I did a number of things this week.
On Monday, Lenore and I brainstormed a few more questions that I could look into. For instance, we wanted to know: If you bound the diameter of the graph, does DSD get bounded as well? My goal was to come up with a few conjectures about DSD and random walks. Later that day, her husband came and gave a mini lecture on operating systems. He spoke about the history of operating systems (briefly covering UNIX and MS-DOS), early programming languages and how they were associated with assemblers and compilers, as well as other computer architecture concepts. It was a very informative session. The highlight for Tuesday was me presenting my mathematical findings to the other students and brainstorming ways to prove the intermediate step I was stuck on. I worked through an example of computing the DSD value for a complete graph of five nodes with varying random walks. After explaining my work, we went through the computation of DSD by hand once again to try to prove my claim. For a five-noded network, we listed the weight corresponding to the start node and other nodes in a table with varying length of random walks (given that initially one node had weight 1 and at each step, the weight at each node was equally distributed to the other nodes):
^{m} - (-1)^{m}) / (4^{m} * 5))
. I then looked at a graph with four nodes and generalized this formula to a graph with n+1 nodes: 2 - 2((n^{m} - (-1)^{m}) / (n^{m} * (n+1))).o-o-o. This weekend, I went to Boston Public Market. There was a farmers market going on when I went, which was a highlight because the produce there was not that pricey (as opposed to grocery stores and other farmers markets I went to). I didn't bring a bag, so I'm thinking of going there again to buy my groceries. The following day, I went on a boat ride tour that showed me lighthouses near and around Gloucester. It was nice seeing them; although, I have to admit, it was equally as nice seeing the expensive summer homes that were next to them. |

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