1. (30 pts for trying out the experiment; 20 pts for your answer) Begin to amend your existing network G by artificially adding extra nodes or edges in any way you like. The goal is to improve the lowest ranked node in G by 15%. For example, if node # 3 in the original G has the lowest rank, say, 0.05, then your amended graph G 0 , which contains G as a subgraph, should be such that the same node now has rank at least 0.0575. Recall that the PageRank is a probability vector, so its entries sum to 1. If this node is to gain probability, then other nodes will have to sacrifice some probability. The extra nodes you add in will have to take away some probabilities from others nodes. Your answer will be either (a) or (b).

(a) If this is possible, show your new graph G0 and give evidence by a comparison between the old scores and the new scores at the original 10 nodes. (Congratulations! You have defied the Google.) Depending on your original network G, it is not obvious at all how you are going to amend the network, i.e., where to place the extra nodes and how to link them. You may have to try many alternatives to gain some insights. (Think about how you can manipulate to attract more people to read your facebook/instagram/personal web pages.)

(b) If you conclude that it is not possible to uplift the score (of this particular node) by 15%, can anything be done at all? Is there a systematic way to increase the score to an upper limit? Justify your conclusion by providing sufficient numerical evidence or, better yet, a theory. (If this is the case, can we trust that Googles recommended selection will never cheat on us?