PageRank Algorithm

Imagine you’re launching a brand-new search engine and need to decide which pages deserve top billing. You’d probably start by counting keywords, but that only tells you so much. Instead, what if you judged a page by how many other reputable sites link to it? That’s the core of PageRank! The web becomes a giant voting network, where each hyperlink casts a vote of confidence. A damping factor keeps this process from spiraling, simulating a “random surfer” who occasionally jumps to a fresh page. In this article, we’ll unpack how PageRank turns link patterns into a powerful ranking signal, and conclude it by elucidating the algorithm that powers it.

What is PageRank?

PageRank is an algorithm developed by Google to rank web pages in search results by measuring the importance of each page based on the number and quality of links it receives. PageRank scores pages by treating every hyperlink as a “vote” of confidence. It not only counts votes but also weighs them by each voter’s own PageRank, so links from important sites matter more. A damping factor (usually 0.85) models a “random surfer” who occasionally jumps to any page, ensuring the scores converge. By combining link quality and quantity, PageRank pushes genuinely authoritative pages to the top, solving the keyword-only bias.

A user enters the query: “Harvard Business School“. The expectation is for the first link to be “http://www.harvard.edu/”.  But what would your algorithm do? It would try to find out pages which have the word “Harvard” a maximum number of times, as “Business” and “School” will come out to be common words. Now, there is a possibility that the “Harvard” keyword might not be repeated multiple times on Harvard’s website. However, websites like Business School Consultants or articles on business school might have this keyword multiple times. This leads these websites to achieve a rank much higher than the actual business school website. This would’ve been the case if the keyword-only algorithm is used standalone. Thanks to the Google PageRank algorithm, which takes into account not only the frequency of keywords but also the quality and quantity of links to a website, such a problem is obviated.

Ranking Mechanism

Imagine a web with only four pages that are linked to each other. Each box below represents a web page. The words in black and italics are the links between pages.

For instance, the web page “Tavish” has three outgoing links to the other three web pages. Now, let’s draw a simpler directed graph of this ecosystem.

Here is how Google ranks a page: The page with the maximum number of incoming links is the most important page. In the current example, the “Kunal Jain” page comes out as the most significant page.

Mathematical Formulation of Google Page Rank

The first step of the formulation is to build a direction matrix. This matrix will have each cell as the proportion of the outflow. For instance, Tavish (TS) has 3 outgoing links, which makes each proportion as 1/3.

Now we imagine that if there were a bot that followed all the outgoing links, what would be the total time spent by this bot on each of these pages? This can be broken down mathematically into the following equation:

A * X = X

Here, A is the proportion matrix mentioned above

X is the probability of the bot being on each of these pages

Clearly, Kunal Jain‘s page in this universe is most important, which goes in the same direction as our intuition.

Teleportation adjustments

Now, imagine a scenario where we have only 2 web pages: A and B. A has a link to B, but B has no external links. In such cases, if you try solving the matrix, you will get a zero matrix. This looks unreasonable as B looks to be more important than A. But our algorithm still gives the same importance to both. To solve this problem, a new concept of teleportation was introduced. We include a constant probability of alpha to each of these pages. This is to compensate for instances where a user teleports from one webpage to another without any link. Hence, the equation is modified to the following:

(1-alpha) * A * X + alpha * b = X

Here, b is a constant unit column matrix. Alpha is the proportion of teleportation. The most common value taken for alpha is 0.15 (but it can depend on different cases).

Other uses of the PageRank Algorithm

Here are some uses of PageRank:

1. Finding how well-connected a person is on Social Media. One of the unexplored territories in social media analytics is network information. Using this network information, we can estimate how influential the user is and, therefore, prioritize our efforts to please the most influential customers. Networks can be easily analyzed using the PageRank algorithm.
2. Fraud Detection in the Pharmaceutical Industry. Many countries, including the US, struggle with a high percentage of medical fraud. The PageRank algorithm can detect such frauds.
3. Understand the importance of packages in any programming language. The PageRank algorithm can also be used to understand the layers of packages used in languages like R and Python.

Conclusion

We’ve seen how PageRank works its magic by treating the web like a vast network of streets, passing “importance” from one page to another through links, dampening and redistributing scores until everything settles into a clear ranking. We broke down the core formula, talked through iterations and the damping factor; techniques used to keep the ranking system honest. While its origins lie in search engines, PageRank’s real power shows up wherever connections matter, from social network analysis and recommendation systems to biology and beyond, making it a versatile tool for understanding any linked system.

Frequently Asked Questions

Tavish Srivastava

Tavish Srivastava, co-founder and Chief Strategy Officer of Analytics Vidhya, is an IIT Madras graduate and a passionate data-science professional with 8+ years of diverse experience in markets including the US, India and Singapore, domains including Digital Acquisitions, Customer Servicing and Customer Management, and industry including Retail Banking, Credit Cards and Insurance. He is fascinated by the idea of artificial intelligence inspired by human intelligence and enjoys every discussion, theory or even movie related to this idea.