Control-Network-Systems/docs/Chapters/EXTRAS.md

Extras

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PageRank

Whenever we need to search for a result on Google, this one of the most famous algorithm used.

First we take some pages and all the links available on that page linking to the other gathered pages:

p1, p2, p3, p4, ..., pn

# Links found on the page
p1: 
- a1 -> p2
- a3 -> p4
...
- al -> p8

p2:
...

Then we make an oriented graph where all the points are pages and links are arrows pointing to the landing page

Variables

• B_p : Set of pages that backlinks to p
• |a| : Cardinality of a | Number of outgoing links from a

Formula

rank_k(p) = \sum_{a \in B_p}\frac{
 rank_{k-1}(a)
}{
|a|
} \in [0, \dots, 1]

Here the state is composed of all ranks for our gathered pages. If | rank_k(p) - rank_{k-1}(p)| < \epsilon we will stop iterating.

At time k = 0 all pages have the same importance that is rank_0(p) = \frac{1}{n}

Economic System