# Properties of Vector Embeddings in Social Networks  
[PERSON]  
Correspondence: [PERSON][PERSON]  
15 July 201715 July  
Word2 Vec [13] model to train node representations. LINE [14] optimizes the objective function to preserve both the local and the global network structures. In extension to these methods, Node2 vec [10] introduces mechanisms to balance the random walk between Breadth-First Search (BFS) sampling and Depth-First Search (DFS) sampling, thereby allowing to preserve more local or global structures. In [15], authors exploit an extension of the Word2 Vec algorithm, the Paragraph Vector [16], to learn embeddings for local regions of the network. These local regions are defined as neighborhoods around a focal node, as it is usually seen in ego-networks [17]. Overall, graph embeddings have shown promising performance for various graph analytic tasks such as node classification [17], clustering [18], link prediction [19], knowledge extraction from semantic networks [20] and network visualization [21]. However, due to the stochastic nature of the random walk, it remains unclear what graph properties are retained exactly in a given set of embeddings. Understanding the rather abstract vectorial representation of a node embedding can be compared to the effort in understanding the structure of neural networks in image processing. One possible solution in the field of image processing, therefore, is to reconstruct exemplary images from particularly activated parts of the neural network [22]. Since neural networks are usually organized into different layers detecting features of different granularity, we can find the (usually abstract) input image that maximizes a neuron of a layer. Hence, the role of a particular element in the neural network can be described in terms of input that maximizes its activation. A similar idea would be quite helpful for graph embeddings when one wants to know the parts of the social network, i.e., the input space for our network-embedding problem, which are preserved. Hence, the abstract vectorial representation of a node can be mapped back in input space to understand the properties observed by the representation. However, to the best of our knowledge, there is no method and corresponding study which investigates how to analyze graph properties preserved by random-walk based graph embeddings. Moreover, when certain graph properties are retained in the vector embeddings, we can try to reconstruct those properties given the embeddings alone. Since random-walk based graph embeddings can be computed in linear time [10], finding a way to reconstruct graph measures from those embeddings with a high computational complexity, like, for example, closeness or betweenness centrality [24], could allow us to find computationally efficient approximations. Given this motivation, we make the following contributions in our work:  
**Contribution 1:** We propose a new methodology to identify graph properties preserved by random-walk based graph embedding methods (Section 4). In particular, we are interested in graph properties explaining the similarity of the local neighborhood of a node. In addition, two nodes are accounted to be similar if they show similar graph characteristics in their local neighborhood, like, for example, similar topological structures (e.g., same degree) or comparable centrality and betweenness distributions. Therefore, we propose to relate the neighborhood-similarity between two nodes to the similarity of the embeddings by solving a learning to rank problem. Hence, the network measure with the highest contribution to the similarity of two nodes can be considered as the highest explanatory factor regarding their relatedness in embedding space. The results show that random-walk based embedding techniques can be grouped into two categories, depending on the properties of the random walk. LINE and DeepWalk walk globally over the graph and thus learn betweenness and eigenvector centrality. Node2 vec and our Paragraph Vector method retain local properties such as degree centrality.