1. What are going to be the ranks of the vertices {1,2,3} in the graph G=(V,E) with E={{1->1},{1->2},{1->3},{2->1},{2->3},{3->2},{3->3}}?
Is this graph an irreducible and aperiodic one?

2. Verify that the largest eigenvalue of stochastic matrices is 1.0!

3. Verify that assuming we start a random walk with equal probability from any of the states of G=(V,E) with V={1,2,3,4} and E={{1->2},{1->3},{1->4},{2->1},{2->4},{3->1},{4->2},{4->3}} then the ranks of node 2, 3 and 4  will have to match across all the iterations of the power method!

4. Download the Zachary karate graph from http://www.cise.ufl.edu/research/sparse/matrices/Newman/index.html and perform the personalized PageRank algorithm in which
a) node 6 is regarded as the 'favored' node,
b) node 16 is regarded as the 'favored' node.