# Properties of Vector Embeddings in Social Networks
## 5 Experiments
### Dataset  
Since our learning-to-rank method is based on estimating node similarities using distributional properties of their neighborhoods, we focus our experiments on data sets providing such structures. Ego-networks serve as possible datasets. In Ego-networks, every graph contains sub-graphs around a focal node called ego. The idea is to break up the large graph into smaller, easier to manage components and study the properties of the subgraphs. The ego-network model allows the small patterns, anomalies, and features to be discovered that would be missed when an entire graph is analyzed [15]. Therefore, we use ego-networks from three major social networking sites: Facebook, Google+, and Twitter, available from the University of Stanford [17]. Table 1 describes the details of the datasets we used in our experiments.  
\begin{table}
\begin{tabular}{l l r r r} \hline \hline  & & **Facebook** & **Twitter** & **Google+** \\ \hline nodes & \(|V|\) & 4,039 & 81,306 & 107,614 \\ edges & \(|E|\) & 88,234 & 1,768,149 & 13,673,453 \\ egos & \(|U|\) & 10 & 973 & 132 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Statistics of social network datasets.  
Moreover, we generate an artificial scale-free graph utilizing the Barabasi-Albert model Barabasi-Albert (2017). The Barabasi-Albert model is an algorithm generating random graphs using a preferential attachment process. The process starts with an initial graph of \(m\) nodes. One new node is added to the network at each time step \(t\in\mathbb{N}\). In more detail, the preferential attachment process works as follows:  
* With a probability \(p\in[0,1]\), this new node connects to \(m\) existing nodes uniformly at random.
* With a probability \(1-p\), this new node connects to \(m\) existing nodes with a probability proportional to the degree of node which it will be connected to.  
We simulate the Facebook graph using the Barabasi-Albert model, which generates 87,516 edges for 4000 nodes. Similar to the Facebook dataset, we divide the Barabasi-Albert graph into 10 ego-networks. The community library in Python (2017) properly clusters the graph into several communities. We consider the node with highest betweenness as ego for each subgraph Barabasi and Albert (2017). For further investigations, we compare degree, closeness, betweenness and eigenvector centrality distribution between one ego-network of the Facebook and one from the artificial graph. Figure 1 demonstrates centrality distribution of two random egos; ego '686' of the Facebook and ego '25' of the artificial dataset. We then require a metric to measure whether two distributions are identical. The Kolmogorov-Smirnov test [PERSON] (1961) is the most popular test to find identical distributions. Therefore, we consider pairs of egos with similar size (number of edges) and use the Kolmogorov-Smirnov test. The Kolmogorov-Smirnov test generates two key values: KS statistic and \(p\)-value. If the KS statistic is small or the \(p\)-value is high, then the distributions of two samples are the same. Table 2 describes details for pairs of ego-networks that have nearly the same number of nodes. For all pairs, distributions are similar since KS statistic is low (around 0.1) and the \(p\)-value is higher than 0.1 [PERSON] and Smirnov (1961).  
Figure 1: Centrality distribution for ego ’686’ in the Facebook graph and ego ’25’ in the artificial graph.