PageRank Explained: The Actual Algorithm and Formula

PageRank is an eigenvector centrality algorithm that measures the structural importance of web pages by analyzing the directed graph of hyperlinks connecting the web. Developed by Larry Page and Sergey Brin at Stanford University, it calculates probability distributions representing a random web surfer clicking links across the internet. Although Google has evolved, PageRank remains the foundational mathematical basis for modern link-based search engine authority.

What PageRank actually is

PageRank is a mathematical algorithm that calculates the objective importance of a webpage based on the link structure of the web. In graph theory, the web is a directed graph where web pages are nodes and hyperlinks are directed edges. PageRank interprets a link from page A to page B as an explicit endorsement of page B by page A.

Not all endorsements carry equal weight in the calculation. A vote from a highly authoritative page confers more credibility than a vote from an obscure personal homepage. In addition, the algorithm dilutes the voting power of a linking page across all its outgoing links. If an authoritative page links to hundreds of other sites, each recipient receives only a fraction of that page’s authority score.

This recursive definition is what makes the system distinct. The importance of a webpage is defined by the importance of the pages linking to it. Calculating this value requires solving for the stationary probability distribution of a Markov chain across billions of interlinked documents. You can review how search engines combine link topology with lexical scoring in our guide to how search engines rank pages.

The original Stanford formula step by step

The original PageRank algorithm was published in 1998 by Larry Page, Sergey Brin, Rajeev Motwani, and Terry Winograd. The formula calculates the score of a target webpage by summing the adjusted scores of all pages that link to it. The standard mathematical expression is structured as follows:

PR(A) = (1 - d) / N + d * Σ [ PR(T_i) / C(T_i) ]

In this equation, PR(A) represents the PageRank score of the target webpage A. The constant d represents the damping factor, which is typically set to 0.85 in information retrieval literature. The variable N represents the total number of web pages indexed in the entire search engine collection.

The summation component iterates across every individual webpage T_i that links to page A. For each linking page, its current PageRank score PR(T_i) is divided by C(T_i), which represents the total number of outbound links leaving that page. If a linking page has ten outbound links, it passes exactly one-tenth of its PageRank score to each linked destination.

The term (1 - d) / N ensures that the sum of all PageRank scores across the entire web graph remains normalized. It represents a baseline probability that a user teleports directly to page A without following any hyperlink. In smaller toy graphs, engineers often simplify the equation to (1 - d) + d * Σ [ PR(T_i) / C(T_i) ] to focus on relative ratios rather than global network fractions.

The random surfer model and the damping factor

The intuitive mechanism behind PageRank is the random surfer model. The algorithm imagines an idealized web user sitting at a browser with no prior knowledge or commercial intent. The surfer visits a random webpage and clicks hyperlinks continuously at random, never using the browser back button.

If a web user clicked links indefinitely, they would eventually become trapped in circular loops or dead ends. To prevent this, the algorithm introduces a damping factor, denoted as d. The damping factor represents the probability that the user continues clicking links on their current page rather than jumping to a completely new URL.

In standard implementations, the damping factor is set to 0.85. This means that at any given step, there is an 85 percent probability that the surfer clicks an available link on the page. Conversely, there is a 15 percent probability that the surfer grows bored, abandons the current link trail, and jumps to a completely random URL chosen from the index.

This random jump mechanism is mathematically vital. It models natural browsing behavior where readers close tabs or enter new addresses into their browser navigation bar. More importantly, it guarantees that the underlying probability distribution reaches mathematical convergence across complex networks.

Rank sinks and dangling nodes: what breaks without damping

Without the damping factor, the mathematical calculations behind PageRank collapse when confronted with real-world web structures. Two specific architectural topologies create fatal errors in raw link voting systems: rank sinks and dangling nodes.

A rank sink occurs when a group of web pages link exclusively to one another with no outbound links pointing back to the broader internet. In a toy network, imagine page B and page C link back and forth continuously. If the damping factor is omitted (d = 1.0), every iteration transfers authority into the loop without letting any score escape. Over successive iterations, pages B and C absorb all the PageRank of the entire web, reducing the score of every external site to zero.

Rank Sink Problem:
[ External Web ] ---> [ Page B ] <====> [ Page C ]
                        (Trapped authority accumulates indefinitely)

A dangling node is a webpage that possesses inbound links but contains zero outbound hyperlinks. Common examples include PDF documents, image files, video assets, and raw data tables. When a random surfer reaches a dangling node, there are no links to click, causing the surfer to vanish and destroying probability mass from the system.

The damping factor solves both problems simultaneously. By granting a 15 percent chance of jumping to an arbitrary page at every step, rank sinks continuously bleed authority back into the open web graph. For dangling nodes, search engines treat the page as linking equally to every other page in the index, preserving probability conservation across calculations. You can explore how search engines extract these document links in our overview of what a web crawler is.

