# Properties of Vector Embeddings in Social Networks
## 4 Approach  
### Problem Statement  
**Problem 1--Explaining Embedding Relatedness:** Let \(G\) be a social network graph where every node \(u\in G\) has an embedding \(Y_{u}\) obtained from random-walk based embedding methods such as \(\text{DeepWalk}(u)\), \(\text{loc}(u)\), \(\text{node2 vec}(u)\), and \(\text{LINE}(u)\). The inner-product \(Y_{u}\cdot Y_{v}\) of two nodes \(u\) and \(v\) determines a similarity relation between the two nodes. Due to the random-walk based creation of the embeddings, we assume that this similarity can be approximated by similar network properties of the neighborhoods \(N(u)\) and \(N(v)\) of node \(u\) and \(v\), respectively. Therefore, we aim to explain the inner-product \(Y_{u}\cdot Y_{v}\) by a weighted linear combination of centrality measures obtained from the neighborhoods \(N(u)\) and \(N(v)\). Degree centrality \(\text{DC}(u)\), closeness centrality \(\text{CC}(u)\), betweenness centrality \(\text{BC}(u)\) and eigenvector centrality \(\text{EC}(u)\) have been chosen as centrality measures due to their importance in social network analysis and due to their well understood properties [35].  
Degree centrality of node \(u\) is simply the degree of \(u\)[36]:  
\[DC(u)=\text{deg}(u).\]  
Closeness centrality of a node is the average sum of the inverse of the distance to other nodes. Formally, closeness is defined as:  
\[CC(u)=\frac{1}{\sum_{v\in V}d(u,v)},\]  
where \(d(u,v)\) is the length of the shortest path between \((u,v)\)[36].  
Betweenness centrality counts the fraction of shortest paths going through a node. Betweenness centrality of a node \(u\) is then formally defined as follows:  
\[BC(u)=\sum_{s\
eq u\
eq t}\frac{\sigma_{s,t}(u)}{\sigma_{s,t}},\]  
where \(\sigma_{s,t}(u)\) is the number of shortest paths between node \(s\in V\) and \(t\in V\) that pass through node \(u\). \(\sigma_{s,t}\) is the number of shortest paths between node \(s\in V\) and \(t\in V\)[36].  
Eigenvector centrality generalizes degree centrality by incorporating the importance of the neighbors. The eigenvector centrality of \(u_{i}\) is a function of its neighbors' centralities. It is proportional to the summation of their centralities:  
\[EC(u_{i})=\frac{1}{\lambda}\sum_{j=1}^{n}A_{ij}EC(v_{j}),\]  
where \(\lambda\) is some fixed constant [36].  
Closeness centrality is widely used to study information flow in social networks [37]. Betweenness and eigenvector centrality are commonly used to detect and investigate community structure in social networks [38].  
**Problem 2--Predicting network properties:** In order to consider single nodes without a neighborhood, we consider the problem of predicting the centrality values \(\text{DC}(v)\), \(\text{CC}(v)\), \(\text{BC}(v)\) and \(\text{EC}(v)\) for a particular node \(v\) based on its embeddings only. We aim to find a nonlinear mapping between the embedding vector and a single centrality property of its node. By obtaining such an approximately correct mapping, we can conclude that the vector space of the embeddings retains the structural information of the network property.  
Overall, both case studies allow us to gain insights on the embedding properties in terms of centrality measures retained and the relatedness of their neighborhoods. In addition, predicting network properties successfully allows us to approximate computationally complex centrality measures more efficiently.