现实世界中，系统是一切事物的基本存在方式。系统，尤其是复杂系统，一般由相互作用的子系统构成，往往是多尺度的且具有层次性。复杂系统的功能涌现和状态演化并非是子系统属性的简单叠加，其内在机制的揭示依赖于系统相变与临界性研究。一般地，复杂自组织系统往往与外界存在能量和物质交换，通常这些系统处于非平衡状态。虽然平衡系统临界性的各种研究方法已经相对成熟，也取得一些重要的研究结果，但是这些方法并不能完全照搬到非平衡系统。重要的是，真实系统的序参量、哈密顿量和能量分布等统计物理量并不已知，如何通过观测数据研究这些系统的集体行为和相变与临界现象是亟待解决的科学问题。从吉普斯统计系综出发，基于观测数据或模拟数据，构建微观态构型和统计系综矩阵，利用本征微观态凝聚和系综矩阵奇异值研究系统临界性，可以研究各类复杂系统非平衡相变。


从系统科学的研究视角来看，在物理学、化学、生物学等科学领域，同一普适类中经典系统与衍生系统之间均存在普适关系，经典系统归纳总结出的建模思想和研究方法也适用于衍生系统，甚至是其他研究领域的复杂系统。目前，本征微观态方法已经推广到平衡系统规则晶格的Ising模型和非平衡系统的经典Vicsek模型，其研究方法和结论可用于高维任意相铁磁系统和多智能体系统。众所周知，同步和传播是复杂系统中``流驱动"动力学的经典研究问题。本质上，两者都是个体状态在系统中传输，其理论研究方法有时也是相通的。关于非平衡复杂系统动力学研究，如耦合振子同步和具有吸收态的疾病传播问题，本征微观态方法尚未涉足，其推广意义不言而喻。一方面，耦合振子系统相变与临界性的本征微观态研究有助于揭示神经系统、心脏系统和大脑系统等振荡系统中结构与功能涌现的关系。另一方面，基于本征性指标的传统疾病传播模型阈值分析更适用于融合个体移动和多种群相互作用的集合种群模型。该方法在两类非平衡系统经典模型的推广，对各自系统对应的实际问题研究具有重要的、开创性的指导意义。基于上述研究背景和本征微观态方法，本文研究了非平衡系统中相位耦合振子同步和SIR疾病传播动力学，开展了如下研究工作：


1、本文提出了相位振子模型微观态构型，将本征微观态方法推广到经典Kuramoto耦合振子系统，给出了微观态奇异值和微观态构型涨落的有限尺度标度。通过系统的数值计算，我们发现本征微观态方法得到的系统临界性与经典平均场方法基本一致。但是，在固有频率规则分布中，实验结果显示振子频率的随机排布并没有影响关联长度的发散性，只是影响了微观态涨落的发散，改变了其标度指数$\gamma$的估计值。与经典方法不同，本征微观态方法得到的Kuramoto模型的临界性质均不满足超标度关系$\gamma=d\nu-2\beta$，这符合系统非平衡性质。另外，系统本征微观态的时空分布还展示出了振子系统的介观结构以及振子的主运动模式，这些是以往经典方法无法获取的时空信息。


2、在Kuramoto模型研究基础上，将本征微观态方法推广到非全同异质阻挫相位振子的幂律耦合模型的奇异态研究中。通过局部序参量粗粒度分析法和Ott-Aantonsen降维法的临界性分析得到了系统相干态涌现的基本条件，通过本征微观态方法还给出了奇异态的本征微观态介观结构，揭示了部分锁相同步态与奇异态之间的本质差别，归纳得出奇异态现象是在弱耦合强度下涌现出的特殊群体现象，其依赖于系统中个体的相互作用结构和系统非平衡基态。


3、将本征微观态方法推广到SIR疾病传播模型，并利用奇异值在临界点处的尺度发散性给出了估计传播阈值的本征性指标。与经典的敏感性指标和变异性指标相比，本征性指标在静态配置网络和真实网络中的估计阈值要更加接近异质平均场和淬火平均场的理论阈值。另外，静态网络研究发现与同配异配性相比，网络平均度(或者网络拓扑维数)对阈值估计的影响更加显著。在周期切换网络中，本征性指标比其他两种指标表现的更加稳定，这说明本征性指标更适合复杂机制下的阈值估计。


4、基于主体模型和集合种群模型，建立了网络化集合种群模型，并提出了非马尔可夫过程的静态迁移节点搜索算法。在理论分析中，通过下一代矩阵法得到了基本再生数$R_0$，理论阈值和本征性指标估计阈值均发现迁移不会直接影响疾病复制能力。在疾病扩散机制研究中，算法随机模拟结果显示当$R_0>1$时迁移会使易感种群出现双稳态，即由亚稳态(无病平衡状态)逐渐演化到稳态(地方病平衡状态)，推动了疾病的大范围传播。最后，利用节点搜索算法和前94个城市新冠确诊病例、人口迁移和政府防控措施等关键因素对新冠疫情传播进行反演，模拟结果的染病者数量和代数分布验证了该算法的准确性以及种群内部接触结构的异质性。此外，模拟结果与实证数据的对比表明在疫情初期中国政府采取的积极防疫措施至少减少了400余万病例。

