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T=01/2000001000000001/2101/3001/2001/2000001/201/3100001/200000001/30 Project Activity 9.6. Use appropriate technology to do the following. Choose several different initial state vectors xo and calculate the vectors in the sequence {Txo} for large values of . (Note that, as state vectors, the entries of xo cannot be negative and the sum of the entries of xo must be 1.) Explain the behavior of the sequence {x,} as n gets large. Do you notice anything strange? What aspects of our seven page internet do you think explain this behavior? Clearly communicate all of the experimentation that you do. . If there is a limit of the sequence {Txo} (in other words, if there is a vector v such that v = lim Txo), we call this limit a steady-state or equilibrium vector. Such a steady-state vector has another important property. Since T is independent of we have Tv =T(lim Tx = lim Tn+1xg =v (9.8) Equation (9.8) shows that a steady state vector v is an eigenvector for T with eigenvalue 1. We can interpret the steady-state vector for T in an important way. Let t; be the fraction of time we spend on page j and let l; be the number of links on page j. Then the fraction of the time that we end up on page i coming from page j is ,. If we sum over all the pages linked to page i we have that ti= Notice that this is essentially the same process we used to obtain x, from x,1 and so we can interpret the steady-state vector v as telling us what fraction of a random web surfers time was spent at each web page. If we assume that the time spent at a web page is a measure of its importance, then the steady-state vector tells us the relative importance of each web page. So this steady-state vector provides the page rankings for us. In other words, The importance of a webpage may be measured by the relative size of the corresponding entry in the steady-state vector for an appropriately chosen Markov chain. Project Activity 9.7. Show that the limiting vector you found in Project.Activity.9.6 is an eigenvector of T with eigenvalue 1.

T=01/2000001000000001/2101/3001/2001/2000001/201/3100001/200000001/30
Project Activity 9.6. Use appropriate technology to do the following. Choose several different initial state vectors xo and calculate the vectors in the sequence {Txo} for large values of . (Note that, as state vectors, the entries of xo cannot be negative and the sum of the entries of xo must be 1.) Explain the behavior of the sequence {x,} as n gets large. Do you notice anything strange? What aspects of our seven page internet do you think explain this behavior? Clearly communicate all of the experimentation that you do. .
If there is a limit of the sequence {Txo} (in other words, if there is a vector v such that v = lim Txo), we call this limit a steady-state or equilibrium vector.
Such a steady-state vector has another important property. Since T is independent of we have
Tv =T(lim Tx = lim Tn+1xg =v
(9.8)
Equation (9.8) shows that a steady state vector v is an eigenvector for T with eigenvalue 1. We can interpret the steady-state vector for T in an important way. Let t; be the fraction of time we spend on page j and let l; be the number of links on page j. Then the fraction of the time that we end up on page i coming from page j is ,. If we sum over all the pages linked to page i we have that
ti=
Notice that this is essentially the same process we used to obtain x, from x,1 and so we can interpret the steady-state vector v as telling us what fraction of a random web surfers time was spent at each web page. If we assume that the time spent at a web page is a measure of its importance, then the steady-state vector tells us the relative importance of each web page. So this steady-state vector provides the page rankings for us. In other words,
The importance of a webpage may be measured by the relative size of the corresponding entry in the steady-state vector for an appropriately chosen Markov chain.
Project Activity 9.7. Show that the limiting vector you found in Project.Activity.9.6 is an eigenvector of T with eigenvalue 1.

Added by Michelle F.

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