what is r. in the pagerank algorithm

I was reading Matlabs Pagerank algorithm and I was hoping that someone can clarify what the r. is within the equation. I have all the other elements just not sure what r.

https://au.mathworks.com/help/matlab/math/use-page-rank-algorithm-to-rank-websites.html#

Thank you!

Answer 1

[Askic V]

Hi Kellie,

you probably mean on the following equation:

r = (1-P)/n + P*(A'*(r./d) + s/n);
% r is a vector of PageRank scores.
% P is a scalar damping factor (usually 0.85), which is the probability that a random surfer clicks on a link on the current page, instead of continuing on another random page.
% A' is the transpose of the adjacency matrix of the graph.
% d is a vector containing the out-degree of each node in the graph. d is set to 1 for nodes with no outgoing edges.
% n is the scalar number of nodes in the graph.
% s is the scalar sum of the PageRank scores for pages with no links.

So, this equation updates the vector of Pagerank scores based on the current value of vector r and other parameters.

Specifically,

r./d

is an element-wise division, which means that each element of r is divided with its corresponding element in vector d (the vector d contains same number of elements like vector r).

[Kellie]

so is r. the values within the inital (V0) vector???? is there code to make r. and d?

[Kellie]

thank you for answering my question so far Askic :)

[Askic V]

No problem @Kellie

This is how I would implement it:

clear all;
clc;
% Example from the documentation
s = [1 1 2 2 3 3 3 4 5];
t = [2 5 3 4 4 5 6 1 1];
names = {'http://www.example.com/alpha', 'http://www.example.com/beta', ...
         'http://www.example.com/gamma', 'http://www.example.com/delta', ...
         'http://www.example.com/epsilon', 'http://www.example.com/zeta'};
G = digraph(s,t,[],names);

P = 0.85; % P is a scalar damping factor (usually 0.85)
pr = centrality(G,'pagerank','FollowProbability', P)

pr = 6×1

0.3210 0.1706 0.1066 0.1368 0.2008 0.0643

plot(G,'Layout','layered', ...
    'NodeLabel',{'1','2','3','4','5','6'})

n = 6; % Number of nodes
d = outdegree(G); % d is a vector containing the out-degree of each node
A = adjacency(G); % A is the adjacency matrix of the graph
r = ones(6,1); % initial values

s = 0; % s is the scalar sum of the PageRank scores for pages with no links
for i = 1:n
    if isempty(outedges(G,i))
        s = s+1;
    end
end
% s should be 1 after this step because of node 6
for i = 1:100 % repeat certain number of times until converge to final values
   r = (1-P)/n + P*(A'*(r./d) + s/n);   
end
r = r/sum(r) % sum must be equal to 1

r = 6×1

0.3210 0.1705 0.1066 0.1368 0.2007 0.0643

[Kellie]

Thank you Askic that has helped so much! one more question, if you are assessing a matrix (A) is it G = digraph(A,[],names);?

Thank you! :)

[Askic V]

Matrix A is an adjacency matrix of a graph G.

Please have a look at this link: https://en.wikipedia.org/wiki/Adjacency_matrix

If you want to create a directed graph based on the adjacency matrix, then you simply use the following syntax:

G2 = digraph(A)

In the example code above, matrix A is simply:

A =

     0     1     0     0     1     0
     0     0     1     1     0     0
     0     0     0     1     1     1
     1     0     0     0     0     0
     1     0     0     0     0     0
     0     0     0     0     0     0

If A(i,j) is 1 that means there is a directed connection from i to j.