Foundations of Data Science

or
p[I − (1 − α)A] = α (1, 1, . . . , 1)
n
p = α n (1, 1, . . . , 1) [I − (1 − α) A]−1 .
Thus, in principle, p can be found by computing the inverse of [I − (1 − α)A] −1 . But
this is far from practical since for the whole web one would be dealing with matrices with
billions of rows and columns. A more practical procedure is to run the random walk and
observe using the basics of the power method in Chapter 3 that the process converges to
the solution p.
For the personalized page rank, instead of restarting at an arbitrary vertex, the walk
restarts at a designated vertex. More generally, it may restart in some specified neighborhood.
Suppose the restart selects a vertex using the probability distribution s. Then, in
the above calculation replace the vector 1 (1, 1, . . . , 1) by the vector s. Again, the computation
could be done by a random walk. But, we wish to do the random walk calculation
n
for personalized pagerank quickly since it is to be performed repeatedly. With more care
this can be done, though we do not describe it here.
5.9 Bibliographic Notes
The material on the analogy between random walks on undirected graphs and electrical
networks is from [DS84] as is the material on random walks in Euclidean space. Additional
material on Markov chains can be found in [MR95b], [MU05], and [per10]. For
material on Markov Chain Monte Carlo methods see [Jer98] and [Liu01].
The use of normalized conductance to prove convergence of Markov Chains is by
Sinclair and Jerrum, [SJ] and Alon [Alo86]. A polynomial time bounded Markov chain
based method for estimating the volume of convex sets was developed by Dyer, Frieze and
Kannan [DFK91].
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