What is PageRank?
PageRank is the name of the link analysis algorithm invented by Sergey Brin and Larry (Lawrence) Page at Stanford University, and simultaneously the formula that was the foundation for establishing Google.
Many people mistakenly believe that PageRank is no longer part of Google's algorithms. It's important to understand that PageRank is one of Google's cornerstones and an absolutely essential part of Google.
In its simplicity, the formula aims to assign numerical values to hyperlinks to measure a page's relative importance. In other words, PageRank is a popularity algorithm based on citation analysis.
Origin of the Name
It was Larry himself who named the formula, hence the name PageRank. It's not called PageRank because the value covers a website as a whole. PageRank accumulates on all pages of a website that Google has access to, not at the domain level.
Common Misconception
PageRank is applied at the page level, which means there's no such thing as, for example, a PageRank-4 domain. It would be called a website with a homepage that has a PageRank of 4 – it's not the domain itself.
The History of PageRank
The Invention
Larry Page and Sergey Brin develop BackRub (later Google) at Stanford University
First iteration of the PageRank algorithm based on citation analysis
Patent Filing
Stanford University files patent for the PageRank algorithm
Patent US6285999B1 describes the mathematical algorithm in detail
Google Foundation
Google Inc. is founded with PageRank as the core algorithm
PageRank becomes the primary ranking factor for search results
Toolbar PageRank
Google Toolbar displays public PageRank values (0-10 scale)
Webmasters can see their pages' PageRank scores in Google Toolbar
Toolbar Discontinued
Google stops updating public PageRank values
PageRank becomes internal Google tool, no longer publicly accessible
Modern Implementation
PageRank integrated with hundreds of other ranking signals
Still fundamental to Google, but combined with machine learning
The Original PageRank Formula
Basic PageRank Formula
PR(A) = PageRank of page A
d = Damping factor (typically 0.85)
PR(Ti) = PageRank of page Ti that links to A
C(Ti) = Number of outbound links from page Ti
Simplified Version
The formula in its original form distributes weight equally among the links found on a page, regardless of whether they are internal or external links.
Value = PageRank + (PageRank from sources × 0.85) / number of linksPractical Example
If you have, for example, 10 outbound links on a page, each link transfers 10% of the PageRank value you have the ability to transfer.
Mathematical Analysis
Matrix Representation
PageRank can be calculated using matrix algebra, where the web is represented as a transition matrix M:
M = d × H + (1-d)/N × J Where: H = Link matrix (hij = 1/L(j) if j links to i, else 0) J = Matrix of all 1s N = Number of pages d = Damping factor
Eigenvector Calculation
The PageRank vector is the dominant eigenvector of transition matrix M:
π = M × πWhere π is the PageRank vector and M is the transition matrix
Convergence Criteria
The algorithm converges when the difference between iterations is sufficiently small:
||π(k+1) - π(k)|| < εWhere ε is typically 10⁻⁶ for high precision
Damping Factor (0.85)
What is the damping factor?
The damping factor (d = 0.85) represents the probability that a user continues clicking on links, rather than starting a new search. In other words, there's an 85% chance of following a link and a 15% chance of jumping to a random page.
With damping factor (d = 0.85)
- ✓ Prevents ranking manipulation
- ✓ Handles dangling nodes (pages without outgoing links)
- ✓ Ensures algorithm convergence
- ✓ Models realistic user behavior
Without damping factor (d = 1.0)
- ✗ Rank sinks (pages that collect all PageRank)
- ✗ Algorithm doesn't always converge
- ✗ More vulnerable to manipulation
- ✗ Unrealistic user model
Mathematical significance of damping factor
Iterative Calculation Process
PageRank is calculated iteratively, where each iteration improves the estimate of all pages' PageRank values:
Iterative Algorithm
1. Initialize: PR⁰(i) = 1/N for all pages i 2. For k = 0, 1, 2, ... until convergence: PR^(k+1)(i) = (1-d)/N + d × Σ(PR^k(j)/L(j)) where j links to i 3. Stop when ||PR^(k+1) - PR^k|| < ε
Matrix-based Calculation
Adjacency Matrix Example
For a simple 4-page network, we can represent the link structure as a matrix:
A B C D A [ 0 1/2 1/2 0 ] B [1/3 0 1/3 1/3] C [1/2 0 0 1/2] D [ 0 1 0 0 ] Matrix H (link transition matrix)
Google Matrix Construction
G = d × H + (1-d)/N × J
Where J is matrix of 1/N values:
A B C D
A [0.25 0.25 0.25 0.25]
B [0.25 0.25 0.25 0.25]
C [0.25 0.25 0.25 0.25]
D [0.25 0.25 0.25 0.25]Final Google Matrix (d=0.85)
A B C D A [0.0375 0.4625 0.4625 0.0375] B [0.3208 0.0375 0.3208 0.3208] C [0.4625 0.0375 0.0375 0.4625] D [0.0375 0.8875 0.0375 0.0375]
Network Visualization
Interactive PageRank Network
Visualization Explanation
- • Node size: Represents PageRank value
- • Arrows: Show link direction
- • Animation: Simulates PageRank flow
- • Colors: Blue = normal, Purple = active iteration
Practical Calculation Examples
Interactive PageRank Calculator
Calculation:
PR(A) = (1-d) + d × (PR(B)/L(B))
PR(A) = (1-0.85) + 0.85 × (5/10)
PR(A) = 0.150 + 0.425
Example 1: Simple calculation
Page A receives a link from page B with PageRank 5.0
Page B has 10 outgoing links
PR(A) = 0.15 + 0.85 × (5.0/10) = 0.15 + 0.425 = 0.575Example 2: Multiple links
Page A receives links from page B (PR=3.0, 5 links) and C (PR=2.0, 2 links)
PR(A) = 0.15 + 0.85 × (3.0/5 + 2.0/2) = 0.15 + 0.85 × 1.6 = 1.51Modern vs. Original PageRank
Original PageRank (1998-2010)
- • Primary ranking factor
- • Publicly accessible (Toolbar)
- • Simple link-based algorithm
- • Vulnerable to manipulation
- • Monthly updates
Modern PageRank (2010+)
- • One of hundreds of factors
- • Internal Google tool
- • Integrated with machine learning
- • Spam-resistant improvements
- • Real-time updates
Modern Improvements
Personalized PageRank:
Adjusted based on user interests and search history
Topical PageRank:
Weighting based on topic relevance and context
TrustRank integration:
Combined with trust signals for spam protection
Temporal factors:
Time-based weighting of links and freshness signals
Limitations and Challenges
Manipulation and spam
- • Link farms and PBN networks
- • Artificial link exchanges
- • Purchased links for manipulation
- • Comment spam and forum spam
Technical challenges
- • Computational complexity for billions of pages
- • Dangling nodes (pages without outgoing links)
- • Spider traps and infinite loops
- • Scaling to real-time updates
Conceptual limitations
- • Focus only on link popularity, not content
- • Bias toward older, established websites
- • Ignores user intentions and context
- • Static model vs. dynamic web
PageRank's Significance Today
Still Fundamental
PageRank is still a core part of Google's algorithm, although it now works in combination with hundreds of other ranking factors. The fundamental idea of link-based authority remains central to how Google evaluates web pages.
Practical implications for SEO
Links are still important:
Quality links from authoritative pages still have high value
Focus on quality:
Get links from relevant, trustworthy sources
Internal links:
Distribute PageRank strategically on your own site
Holistic approach:
Combine link building with content and technical SEO
Want to master modern link building?
Now that you understand the mathematical background of PageRank, you can learn to apply this knowledge practically in modern SEO and link building strategies.
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