In the real world, systems are the fundamental forms of everything. Systems, especially complex systems, are generally composed of interacting subsystems and are often multi-scale and hierarchical. The function emergence and state evolution of complex systems are not simply the superposition of the properties of subsystems. The discovery of their internal mechanisms depends on the study of phase transition and criticality. In general, complex self-organizing systems, exchanging energy and material with the outside, are usually in a non-equilibrium state. Although there exist various and mature research methods about the criticality and some important results obtained in equilibrium systems, these methods can not be directly applied to to non-equilibrium systems. It is important that in some real systems, the order parameters, hamiltonian and energy distribution are not known. How to study the collective behavior, phase transition and critical phenomena of these systems just through observational data is an urgent scientific problem. Based on the Gipps statistical ensemble, the microstate configuration and statistical ensemble matrix are constructed based on the observed or simulated data, and the criticality of the system is studied through the condensation of the eigen microstates and the singular values of the ensemble matrix, which will open a new channel for the study of non-equilibrium phase transitions in real complex systems.


From the perspective of systems science, in physics, chemistry, biology and other scientific fields, there always exist universality between classical systems and their derived systems in the same universal class. The modeling ideas and research methods summarized by classical systems can also be applied to derived systems, even complex systems in other research fields. At present, the Eigen Microstate(EM) method has been extended to the 1d and 2d Ising model in equilibrium system and the classical Vicsek model in non-equilibrium system. The research method and conclusion can be applied to high-dimensional arbitrary phase ferromagnetic system and multi-agent system. It is well known that synchronization and transmission are classical problems belonging to ``flow-driven dynamics" in complex systems. In essence, both of them study the transmission of individuals' state, and sometimes their theoretical research methods are similar. For these two kinds of non-equilibrium complex systems, the coupled oscillator systems and transmission system with absorbing state, this method has not been introduced, and its popularization significance is self-evident. On one hand, the study of the eigen microstate about the phase transition and criticality in coupled oscillator systems is helpful to reveal the relationship between structure and function emergence in oscillatory systems such as nervous system, heart system and brain system. On the other hand, the threshold estimation of the traditional disease transmission model based on the eigen measure is more suitable for the metapopulation model integrating individual mobility and multi-population interaction. The extension of EM method to the classical models in non-equilibrium systems has important and pioneering guiding significance for the study of practical problems. Based on the above research background and the EM method, we conduct the following study:


1. We propose the microstate configuration of the phase oscillator model, and extend the EM method to the classical Kuramoto coupled oscillator system, and give the finite-scaling of singular values of eigen microstate and the fluctuations of the microstate configuration. Numerical results find that the criticality obtained by the EM method are consistent with the classical mean-field method. However, under the regular distribution, the scaling exponent $\nu$ corresponding to the correlation length is the same as the random distribution, but only affects the divergence of the microstate fluctuations, and then changes the estimated value of its scaling exponent $\gamma$. Different from the classical method, the critical properties of the Kuramoto model obtained by the EM method do not satisfy the hyperscaling relation $\gamma= D \nu-2\beta$, which is consistent with the non-equilibrium system property. In addition, the spatial and temporal distribution of the eigen microstates can also show the mesoscopic structure and the main motion mode of oscillators, which cannot be obtained by classical mean-field methods.


2. Based on the study of the classical Kuramoto model, the EM method is extended to the study of the chimera states of nonidentical power-law coupled oscillators with heterogeneous phase lag. The conditions emerging coherent states are obtained by the coarse-grain analysis of local order parameters and the criticality analysis of Ott-Aantonsen reducing dimensions method. Further，the EM method gives eigen microstate mesoscopic structure of chimera states, and reveals that the essential difference between the chimera state and phase-locked synchronization, which supports that the chimera state is a special group phenomenon emerging under weak coupling strength, and it depends on the interaction structure of individuals and the non-equilibrium ground state.


3. We introduce the EM method the SIR disease transmission model, and devise the eigen measure for estimating the transmission threshold by the scaling divergence of the singular values near the critical point. Compared with the classical susceptibility measure and variability measure, the estimated threshold by eigen measure in the static configuration network and the real network is closer to the theoretical threshold of the heterogeneous mean-field and the quenched mean-field. In addition, compared with assortativity, network mean degree (or network topology dimension) has a more significant impact on threshold estimation. In the periodic switching double-layer network, the eigen measure is more stable than the other two measures, which indicates that our measure is more suitable for threshold estimation under complex mechanism.


4. Based on the agent-based and metapopulation model, we establish the networked metapopulation model and propose the node-search algorithm of static network migration ansatz based on non-Markov process. In theoretical analysis, we obtain the basic reproduction number $R_0$ by the next generation matrix method, and find that migration doesn't directly affect the disease reproduction ability. In the study of epidemics diffusion mechanism, stochastic simulations show that when $R_0>1$, the susceptible population will evolve from metastable state (disease-free equilibrium state) to steady state (endemic equilibrium state), and then increase the spreading range of disease. Finally, the node search algorithm combining some key factors, such as confirmed cases in top 94 cities, population migration and government prevention and control measures, are used to re-exercise the COVID-19 in Wuhan. The infected number and generation distribution of simulated results verify the accuracy of our algorithm and diffusion mechanism as well as the heterogeneity of contact network within populations. In addition, the results estimate that the positive measures taken by the Chinese government prevented at least 4 million people from being infected in the early stages in Wuhan.