ps4 - Networks Fall 2011 David Easley and Eva Tardos...

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Networks: Fall 2011 Homework David Easley and Eva Tardos Due before class on October 28, 2011 As noted on the course home page, homework solutions must be submitted by upload to the CMS site, at . This means that you should write these up in an electronic format (Word files, PDF files, and most other formats can be uploaded to CMS). Homework will be due at the start of class on the due date, and the CMS site will stop accepting homework uploads after this point. We cannot accept late homework except for University- approved excuses (which include illness, a family emergency, or travel as part of a University sports team or other University activity). Reading: The questions below are primarily based on the material in Chapters 11, 13, 14, and 15 of the book. (1) Consider a trading network with intermediaries in which there is one seller S , two buyers B 1 ,B 2 and two traders (intermediaries) T 1 ,T 2. The seller is allowed to trade with either trader. The buyers can each trade with only one of the traders: buyer B 1 can only trade with trader T 1; and buyer B 2 can only trade with trader T 2. The seller has one unit of the object and values it at 0; the buyers are not endowed with the object. Buyer B 1 values a unit at 3 and buyer B 2 values a unit at 1. (a) Draw the trading network, with the traders as squares, the buyers and the seller as circles, and with edges connecting nodes who are able to trade with each other. Label each node as S , B 1, B 2, T 1 or T 2. (b) Find Nash equilibrium bid and ask prices for this trading network. How much profit do the traders make? (c) Suppose now that we add edges representing the idea that each buyer can trade with each trader. Find a Nash equilibrium in this new trading game. What happens to trader profits? Why? (2) Consider the graph of Figure 1. Assume this is a part of the Web where nodes are Web pages, and edges are pointers. For example, the page corresponding to node 1 has a single pointer to the page corresponding to node 8, while the page corresponding to node 2 has no pointers, but that page is pointed to by the page corresponding to node 6. The web graph represents Web pages and pointers by a snapshot of the Web at a particular time, and the giant component evolves as time goes on. This question explores how the giant strongly connected 1
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7 6 8 11 2 12 13 14 9 3 4 5 10 15 16 17 1 18 Figure 1: The network of Web pages for Question (2). component of the Web changes over time. To simplify the questions, here we consider the
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ps4 - Networks Fall 2011 David Easley and Eva Tardos...

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