Lecture 7- Centrality continued

The lecture started with an outline of previous lecture. Then EigenVector Centrality was talked about. In the above concept if a node is connected to more central node then the node also assumes some importance. So, in case of Eigen Vector centrality a vertex is given a score proportional to the sum of its neighbor’s scores. Eigen Vector centrality can be calculated in case of directed graphs as well.

Next, Katz centrality is talked about. In this concept each vertex is given a certain amount of centrality without considering its neighbors and their centrality. The equation for Katz centrality is

X= b(I - aA)^-1.1

Where I is an identity matrix, 1 is a matrix of all 1’s, a is the scaling vector and b is the extent to which we weigh the centrality of people ego is tied to.

Next concept is that of page rank. The gist behind this concept is that if numerous pages on the web are referring to a particular web page then the later is said to have high importance, but, we cannot consider all the referring pages to be equal. Therefore, page rank uses the concept called as tracking a drunk. If a drunk person is walking through a network he would be spending sometime at each node which is proportional to the importance of that node. The drunk person must be allowed to teleport to some other node with some probability. This is similar to a websurfer visiting various web pages spending some time viewing each webpage and then clicking on a link in that web page and then after a while the person can move to some random webpage with some probability (teleporting). Using this concept the formulation for page rank algorithm is written.

The next topic discussed was Hubs and Authorities. Hubs are nodes that refer to good webpages and Authorities are webpages referred to by good Hubs. In the Hyperlink Induced Topic Search algorithm (HITS), we can start with a set of pages matching a particular query. We expand this set by following all the links on the initial set of web pages. We construct a transition matrix E where in E_ij=1/n_i, where there is a link between I and j and n_i is the number of links from i. The authority and the hub scores can be calculated as a’=E^Th and h’= Ea.