Worked calculation on a four-page web graph

Walking through a concrete numerical calculation on a small network illustrates how PageRank scores converge through power iteration. Consider a toy web graph consisting of four distinct pages: A, B, C, and D. The network links are structured as follows:

Toy Network Link Map:
- Page A links to: Page B, Page C (2 outbound links)
- Page B links to: Page C (1 outbound link)
- Page C links to: Page A (1 outbound link)
- Page D links to: Page C (1 outbound link)

In this network, Page D receives no inbound links, while Page C receives inbound links from A, B, and D. We initialize every page with an equal starting score of 1.000. We will set the damping factor d to 0.85 and use the standard simplified equation PR(P) = 0.15 + 0.85 * Σ [ PR(in) / C(in) ].

Iteration 1

We compute the incoming values based on the initial starting scores of 1.000 for all nodes. Every page distributes its current score evenly across its outbound connections:

PR(A) = 0.15 + 0.85 * [ PR(C) / 1 ]
      = 0.15 + 0.85 * [ 1.000 / 1 ] = 1.000

PR(B) = 0.15 + 0.85 * [ PR(A) / 2 ]
      = 0.15 + 0.85 * [ 1.000 / 2 ] = 0.15 + 0.425 = 0.575

PR(C) = 0.15 + 0.85 * [ PR(A) / 2 + PR(B) / 1 + PR(D) / 1 ]
      = 0.15 + 0.85 * [ 1.000 / 2 + 1.000 / 1 + 1.000 / 1 ]
      = 0.15 + 0.85 * [ 0.500 + 1.000 + 1.000 ]
      = 0.15 + 0.85 * 2.500 = 0.15 + 2.125 = 2.275

PR(D) = 0.15 + 0.85 * 0 = 0.150

After the first iteration, Page C has surged to 2.275 because it receives incoming links from three different pages. Page D drops to the minimum baseline of 0.150 because no other page points to it.

Iteration 2

We feed the scores from Iteration 1 into the next calculation cycle. The updated values flow through the exact same link structure to recalculate node weights:

PR(A) = 0.15 + 0.85 * [ 2.275 / 1 ] = 0.15 + 1.934 = 2.084

PR(B) = 0.15 + 0.85 * [ 1.000 / 2 ] = 0.15 + 0.425 = 0.575

PR(C) = 0.15 + 0.85 * [ 1.000 / 2 + 0.575 / 1 + 0.150 / 1 ]
      = 0.15 + 0.85 * [ 0.500 + 0.575 + 0.150 ]
      = 0.15 + 0.85 * 1.225 = 0.15 + 1.041 = 1.191

PR(D) = 0.15 + 0.85 * 0 = 0.150

In Iteration 2, Page A increases dramatically because it absorbs the high score of Page C from the prior round. Page C drops temporarily because its incoming sources (A, B, and D) were previously scored before C’s boost could propagate back to them.

Final Converged Scores

As iterations continue, the numerical shifts between rounds shrink until the values stabilize completely. The table below shows the progression across successive rounds until full mathematical convergence is achieved.

Page C finishes with the highest score (1.540) because it receives links from three nodes, including Page A. Page A finishes second (1.459) because it receives all the concentrated voting power of Page C. Page D finishes last with the minimum teleport score of 0.150. You can study how developers implement these matrix iterations in code in our tutorial on building search components in Python.

From random surfer to reasonable surfer

The original random surfer model assumes that users select outgoing links with completely uniform probability. If a webpage contains forty navigation links, ten footer copyright links, and one prominent editorial link inside the main article, the original formula assigns an identical 1-in-51 probability to each link. This mechanical assumption created an immediate vulnerability that webmasters exploited through link stuffing and footer networks.

In 2004, Google filed a breakthrough patent titled “Ranking based on user behavior and properties of links”, authored by engineer Jeffrey Dean. This model became known throughout the industry as the reasonable surfer model. It replaced uniform probabilities with weighted probabilities based on human visual attention and interaction likelihood.

Under the reasonable surfer model, the probability that a link transfers PageRank depends on observable DOM characteristics. Links placed prominently within the main body copy in large typography receive high transition probabilities. Links buried in footer copyright notices, legal disclaimers, or terms of service sidebars receive drastically reduced transition probabilities.

Reasonable Surfer Link Weighting:
- High Weight: Main body editorial link, high contrast, top of page
- Medium Weight: Contextual sidebar link, related articles section
- Low Weight: Footer copyright link, site-wide privacy policy, small font

This evolution fundamentally changed link building and site architecture. Search engines no longer treat links as purely mathematical tokens. The surrounding context, visual prominence, anchor text relevance, and user click propensity directly dictate the volume of equity transmitted across the link graph.

Toolbar PageRank vs internal PageRank

Much of the public confusion regarding PageRank stems from the distinction between internal algorithmic scoring and the historical Google Toolbar. In 2000, Google launched an Internet Explorer browser add-on that displayed a green metric bar labeled PageRank, scored on a logarithmic scale from 0 to 10.

