Convergence of HITS hyperlinkinduced topic search algorithm by

Convergence of HITS (hyperlink-induced topic search) algorithm by Victor Boyarshinov

• Was first introduced by Jon M. Kleinberg (1998). • Assumption: a topic can be roughly divided into pages with good coverage of the topic, called authorities, and directory-like pages with many hyperlinks to useful pages on the topic, called hubs. • And the goal of HITS is basically to identify good authorities and hubs for a certain topic which is usually defined by the user's query. • Given a user query, the HITS algorithm first creates a neighborhood graph for the query. Then, an iterative calculation was performed on the value of authority and value of hub.

For each page p , the authority and hub values are computed as follows: The authority value of page p is the sum of hub scores of all the pages that points to p The hub value of page p is the sum of authority scores of all the pages that p points to

Algorithm Complexity Time complexity of one iteration of the HITS algorithm is O(|E(G)|). Experimental Data • The algorithm was tested on uniformly generated directed random graphs. • The edge probability value was tuned so that expected value for out-degree of every vertex was 10. Convergence Criterion Iterations were performed until sum of absolute values of weight changes fall below constant threshold (0. 0000001).

Goal of the Experiments: Determine how number of iterations increases with size of graph if average degree of a vertex remains the same (10). Experiments results

The reason why attempt of convergence rate estimating failed is insufficient number of tests (computationally expensive!) Another set of test examples was generated as follows: take two disjoint uniformly generated random graphs of size n, take any vertex v from the first graph, vertex u from the second graph and connect the components by adding edges (u, v) and (v, u). Experiments results: