Lecture 12 : Random Walks and Graph Centrality

We now present a different type of connection between graphs and eigenvalues. In several applications, we wish to find influential vertices in a graph. For instance, suppose we model Twitter as a directed graph where there is an edge from user i to user j if i follows j. Then how can we measure importance of a user in the graph? One way of measuring it is look at the in-degree of a user, or the number of users following this user. However, such a measure does not give different amounts of importance to the different followers – if I have a follower who himself has a large number of followers, then I must be really important! In sociology, importance of a user in a social network is termed centraility. As discussed above, degree centrality measures the importance of a user in terms of its degree in the graph. The concept of eigenvector centrality or Bonacich centrality generalizes this as follows: Let πi denote the importance of user i in a directed social network, and let A denote the adjacency matrix, where aij = 1 if there is an edge from i to j. . A user’s importance proportional to the sum of the importance of his or her neighbors.