The Toolbar PageRank metric was merely a coarse public export of Google’s internal calculations. While Google updated its internal link graph continuously in real time, the public toolbar was updated manually only every three to four months. A score of PR 6 did not mean a page had six points; it meant the page sat within an exponential bracket of logarithmic link authority.

The public score created an aggressive international market for link purchasing, paid reciprocal directories, and blog comment spam. Webmasters focused exclusively on trading links with high public numbers regardless of topical relevance. In response, Google introduced the nofollow attribute in 2005 to devalue untrusted commercial links.

Google gradually reduced public updates before permanently disabling the Toolbar PageRank API in March 2016. However, retiring the public display did not mean retiring the algorithm. Google eliminated the score indicator to prevent market manipulation, while keeping the underlying mathematical calculation active inside core ranking clusters.

Where PageRank fits in modern Google search

Modern search engines do not rank search results using PageRank alone. In the late 1990s, when the web was small and spam was primitive, link counts provided an exceptional quality filter. In modern search retrieval, PageRank functions as one of hundreds of interconnected signals inside a multi-tier machine learning architecture.

During first-pass retrieval, search engines scan inverted indexes using fast lexical algorithms like BM25 to locate candidate documents matching query tokens. You can examine how BM25 normalizes keyword frequencies in our detailed guide comparing BM25 vs TF-IDF. Once candidate documents are retrieved, PageRank provides a query-independent baseline score representing structural authority.

A page with outstanding lexical relevance but near-zero link authority often struggles to rank for competitive commercial queries. Conversely, a page with massive link authority can rank for competitive terms even if its on-page keyword optimization is minimal. The search engine balances textual relevance against link graph centrality to separate credible sources from superficial matches.

In later ranking phases, modern neural networks and machine learning rerankers evaluate the link graph in combination with user intent signals, freshness, and quality rater standards. Link authority establishes whether an organization is recognized by independent peers across the web. You can learn how to structure site authority effectively in our beginner reference explaining what SEO is.

Frequently asked questions

What is Google PageRank?

Google PageRank is a link analysis algorithm that measures the authority of a webpage by counting the quantity and quality of links pointing to it. It models a random web surfer to calculate the probability that a user arrives at a specific page. Google uses it to evaluate structural trust across the web.

Does Google still use PageRank today?

Google still uses modernized versions of PageRank within its core ranking algorithms. Google retired the public Toolbar PageRank score in 2016 to discourage link manipulation, but company representatives and patent filings consistently confirm that internal link graph analysis remains active. Modern implementations incorporate the reasonable surfer model to evaluate link placement and user engagement.

What is the damping factor in PageRank?

The damping factor is a mathematical constant, typically set to 0.85, representing the probability that a web user continues clicking links rather than jumping to a random page. It prevents calculations from getting trapped in circular link loops and ensures that PageRank calculations reach mathematical convergence across billions of pages.

Can internal links pass PageRank?

Internal links pass PageRank between pages on the same domain exactly like external backlinks. A logical site architecture distributes equity from high-authority landing pages down to deeper articles. Search engines evaluate internal link graphs to determine which subpages on a website are structurally most important.

What is the difference between PageRank and domain authority?

PageRank is an actual mathematical algorithm created by Google that scores individual URLs. Domain Authority is a third-party predictive metric developed by SEO software companies like Moz or Ahrefs to estimate organic ranking strength. Google does not use third-party domain authority metrics in its search algorithms.

Do nofollow links pass PageRank?

Nofollow links do not pass PageRank to the destination URL. Google introduced the nofollow attribute in 2005 to allow webmasters to link to paid advertisements, user comments, or untrusted sources without transferring authority. While search engines may use nofollow links for discovery, they do not convey link equity.

What is a rank sink in PageRank?

A rank sink is a group of interlinked web pages that contain no outbound links pointing to external sites. Without a damping factor, rank sinks trap link equity, continuously absorbing authority until all external pages score zero. The damping factor prevents this failure by allowing authority to escape through random teleports.

How can I check my website PageRank score?

You cannot check your website’s actual Google PageRank score because Google retired the public Toolbar PageRank API in 2016. Third-party software tools calculate proprietary proxy metrics like URL Rating or Page Authority, but these are independent estimates. Google does not publish internal link authority scores for any webpage.

Sources

• Google InfoLab: The PageRank Citation Ranking: Bringing Order to the Web, ilpubs.stanford.edu:8090/422/1/1999-66.pdf
• Stanford University: The Anatomy of a Large-Scale Hypertextual Web Search Engine, infolab.stanford.edu/pub/papers/google.pdf
• Journal of the ACM: Authoritative Sources in a Hyperlinked Environment (Jon M. Kleinberg), cs.cornell.edu/home/kleinber/auth.pdf
• United States Patent and Trademark Office: Ranking Search Results Based on User Behavior and Link Properties, Patent US7716225B1