Example Data Table
| From | To | Weight | Meaning |
|---|---|---|---|
| A | B | 1 | A points to B |
| A | C | 1 | A points to C |
| B | C | 1 | B points to C |
| C | A | 1 | C points back to A |
| D | C | 1 | D contributes rank into C |
Formula Used
PageRank assigns each node a probability-like score based on incoming influence. For a network with N nodes and damping factor d:
Here P(i → j) is the transition probability from node i to j. If node i has outgoing weights w(i→k), then P(i → j) = w(i→j) / Σₖ w(i→k). Dangling nodes are handled using your selected option.
How to Use This Calculator
- Enter node labels like A,B,C, or set a node count.
- Paste edges as from,to per line, with optional weight.
- Adjust damping, tolerance, and max iterations for precision.
- Choose how dangling nodes should be treated.
- Click Calculate PageRank to view scores above the form.
Notes for Physics Use Cases
- Model transport, interaction, or influence on a directed graph.
- Use weights to represent rates, coupling strengths, or transition likelihoods.
- Compare node rankings across different parameter settings.
Article
1) What PageRank Measures in Networked Physics
PageRank treats a directed network as a flow process where each node holds a share of “importance” and passes it along outgoing links. In physics-style models, nodes can represent states, detectors, modes, particles, regions, or agents, while edges represent transitions, couplings, or transport pathways. The final score behaves like a steady-state distribution after repeated redistribution.
2) Damping Factor and Random-Reset Probability
The damping factor d controls how strongly the walk follows links versus resetting uniformly. A common value is d = 0.85, meaning 85% of the time the process follows the network and 15% performs a reset. Smaller d smooths scores and speeds convergence, while larger d emphasizes link structure and can slow convergence.
3) Weighted Links as Rates or Coupling Strengths
This calculator supports optional positive weights per edge. For an outgoing node i, each weight w(i→j) is normalized by the sum of all outgoing weights, producing transition probabilities P(i→j) = w(i→j)/Σk w(i→k). Use weights to encode measured rates, interaction strengths, or empirical transition frequencies.
4) Dangling Nodes and Conservation of Flow
Dangling nodes have no outgoing edges, which breaks strict conservation unless handled. The “Redistribute uniformly” option conserves total flow by spreading dangling mass across all nodes each iteration. “Ignore (then normalize)” can be useful for diagnostics, but it may distort intermediate totals until final normalization.
5) Convergence Controls: Tolerance and Iterations
PageRank is computed by iterative updates until change is small. The calculator uses an L1 difference threshold: when the total absolute change drops below the tolerance, it stops. Typical tolerances range from 1e-6 (fast) to 1e-10 (strict). For small graphs, convergence often occurs within 30–200 iterations, depending on d and connectivity.
6) Scaling with Network Size
Runtime grows primarily with the number of edges E and iterations T. Each iteration distributes rank along outgoing links, so the work is roughly proportional to E × T. Sparse networks (few edges per node) typically compute quickly even for dozens of nodes, while dense graphs may require fewer iterations but more per-iteration work.
7) Interpreting the Output Table
Scores sum to 1.00 after normalization, making comparisons straightforward. The “Share” column expresses the same score as a percentage for quick ranking. Outdegree and indegree help explain structure: a node can rank highly either by collecting many inbound links or by receiving weight from a few highly-ranked sources.
8) Practical Workflow for Experiments
Start with your measured or simulated interaction list, then run a baseline using d = 0.85, tolerance 1e-8, and 200 iterations. Modify weights to reflect updated measurements and rerun to observe score shifts. Export CSV for spreadsheets and PDF for reports, ensuring the same parameters are documented with every run.
FAQs
1) What input format should I use for edges?
Enter one edge per line as from,to. You may add a third value from,to,weight where weight is a positive number.
2) Why is the damping factor usually 0.85?
It balances following network structure with occasional resets, improving stability. In practice, 0.85 often provides good sensitivity to links without making convergence too slow.
3) What does “Final L1 Δ” mean?
It is the total absolute difference between consecutive score vectors. Smaller values indicate the iterative solution has stabilized under the chosen tolerance.
4) How should I set tolerance for reliable results?
Use 1e-8 as a strong default. For quick exploration use 1e-6. For highly sensitive comparisons, tighten to 1e-10 and increase iterations.
5) What are dangling nodes and why do they matter?
Dangling nodes have no outgoing edges. If unhandled, rank “leaks” from the system. Redistributing dangling mass preserves a valid probability-like score distribution.
6) Can I model physical transition rates with weights?
Yes. Weights are normalized into transition probabilities, so larger weights increase the fraction of flow passed along that edge each iteration.
7) Why do my rankings change when I add or remove edges?
PageRank depends on the full network structure. Adding an edge changes how influence circulates, which can raise some nodes while lowering others, even if their direct connections stay the same.