# Deep Cross-Network Alignment with Anchor Node Pair Diverse Local Structure
## 1 Introduction  
Network analysis, particularly cross-network analysis, has recently been gaining attention in various study domains due to the exceptional rise of the diversified information that has produced a significant volume of networks. The key to transferring information across multiple networks is to identify hidden inter-network links (i.e., anchor links) [1, 2, 3], which connect nodes that are shared among different networks (i.e., anchor nodes) [4]. For example, a person's account nodes in different social networks can be regarded as anchor nodes, and the corresponding relationship between these two accounts can be regarded as anchor links. The process of finding corresponding anchor nodes is also called network alignment (NA). Comparative studies of specific tasks, such as cross-network recommendation [5], mutual community detection [6], and genetic disease classification [7], can be conducted at the systems level with help of NA. Various NA methods have been proposed and most of them are based on topological similarity, and sometimes the attribute similarity is fused to determine the corresponding relationship between nodes. Inspired by network embedding [8, 9, 10], which has the potent ability to represent network structures into a low-dimension embedding, numerous embedding-based network alignment approaches have been presented [11, 12, 13]. Recent embedding-based network alignment methods first, typically, learn node representations before learning mapping functions over latent representation spaces to decrease the disparity in node representations across different networks [14], whereas learning mappingfunctions usually use labeled anchor node pairs as supervision information [12; 15; 16; 17; 18]. Although there has been notable progress in the field of network alignment, current methods still have several deficiencies. One of the most significant limitations is that most methods tend to overlook the diverse local structure of anchor node pairs. Consequently, these approaches are often only suitable for specific alignment cases and fail to apply in some real-world scenarios. Specifically, anchor node pairs' local structure, i.e., the connection status of anchor node pairs across networks can be roughly divided into four situations according to labeled anchor nodes, as shown in Figure 1. Equivalence connection status. As shown in Figure 1a, anchor node \(v_{1}\) \"replicates\" its interaction in network1 to network \(G^{2}\), i.e., \(v_{1}\) tends to maintain a similar interaction with the same neighbors. This connection status demonstrates that an anchor node \(v_{1}\) in different networks has a similar local topology structure (such as node degree) and connects to similar neighborhoods. Non-equivalence connection status. As shown in Figure 1b, the activity of anchor node \(v_{1}\) in the two networks differs a lot, i.e., \(v_{1}\) has different participation in the two networks and interacts with different neighbors in the two networks. This connection status demonstrates that anchor node \(v_{1}\) has a different local topology structure and connects the different neighborhood nodes as well. Containment connection status. As shown in Figure 1c, anchor node \(v_{1}\) chooses a part of the same neighbors in \(G^{1}\) to interact in \(G^{2}\), i.e., \(v_{1}\) only \"replicates\" part of the neighbors in \(G^{1}\) to \(G^{2}\) and follow some new friends in \(G^{2}\). This connection status demonstrates that anchor node \(v_{1}\) in different networks has different local topology structures but connects similar neighborhood nodes. Non-containment connection status. As shown in Figure 1d, anchor node \(v_{1}\) divides its time/interest equally between the two networks but interacts with a different neighborhood. This connection status demonstrates that anchor node \(v_{1}\) has a similar local topology structure but connects the different neighborhood nodes.