| Since the beginning of this century, complex network has received much attention in various scientific fields. The advances in the research of complex networks not only offer a new perspective for people to understand the complex-ity of practical systems, but also provide an ideal framework to model them. At the same time, various methods and tools have been developed to help us effec-tively analyzing and controlling the behaviors of complex system. At present,following subjects are particularly concerned about: (?) The influences of net-work structures on the function of systems (forward problem) have always been the focus of related researches. When a network system is formed by simple dynamical units though a complex network, new collective behaviors, which network nodes do not possess, can emerge and such emergence behavious have attracted great interest of researchers from various interdisciplinary fields, due to their practical and theoretical significance, (?) How to reconstruct a net-work from the measured data (inverse problem) has attracted more and more attention recently. With the technical advancement of measurement and great improvement of storage capability and computing power, we have entered an"big data" era. In all areas, especially in the biological and social fields, the data are accumulating at all times. How to dig out as much information as possible from these data, especially to recover the dynamical mechanism and network structure hidden beneath the data, has always been the focus of attentions.In the researches of forward problem A first-order phase transition named explosive synchronization (ES) has been identified in various complex net-works composed of oscillatory nodes. Explosive synchronization refers to novel and interesting phenomena of transitions from incoherent states where differ-ent nodes have diversely distributed frequencies to coherent ones with all nodes having an identical frequency, or vice versa, and many works have argued that certain parameter conditions on local dynamics and network structures are re-quired as the causes of explosive synchronization. Therefore, the phenomenon profoundly reveals the impact of network structure on its functions. In the study of inverse problems, The role of noise in network reconstruction has recently intrigued broad interest. In network reconstructions of practical system, various difficulties always appear jointly: unknown nonlinearity of network dynamics,unknown complexity of interaction structures between nodes and unknown and strong noises during data production. In some cases, even the data of part nodes cannot be reached. Therefore, it turns to be crucial to develop effective methods to overcome these difficulties. On these subjects, we have made the following progress:(I) We find that explosive synchronization and the equired frequency-degree correlation can be self-organized in a very wide range of oscillatory networks and therefore the parameter constraints for ES required by all pre-vious ES works can be completely lifted. The abrupt ES transitions are natural and general emergent phenomena of heterogeneous oscillatory networks that universally appear around Hopf bifurcation. In the discussion, we first focus on the ES phenomena of the diffusively coupled complex Ginzburg-Landau equations (CGLE) and then directly compute the general network of reaction-diffusion equations to confirm the predictions from CGLE networks. We also show (through both theoretical derivations and numerical verifications) how to identify parameters of general RDE networks to the scaled control parameters of CGLE through the order parameter representation, and thus the ES phenom-ena can be predicted in realistic systems for practically controllable parameters.Especially, the parameter region for ES processes are explored and it will be an excellent and useful guidance for experimentalists about where and how to ex-plore ES events, and also excellent predictions about where high ES risks exist in realistic systems whenever these events are desirable or harmful.(II) We study the problem of inferring noise-driven nonlinear dynamic networks with measurable data of node variables only. A high-order correla-tion computation (HOCC) method is proposed to unifiedly treat nonlinear dy-namic structures, coupling topologies and statistics of additive and multiplica-tive noises in networks. This method treats network reconstruction by jointly considering three facts: choosing suitable basis and correlator vectors to expand nonlinear terms of networks; adjusting correlation time difference to decorre-late noise effects; and applying high-order correlations to derive linear matrix equations to infer nonlinear structures, topologies and noise correlation matri-ces. The HOCC algorithm has been theoretically derived, and its predictions are well confirmed by numerical results. Moreover, the problems how to treat measurement noises and colored noises and how reconstruction errors of the HOCC method depend on data length, system size and choice of basis sets are also discussed.(III) We propose a method for inferring networks driven by strong noise when some node variables are hidden. By jointly applying three approaches,namely using high-order correlations to treat nonlinearities, using two-time cor-relations to decorrelate noise effects, and using second-order derivatives to deal with hidden variables, we are able to successfully reconstruct noise-driven nonlinear networks with some hidden variables. In addition, we are particu-larly concerned about the reconstruction of dynamical networks driven by color noises. On one hand, we can regard it as a special case of the inverse problem with hidden variables, by considering the dynamics of both networks and the colored noise together; On the other hand, we derive iterating multiple non-linear matrix equations to depict network topologies and statistics for colored noise. The validity of above methods is confirmed by numerical results. |
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