Structure of Neighborhoods in a Large Social Network

We present here a method for analyzing the neighborhoods of all the vertices in a large graph. We first give an algorithm for characterizing a simple undirected graph that relies on enumeration of small induced subgraphs. We make a step further in this direction by identifying not only subgraphs but also the positions occupied by the different vertices of the graph, being thus able to compute the roles played by the vertices of the graph. We apply this method to the neighborhood of each vertex in a 2.7 M vertices, 6 M edges mobile phone graph. We analyze how the contacts of each person are connected to each other and the positions they occupy in the neighborhood network. Then we compare the intensity of their communications (duration and frequency) to their positions, finding that the two are not independent. We finally interpret and explain the results using social studies on phone communications.