his example shows how to use a PageRank algorithm to rank a collection of websites. Although the PageRank algorithm was originally designed to rank search engine results, it also can be more broadly applied to the nodes in many different types of graphs. The PageRank score gives an idea of the relative importance of each graph node based on how it is connected to the other nodes.
Theoretically, the PageRank score is the limiting probability that someone randomly clicking links on each website will arrive at any particular page. So pages with a high score are highly connected and discoverable within the network, and it is more likely a random web surfer will visit that page.
Algorithm Description At each step in the PageRank algorithm, the score of each page is updated according to,
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. In other words, the rank of each page is largely based on the ranks of the pages that link to it. The term A'*(r./d) picks out the scores of the source nodes that link to each node in the graph, and the scores are normalized by the total number of outbound links of those source nodes. This ensures that the sum of the PageRank scores is always 1. For example, if node 2 links to nodes 1, 3, and 4, then it transfers 1/3 of its PageRank score to each of those nodes during each iteration of the algorithm. also see https://in.mathworks.com/help/matlab/examples/use-page-rank-algorithm-to-rank-websites.html
