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Unformatted text preview: 20 “mcsftl” — 2010/9/8 — 0:40 — page 533 — #539 Random Walks Random Walks are used to model situations in which an object moves in a sequence of steps in randomly chosen directions. Many phenomena can be modeled as a random walk and we will see several examples in this chapter. Among other things, we’ll see why it is rare that you leave the casino with more money than you entered with and we’ll see how the Google search engine uses random walks through the graph of the worldwide web links to determine the relative importance of websites. 20.1 Unbiased Random Walks 20.1.1 A Bug’s Life There is a small ﬂea named Stencil. To his right, there is an endless ﬂat plateau. One inch to his left is the Cliff of Doom, which drops to a raging sea filled with ﬂeaeating monsters. Cliff of Doom 1 inch Each second, Stencil hops 1 inch to the right or 1 inch to the left with equal probability, independent of the direction of all previous hops. If he ever lands on the very edge of the cliff, then he teeters over and falls into the sea. So, for example, if Stencil’s first hop is to the left, he’s fishbait. On the other hand, if his first few hops are to the right, then he may bounce around happily on the plateau for quite 534 “mcsftl” — 2010/9/8 — 0:40 — page 534 — #540 Chapter 20 Random Walks oops... some time. Our job is to analyze the life of Stencil. Does he have any chance of avoiding a fatal plunge? If not, how long will he hop around before he takes the plunge? Stencil’s movement is an example of a random walk . A typical onedimensional random walk involves some value that randomly wavers up and down over time. The walk is said to be unbiased if the value is equally likely to move up or down. If the walk ends when a certain value is reached, then that value is called a boundary condition or absorbing barrier . For example, the Cliff of Doom is a boundary condition in the example above. Many natural phenomena are nicely modeled by random walks. However, for some reason, they are traditionally discussed in the context of some social vice. For example, the value is often regarded as the position of a drunkard who randomly staggers left, staggers right, or just wobbles in place during each time step. Or the value is the wealth of a gambler who is continually winning and losing bets. So discussing random walks in terms of ﬂeas is actually sort of elevating the discourse. 20.1.2 A Simpler Problem Let’s begin with a simpler problem. Suppose that Stencil is on a small island; now, not only is the Cliff of Doom 1 inch to his left, but also there is another boundary condition, the Pit of Disaster, 2 inches to his right! For example, see Figure 20.1 In the figure, we’ve worked out a tree diagram for Stencil’s possible fates. In 535 “mcsftl” — 2010/9/8 — 0:40 — page 535 — #541 20.1. Unbiased Random Walks 1=4 1=16 1=2 1=8 1=2 1=2 1=2 1=2 1=2 1=2 1=2 1=2 : : : : : : Cliff of Doom Pit of Disaster Figure 20.1 Figure 20....
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This note was uploaded on 01/19/2012 for the course CS 6.042J / 1 taught by Professor Tomleighton,dr.martenvandijk during the Fall '10 term at MIT.
 Fall '10
 TomLeighton,Dr.MartenvanDijk
 Computer Science

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