Multilayer networks composed of intralayer edges and interlayer edges are an important type of complex networks. Considering the heterogeneity of nodes and edges, it is necessary to design more reasonable and diverse community detection methods for multilayer networks. Existing research on community detection in multilayer networks mainly focuses on multiplexing networks (where the nodes are homogeneous and the edges are heterogeneous), but few studies have focused on heterogeneous multilayer networks where both nodes and edges represent different semantics. In this paper, we studied community detection on heterogeneous multilayer networks and proposed a motif-based detection algorithm. First, the communities and motifs of multilayer networks are defined, especially the interlayer motifs. Then, the modularity of multilayer networks based on these motifs is designed, and the community structure of the multilayer network is detected by maximizing the modularity of multilayer networks. Finally, we verify the effectiveness of the detection algorithm on synthetic networks. In the experiments on synthetic networks, comparing with the classical community detection algorithms (without considering interlayer heterogeneity), the motif-based modularity community detection algorithm can obtain better results under different evaluation indexes, and we found that there exists a certain relationship between motifs and communities. In addition, the proposed algorithm is applied in the empirical network, which shows its practicability in the real world. This study provides a solution for the investigation of heterogeneous information in multilayer networks.

The phenomenon of load redistribution in complex networks has garnered extensive attention due to its profound impact and widespread occurrence. In recent years, higher-order structures have offered new insights into understanding the structures and dynamic processes of complex networks. However, the influence of these higher-order structures on the dynamics of load redistribution, cascade failures, and recovery processes remains to be fully explored. In this study, we propose the load redistribution model with higher-order structures and recovery strategies of cascade failure based on functional upgrading and reconstruction mechanisms. In the cascading failure process with load redistribution and higher-order recovery strategies, we find that higher-order structures can induce a discontinuous phase transition at the low proportion of load redistribution, and the dynamic process displays a dual character of being robust yet fragile. These findings have been examined in both real and classical modeled networks. Interestingly, the largest connected component exhibits three distinct modes as the attack ratio increases at high densities of higher-order structures and recovery mechanisms.

Link prediction has a wide range of applications in the study of complex networks, and the current research on link prediction based on single-layer networks has achieved fruitful results, while link prediction methods for multilayer networks have to be further developed. Existing research on link prediction for multilayer networks mainly focuses on multiplexed networks with homogeneous nodes and heterogeneous edges, while there are relatively few studies on general multilayer networks with heterogeneous nodes and edges. In this context, this paper proposes a method for heterogeneous multilayer networks based on motifs for link prediction. The method considers not only the effect of heterogeneity of edges on network links but also the effect of heterogeneous and homogeneous nodes on the existence of links between nodes. In addition, we use the role function of nodes to measure the contribution of nodes to form the motifs with links in different layers of the network, thus enabling the prediction of intra- and inter-layer links on heterogeneous multilayer networks. Finally, we apply the method to several empirical networks and find that our method has better link prediction performance than several other link prediction methods on multilayer networks.