what is r. in the pagerank algorithm

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Kellie 2023-4-10

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评论： Askic V 2023-4-11

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!

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Answer 1

Askic V 2023-4-10

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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).

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Askic V 2023-4-11

编辑：Askic V 2023-4-11

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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

% 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 2023-4-11

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 2023-4-11

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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 =
   1     0     0     1     0
   0     1     1     0     0
   0     0     1     1     1
   0     0     0     0     0
   0     0     0     0     0
   0     0     0     0     0