35 16 special examples for a concrete example

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Unformatted text preview: is described by giving two things: (i) a set of vertices V (which we suppose is a finite set) and (ii) an adjacency matrix A(u, v ), which is 1 if there is an edge connecting u and v and 0 otherwise. By convention we set A(v, v ) = 0 for all v 2 V . 3 3 4 @ @ @ 3 3 4 3 5 3 3 @ @ @ 3 3 The degree of a vertex u is equal to the number of neighbors it has. In symbols, X d(u) = A(u, v ) v since each neighbor of u contributes 1 to the sum. To help explain the concept, we have indicated the degrees on our example. We write the degree this way to make it clear that (⇤) p(u, v ) = A(u, v ) d(u) defines a transition probability. In words, if Xn = u, we jump to a randomly chosen neighbor of u at time n + 1. It is immediate from (⇤) that if c is a positive constant then ⇡ (u) = cd(u) satisfies the detailed balance condition: ⇡ (u)p(u, v ) = cA(u, v ) = cA(v, u) = ⇡ (v )p(u, v ) P Thus, if we take c = 1/ u d(u), we have a stationary probability distribution. In the example c = 1/40. 35 1.6. SPECIAL EXAMPLES For a concrete example, consider Example 1.34. Rand...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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