The calculation of PageRanks from the Google matrix relies on several deep theorems of mathematics:
- Since $G$ is a positive stochastic matrix (that is, every entry is positive and each row sums to $1$), an important theorem of linear algebra, proved by Oskar Perron in $\it{1907}$, guarantees that there is a unique positive probability vector $\bf{v}$ that satisfies ${\bf{v}}= {\bf{v}}G$.
- The theory of Markov chains guarantees to produce PageRank scores with approximately $m$ digits of accuracy, the number of iterations should be around $m/\lg\left ( 1/d \right )$.
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