# Deep Cross-Network Alignment with Anchor Node Pair Diverse Local Structure
## 1 Introduction
We present the analysis for labeled anchor node pairs' diverse local structure in two real-world social network datasets collected from Twitter and Foursquare [3] in Figure 2. It is observed that the proportion of commonly labeled anchor node neighbors for anchor nodes in different networks is differential. Figure 1: An example of anchor node pairs’ diverse connection status. Given two networks, the relationships between users within each network are represented by solid black lines, while the yellow dashed line connects the common users (i.e., anchor nodes) across the networks. (**a**) Equivalence connection status. (**b**) Non-equivalence connection status. (**c**) Containment connection status. (**d**) Non-containment connection status. However, most embedding-based network alignment methods will result in node pairs that are more likely to present an equivalence connection status. Specifically, these methods generally hold the assumption that anchor node pairs have similar local structures and/or neighbors across different networks (i.e., the situation as shown in Figure 1a,c). For example, IONE [11] preserves the proximity of users that exhibit \"similar\" sets of followers and followees in the embedded space; FRUI-P [19] extracts the friend feature of each user in a social network, constructs a corresponding friend feature vector, and, subsequently, computes the similarities among all potential identical users between two social networks. In addition, employing network embedding techniques, such as DeepWalk and GCN, to represent each node as a point vector, and using labeled anchor nodes to learn mapping functions will lead to align most nodes in one network with the labeled anchor nodes in the other network. Furthermore, it is difficult to distinguish the labeled anchor nodes and their neighbors due to the neighboring nodes are embedded in closed forms in most network embedding. As shown in Figure 1c, if we use \(v_{1}\) to learn the mapping function, \(v_{2}\), \(v_{4}\), and \(v_{5}\) are more likely to be close to \(v_{1}\) in \(G^{2}\). Since the representations of \(v_{1}\) and \(v_{2}\) are similar in \(G^{2}\), it is difficult for \(v_{2}\) in \(G^{1}\) to match corresponding nodes in \(G^{2}\). To address the above problems, we propose a novel model, namely DCSNA, which adjusts to Diverse anchor nodes Connection Status for Network Alignment. Firstly, considering the different impacts of labeled anchor nodes and other nodes in the neighborhood among node pairs between different networks, we design a masked network encoder to learn node representations by removing some edges to mitigate the influence of some specific nodes. Not all neighboring nodes are equally informative for alignment, as some nodes interact with many nodes across networks. Our encoder selectively incorporates only the relevant neighbors to learn more effective representations. Secondly, to better distinguish a node and its neighbors, we represent each node as a Gaussian distribution instead of a point vector. Thus, we obtain node representations with mean and variance embeddings, where the mean embeddings present the position of the nodes and variance embeddings present the uncertainties of the representations. Meanwhile, compared to representing nodes as deterministic point vectors, using Gaussian distribution to represent nodes could tolerate the network's structural diversity to a large extent which may be the reason for anchor node pairs' diverse connection status, and benefit for network alignment. Figure 2: The common neighbors’ analysis of anchor node pairs in real social networks (Twitter–Foursquare). For a given pair of anchor nodes \((v,u)\), \(r\) is the ration of their common labeled anchor node neighbors to \(v/u\)’s neighbors. The abscissa represents the \(r\) in Twitter, and the ordinate represents \(r\) in Foursquare. Thirdly, because of the diverse connection status of anchor node pairs, it is difficult to generalize the learned information on the labeled anchor node pairs to other node pairs.