeated Averaging","volume":"14","author":"Chatterjee","year":"1977","journal-title":"J. Appl. Probab."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"1644","DOI":"10.1109\/TSP.2013.2238935","article-title":"Discrete Signal Processing on Graphs","volume":"61","author":"Sandryhaila","year":"2013","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_75","doi-asserted-by":"crossref","unstructured":"Gily\u00e9n, A., Su, Y., Low, G.H., and Wiebe, N. (2019, January 23\u201326). Quantum Singular Value Transformation and Beyond: Exponential Improvements for Quantum Matrix Arithmetics. Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC 2019), Phoenix, AZ, USA.","DOI":"10.1145\/3313276.3316366"},{"key":"ref_76","doi-asserted-by":"crossref","unstructured":"Griffiths, D.J., and Schroeter, D.F. (2018). Introduction to Quantum Mechanics, Cambridge University Press. [3rd ed.].","DOI":"10.1017\/9781316995433"},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"051127","DOI":"10.1103\/PhysRevE.80.051127","article-title":"Critical behavior of the Ising model in annealed scale-free networks","volume":"80","author":"Lee","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1103\/RevModPhys.53.385","article-title":"Random-matrix physics: Spectrum and strength fluctuations","volume":"53","author":"Brody","year":"1981","journal-title":"Rev. Mod. Phys."},{"key":"ref_79","unstructured":"Mehta, M.L. (2004). Random Matrices, Academic Press. [3rd ed.]."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1137\/130934283","article-title":"Approximating Spectral Densities of Large Matrices","volume":"58","author":"Lin","year":"2016","journal-title":"SIAM Rev."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"034087","DOI":"10.1103\/PhysRevApplied.15.034087","article-title":"Scalable Spin-Glass Optical Simulator","volume":"15","author":"Pierangeli","year":"2021","journal-title":"Phys. Rev. Appl."},{"key":"ref_82","doi-asserted-by":"crossref","unstructured":"Gomes, A., Moura, A., de Ara\u00fajo, C., and Raposo, E. (2023). L\u00e9vy Statistics and Spin Glass Behavior in Random Lasers, Jenny Stanford Publishing.","DOI":"10.1201\/9781003336181"},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"1194","DOI":"10.3390\/e12051194","article-title":"Quantum Entropy and Its Applications to Quantum Communication and Statistical Physics","volume":"12","author":"Ohya","year":"2010","journal-title":"Entropy"},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"559","DOI":"10.3390\/e14030559","article-title":"Entropy and the Complexity of Graphs Revisited","volume":"14","author":"Mowshowitz","year":"2012","journal-title":"Entropy"},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"3451","DOI":"10.1007\/s40747-022-00679-9","article-title":"Eigenvalue-based entropy and spectrum of bipartite digraph","volume":"8","author":"Sun","year":"2022","journal-title":"Complex Intell. Syst."},{"key":"ref_86","first-page":"041062","article-title":"Spectral Entropies as Information-Theoretic Tools for Complex Network Comparison","volume":"6","author":"Domenico","year":"2016","journal-title":"Phys. Rev. X"},{"key":"ref_87","doi-asserted-by":"crossref","unstructured":"Scholes, G. (2023). Large Coherent States Formed from Disordered k-Regular Random Graphs. Entropy, 20.","DOI":"10.3390\/e25111519"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/10\/1016\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,30]],"date-time":"2025-09-30T04:27:31Z","timestamp":1759206451000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/10\/1016"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,27]]},"references-count":87,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2025,10]]}},"alternative-id":["e27101016"],"URL":"https:\/\/doi.org\/10.3390\/e27101016","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,27]]}}}