Percolation phase transition in the heterogeneous multi-coupled interdependent network

In previous studies, complex system models have often been set up in an oversimplified manner and the failure mechanism for nodes affected by each other is not correspond to reality. Accordingly, we develop a new model of heterogeneous multi-coupled interdependent networks and analyze the change in the giant component of the network after the attack in conjunction with the load redistribution mechanism. We first analyze the cascading failure process of the network analytically by recursive methods, estimate the critical phase transition threshold and the size of the giant component of the remaining network after network percolation, and find that the threshold of the WS sub-network is positively related to the average degree, and the BA sub-network is related to the power exponent ?????? of the distribution. When ?????? is less than or equal to 2, there is always a giant component in the network. After ?????? is greater than 3.4788 & BULL;& BULL;& BULL;, there is no giant component. When ?????? is between 2 to 3.4788 & BULL;& BULL;& BULL;, we found that the threshold of the sub-network is more influenced by the proportion of attacks (??????1 and ??????2), and when ??????1 is larger, the sub-networks interact with each other, resulting in a smaller threshold for the BA sub-network, while the threshold of the WS sub-network is smaller in other cases. Finally, the cross-sectional comparison illustrates that as ?????? increases to 3.4788 & BULL;& BULL;& BULL;, the robustness of the BA sub-network increases, contrary to the findings for single-layer networks. Overall, the heterogeneous multi-coupled interdependent network is more robust, and the giant component is greatly affected by the low robust sub-network, but its existence is related to the stock of nodes.