| The complex network is one of the most popular topics in current research,which provides abstract mathematical models and analytical methodologies for the study of numerous kinds of actual systems.Since the interactions between nodes can shape the behavior and function of the network,more and more researchers are concentrating on the analysis and discussion of the network structure.However,due to the limitation of measurement techniques and measurement costs,people usually lack direct access to the network structure of actual systems.With the rapid advancement of data-collecting and processing capabilities,network reconstruction has emerged as the most effective solution to this challenge.Network reconstruction refers to the process of inferring node interactions from time series data generated from the network.In the study of network reconstruction,the nonlinearity of the system and the complexity of the network structure were the initial concerns of many researchers.But as the studies progressed,various problems in the actual system brought new challenges to the network reconstruction.Only a portion of the variables and nodes can typically be measured during the data collecting process for actual systems,and the other unmeasurable nodes and variables are what we call hidden nodes and hidden variables.Since these nodes and variables are not isolated in the network but will have an influence on the measurable nodes and variables,these non-negligible unknown effects will frequently result in considerable deviations in conventional network reconstruction methods.In addition,there are also problems arising from both nonlinear dynamics and noise influence in the actual system,and all these complicated situations are intertwined,making the implementation of network reconstruction more difficult.Therefore,it is a challenging and practical topic to achieve network reconstruction under hidden information of lacking data of variables and nodes.In this paper,we mainly achieve the following advancements in research on this topic:We propose a network reconstruction method for inferring node interactions in nonlinear systems with linearizable hidden variables.By selecting a proper basis and using it as the expanded variables,we achieve the linearization of the original nonlinear dynamical system;based on the linear dynamical functions of hidden variables,we substitute the hidden variables into functions of measurable variables,thus eliminating the unknown effect of hidden variables.In addition,since the hidden variables create new unknowns in the dynamical equations,we construct a new equation by differentiating the original dynamics,thus solving the underdetermined equations of the system.We validate the reconstruction efficiency of the method in various systems.The numerical results demonstrate that our proposed method can accurately reconstruct systems containing linearizable hidden variables and still has good robustness in systems that do not strictly satisfy the linearizability condition.Moreover,the comparison with the reconstruction method based on linearizable hidden variables (RM-LHV method) also shows that our method can significantly improve the reconstruction accuracy for nonlinear systems by introducing expanded variables.For dynamical networks containing hidden nodes,we propose a detection method for hidden nodes in the network based on random variable resetting.We first reconstruct the local dynamics of the measurable nodes and the couplings between them in the network,taking advantage of the characteristics of the random variable resetting method that are unaffected by hidden nodes and noise;next,we compute the time series comprising the interactions from hidden nodes and white noise based on the dynamical equations of the measurable nodes;finally,we eliminate the noise influence by calculating the autocovariance of the time series,thus giving the criterion for identifying the hidden nodes in the network.Our proposed detection method is theoretically independent of the proportion of hidden nodes,network type,network size,and other factors because of the characteristics of the random variable resetting method,At the same time,our proposed criterion can directly reflect the statistical characteristics of the interactions from hidden nodes,thus reducing the influence of other interactions in the network on the detection results.We verify the accuracy of the theoretical derivation through numerical simulation results in several discrete and continuous systems,and demonstrate the robustness of our detection method under different conditions.On the basis of the reconstruction method based on linearizable hidden variables (RM-LHV method),we propose a theoretical framework for reconstructing the effective connections of neural networks from the experimental neural signals.Due to the inaccessibility of high-dimensional variable during neural signal measurement,we choose the RM-LHV method to accomplish the network reconstruction,which can effectively address the influence of hidden variables in the system.However,since the RM-LHV method assumes that the system can be linearly expanded near the stable point,and its reconstruction accuracy is influenced by noise and hidden nodes,we propose the following network reconstruction framework to reduce the reconstruction deviations introduced by differences between the theoretical model and the actual system:we first use data preprocessing to eliminate inaccurate data and minimize the influence of noise,then we use the RM-LHV method to reconstruct the pre-processed data,and finally,we use Gaussian mixture clustering to identify the connections of the network.We theoretically investigate the rationality and error causes of the proposed network reconstruction framework.Using the LFP data from the rat hippocampus,we employ this theoretical framework to reconstruct the effective connections between relevant regions,and the reconstruction results are essentially comparable with the structural properties of the actual neural network of C.elegans. |
