| As human society develops,various complex systems such as communication systems,transportation systems,and the internet are rapidly advancing.Complex systems share common characteristics of self-organization,self-adaptation,and evolution.Therefore,researchers based on the network began using the methods of network science to study complex systems.The higher-order structure of a network is a research hotspot in complex networks,which is closely related to the efficient transmission of information and other functions in the network.Disrupting the connectivity of the higher-order network can cause significant damage to the network’s functionality.At the same time,it has been pointed out that the functions of complex networks depend on the giant components of the original network(also known as a low-order network).Therefore,the network’s function will be influenced by both the low-order network and its corresponding higher-order network.However,traditional research has separated the higher-order and low-order networks and ignored their interactions.To investigate this problem,a network model combining a directed low-order network and a higher-order network(referred to as the higher-low-order coupling network,ILH)is established in this study.The robustness of this model is explored using percolation theory.Subsequently,a greedy algorithm is employed to enhance the network’s robustness by modifying the network structure while preserving the network degree distribution.Finally,this paper examines the robustness of higher-low-order coupling network with different coupling strengths using three coupling methods.The main research content of this paper is as follows:(1)When some nodes in the low-order network fail,the giant components of the low-order network will shrink,which will lead to changes in the structure of the higher-order network,and changes in the higher-order network will in turn affect the low-order network.This process will happen repeatedly,and the propagation of faults may eventually bring down the entire network.Most complex networks in the real world are directed networks.Therefore,this paper proposes a network model that couples a directed low-order network with its corresponding higher-order network,and studies the robustness of the ILH based on percolation theory.By removing nodes in the network and observing the connectivity of the remaining nodes,the ability of the network to maintain normal operation when under attack is studied.Use the algorithm to generate directed random networks,scale-free networks and small-world networks,and study the robustness of ILH on these three classic networks.The results of network percolation show that characterize the first-order phase transition.Then,experiments were carried out on 14 real networks such as social networks,communication networks,and neural networks.The results show that the robustness of the ILH is significantly weaker compared to the original network.(2)Through research,it has been found that ILH has weak robustness and is more prone to failure when under attack,with a poorer ability to maintain normal network function.To address this issue,without adding additional nodes and edges,the edges in the network are reconnected,and a greedy algorithm is used to change the network’s topology,taking the robustness of both higher-order and low-order networks as optimization criteria,to make the network more robust and effectively enhance the robustness of the higher-low-order coupling network.(3)This paper proposes a higher-low-order coupling network model with adjustable coupling strength.Three methods,namely random coupling,degree-correlated coupling(prioritizing high-degree nodes and prioritizing low-degree nodes),and localized coupling,are utilized to adjust the coupling strength of the network.Two robustness evaluation metrics are employed to investigate the network’s robustness.Verification experiments are conducted on random networks,scale-free networks,small-world networks,and real-world networks.The results demonstrate that the higher-low-order coupling network exhibits complex percolation behavior as the coupling strength varies.Moreover,the network’s robustness decreases as the coupling strength increases. |
