Measure influence across paths in complex graphs quickly. Adjust damping and baseline to match physics. See ranked nodes above inputs, then download files today.
Katz centrality assigns each node a score that sums contributions from all walks ending at that node, with longer walks exponentially damped. In matrix form:
x^{(k+1)} = α A x^{(k)} + β 1x = (I − αA)^{-1} (β 1)
For many graphs, choosing α < 1/ρ(A) helps ensure convergence, where ρ(A) is the spectral radius of A.
Example directed, unweighted network using an edge list.
| From | To | Weight |
|---|---|---|
| A | B | 1 |
| A | C | 1 |
| B | C | 1 |
| C | D | 1 |
| D | A | 1 |
Try alpha = 0.1, beta = 1, and enable normalization to compare node influence.
Katz centrality measures how strongly a node is connected to all others through paths of every length. In physics-style networks, nodes often represent sites, particles, sensors, or states, while edges encode interaction strength, transition likelihood, or coupling. Higher scores indicate nodes that receive many short and moderately long influences.
The calculator uses an iterative walk-summation update x^{(k+1)} = αAx^{(k)} + β1.
The damping factor α down-weights longer walks, so a two-step connection contributes less than
a direct connection. This is useful when transport or signal propagation decays with distance or time.
For stable Katz scores, α is commonly chosen below 1/ρ(A), where ρ(A) is the spectral radius.
This page estimates ρ(A) via a power iteration and displays it in Results. If you see a warning,
reduce α to improve convergence and avoid runaway amplification.
The baseline term β injects a uniform “source” into every node. In physical analogies, it can represent
constant background excitation, uniform injection, or a prior offset that prevents isolated nodes from collapsing to zero.
Setting β=0 removes this bias, emphasizing connectivity only.
Many physical systems are directional: flows, causal influence, or biased transitions. Enable “Directed graph” to preserve directionality. For heterogeneous couplings, enable “Weighted edges” and provide a third column weight. Weights can model conductance, coupling constants, interaction frequencies, or transition rates.
Iterations stop when the maximum absolute change (Max Δ) falls below your tolerance. Tight tolerances (e.g., 1e-8)
provide stable rankings but may need more iterations. If convergence is slow, lower α, relax tolerance, or enable normalization.
The “Convergence trace” table helps diagnose stability.
Raw Katz scores depend on scaling of A, α, and β. Normalization rescales scores by the maximum absolute value,
placing results in a comparable range. This is practical when comparing runs across different experimental conditions, network sizes,
or when weights are in different units.
Katz centrality appears in diffusion and transport networks, lattice-like interaction graphs, oscillator coupling maps, contact networks for spreading processes, and inference graphs in statistical mechanics. It highlights nodes that are not only well-connected locally, but also reachable through many alternative routes, which often correlates with robustness and influence.
Use a matrix when you already have an N×N numeric adjacency table. Use an edge list when nodes are labeled, when data is sparse, or when you want to enter weights per connection.
Start small, like 0.05 to 0.2, then check the 1/ρ(A) hint shown in Results. Keeping alpha below that threshold usually improves stability and prevents diverging scores.
Beta is a uniform background injection. It can represent constant excitation, prior influence, or a baseline field that keeps every node active even if it has few connections.
Non-convergence typically happens when alpha is too large, weights are extreme, or tolerance is very tight. Reduce alpha, increase max iterations, or relax tolerance to reach a stable fixed point.
Negative weights are accepted in matrix mode, but they may produce harder-to-interpret scores and can destabilize iterations. If modeling inhibition, keep alpha small and verify results with sensitivity checks.
Normalization rescales scores so the maximum absolute score becomes 1. It preserves ranking in most cases, but makes output easier to compare across different network sizes and weight scales.
When the method converges, the fixed point matches the closed-form solution. Iteration avoids matrix inversion, which is useful for larger graphs and allows you to monitor stability using the Max Δ trace.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.