Calculator Inputs
Use weighted or binary adjacency values. The calculator normalizes each row into transition probabilities before applying damping.
Example Data Table
The sample below matches the built-in example button. Scores assume damping 0.85 with uniform teleportation.
| Node | Outgoing Links | Approx Rank Score | Approx Share |
|---|---|---|---|
| A | B, C | 0.379734 | 37.973% |
| B | C | 0.198887 | 19.889% |
| C | A | 0.383879 | 38.388% |
| D | A, C | 0.037500 | 3.750% |
Formula Used
1. Row normalization
For each row i, convert outgoing weights into probabilities:
Pij = aij / Σ aik
2. Dangling rows
If row i has no outgoing weight, replace it with either a uniform probability row or the teleportation vector.
3. Google matrix
G = αP + (1 - α)1vT
Here, α is the damping factor and v is the teleportation vector.
4. Power iteration
π(k+1) = π(k)G
5. Convergence check
Δ = ||π(k+1) - π(k)||1
The dominant eigenvalue of a valid Google matrix is one, and the stationary vector gives the long-run ranking distribution.
How to Use This Calculator
- Choose the number of nodes in your network.
- Enter labels for each node, separated by commas.
- Fill the adjacency matrix using nonnegative link weights.
- Set the damping factor, tolerance, and maximum iterations.
- Select uniform or custom teleportation and initial vectors.
- Choose how dangling rows should be handled.
- Press the calculate button to generate matrices and rankings.
- Use the CSV and PDF buttons to export your results.
FAQs
1. What does this calculator measure?
It builds a stochastic transition matrix, applies damping, forms the Google matrix, and estimates the stationary ranking vector through iterative convergence.
2. Can I use weighted links instead of zeros and ones?
Yes. Any nonnegative values are allowed. Each row is normalized automatically, so larger values create stronger transition probabilities.
3. What is a dangling row?
A dangling row has no outgoing weight. The calculator replaces it with either a uniform row or the teleportation vector to preserve a valid stochastic process.
4. Why is the damping factor usually 0.85?
It balances link-following behavior with random teleportation. This often improves stability and prevents rank sinks from dominating the long-run distribution.
5. Why does the calculator use power iteration?
Power iteration is efficient for finding the stationary distribution of stochastic matrices. It is practical, interpretable, and easy to monitor with convergence history.
6. What does the residual value mean?
Residual measures how close the final rank vector is to an invariant distribution. Smaller values indicate a better numerical match to the stationary solution.
7. Why might two nodes have similar scores?
Similar link patterns, comparable incoming influence, and balanced teleportation weights can produce near-equal steady-state probabilities for multiple nodes.
8. When should I use a custom teleportation vector?
Use a custom vector when some nodes should receive stronger baseline visitation probability, such as trusted pages, priority states, or biased navigation assumptions.