The higher-order structure of networks is a hot research topic in complex networks. It has received much attention because it is closely related to the functionality of networks, such as network transportation and propagation. For instance, recent studies have revealed that studying higher-order networks can explore hub structures in transportation networks and information dissemination units in neuronal networks. Therefore, the destruction of the connectivity of higher-order networks will cause significant damage to network functionalities. Meanwhile, previous works pointed out that the function of a complex network depends on the giant component of the original(low-order) network. Therefore, the network functionality will be influenced by both the low-order and its corresponding higher-order network. To study this issue, we build a network model of the interdependence of low-order and higher-order networks (we call it ILH). When some low-order network nodes fail, the low-order network’s giant component shrinks, leading to changes in the structure of the higher-order network, which further affects the low-order network. This process occurs iteratively; the propagation of the failure can lead to an eventual network crash. We conducted experiments on different networks based on the percolation theory, and our network percolation results demonstrated a first-order phase transition feature. In particular, we found that an ILH is more fragile than the low-order network alone, and an ILH is more likely to be corrupted in the event of a random node failure.

Malicious attacks can cause significant damage to the structure and functionality of complex networks. Previous research has pointed out that the ability of networks to withstand malicious attacks becomes weaker when networks are coupled. However, traditional research on improving the robustness of networks has focused on individual low-order or higher-order networks, lacking studies on coupled networks with higher-order and low-order networks. This paper proposes a method for optimizing the robustness of coupled networks with higher-order and low-order based on a simulated annealing algorithm to address this issue. Without altering the network’s degree distribution, the method rewires the edges, taking the robustness of low-order and higher-order networks as joint optimization objectives. Making minimal changes to the network, the method effectively enhances the robustness of coupled networks. Experiments were conducted on Erdős–Rényi random networks (ER), scale-free networks (BA), and small-world networks (SW). Finally, validation was performed on various real networks. The results indicate that this method can effectively enhance the robustness of coupled networks with higher-order and low-order.

Link prediction is a hot issue in information filtering. Link prediction algorithms, based on local similarity indices, are widely used in many fields due to their high efficiency and high prediction accuracy. However, most existing link prediction algorithms are available for unweighted networks, and there are relatively few studies for weighted networks. In the previous studies on weighted networks, some scholars pointed out that links with small weights play a more important role in link prediction and emphasized that weak-ties theory has a significant impact on prediction accuracy. On this basis, we studied the edges with different weights, and we discovered that, for edges with large weights, this weak-ties theory actually does not work; Instead, the weak-ties theory works in the prediction of edges with small weights. Our discovery has instructive implications for link predictions in weighted networks.