Enter adjacency matrix
Use spaces or commas. Each row is a new line. Values may be 0/1 or weights.
Example data table
This example corresponds to the prefilled 4×4 adjacency matrix.
| Node | Connected to | Degree |
|---|---|---|
| 1 | 2, 3 | 2 |
| 2 | 1, 3, 4 | 3 |
| 3 | 1, 2, 4 | 3 |
| 4 | 2, 3 | 2 |
Formula used
- Adjacency matrix A stores edges/weights between nodes.
- Degree matrix D is diagonal, with \(D_{ii}=\sum_j A_{ij}\).
- Combinatorial Laplacian: \(L = D - A\).
- Normalized (symmetric): \(L_{sym}= I - D^{-1/2} A D^{-1/2}\).
- Random-walk: \(L_{rw}= I - D^{-1} A\).
For undirected graphs, L is symmetric and its eigenvalues describe connectivity and diffusion behavior.
How to use this calculator
- Paste an NxN adjacency matrix using spaces or commas.
- Select directed/undirected and weighted/unweighted behavior.
- Choose a Laplacian variant based on your analysis goal.
- Press Calculate to see results above the form.
- Use Download CSV or Download PDF for reporting.
Tip: For noisy data, enable symmetrization when analyzing undirected networks.
Graph Laplacian in network physics
The graph Laplacian is a cornerstone operator for translating a discrete network into physics-ready mathematics. In diffusion, heat flow, and vibration analogies, the Laplacian links local connections to global behavior. Given a measured adjacency matrix, this calculator builds the degree matrix, constructs a chosen Laplacian variant, and reports key spectral quantities used in modeling transport, synchronization, and stability.
What the adjacency matrix represents
Each entry Aij encodes the interaction between node i and node j. In experimental graphs, A may arise from coupling strengths, contact rates, or similarity scores. If you choose “unweighted,” every nonzero entry becomes 1, which is useful when only topology matters. For weighted graphs, numerical magnitudes are preserved to reflect stronger or weaker links.
Degree matrix and conservation intuition
The degree of node i is the row-sum di=∑jAij, stored on the diagonal of D. In combinatorial form, L=D−A, each row of L sums to zero for undirected graphs. That structure mirrors conservation laws: net outflow equals accumulated flow, a property exploited in discretized continuum operators and mass-balance models.
Choosing a Laplacian variant
Use the combinatorial Laplacian when units in A have physical meaning and you want a direct stiffness-like operator. Use the symmetric normalized Laplacian for scale-invariant comparisons across nodes of different degree. Use the random-walk Laplacian to study Markov dynamics where transition probabilities depend on outgoing weights.
Spectral outputs that matter
Eigenvalues and eigenvectors summarize connectivity. For an undirected graph, eigenvalues are real and nonnegative. The number of (near) zero eigenvalues indicates the number of connected components. The second-smallest eigenvalue, often called the Fiedler value, quantifies algebraic connectivity and is widely used in clustering, robustness, and diffusion speed estimates.
Normalized bounds and stability notes
For undirected graphs, the symmetric normalized Laplacian has eigenvalues within [0, 2], a convenient numerical scale for stability comparisons. Large weights can produce large combinatorial eigenvalues, which may require tighter precision when exporting. If your matrix is noisy or nearly symmetric, enabling symmetrization can reduce artifacts in spectral results.
Practical modeling workflow
Start by deciding whether direction matters. If you are modeling bidirectional couplings (springs, resistors, undirected interactions), use undirected mode and consider symmetrization. If you are modeling flows or causal influence, keep direction. Then choose unweighted vs weighted based on whether magnitudes encode meaningful physics. Finally, export CSV/PDF to document matrices and spectra in reports.
Computation, size limits, and interpretation
This page computes eigenvalues using a client-side approach intended for small to medium matrices. For very large networks, spectral decompositions scale roughly cubically with node count, so consider sampling, sparsity methods, or dedicated numerical tools. Always validate that A is square, that self-loops are intentional, and that units are consistent before drawing physical conclusions.
FAQs
1) What is the difference between combinatorial and normalized Laplacians?
The combinatorial form is L=D−A and preserves weight units. The symmetric normalized form rescales by degrees, improving comparisons across uneven node degrees and bounding eigenvalues for undirected graphs.
2) Why do I see a zero eigenvalue?
A zero eigenvalue is expected: it corresponds to a constant eigenvector on each connected component. Multiple zeros typically indicate multiple disconnected components or nearly disconnected regions in the graph.
3) When should I use the random-walk Laplacian?
Use it for Markov or diffusion processes where transition probability from node i depends on its outgoing degree. It directly links to random-walk mixing, hitting times, and steady-state behavior.
4) Should I symmetrize my matrix?
If the network is intended to be undirected but measurements introduce asymmetry, symmetrization can improve stability and yield real spectra. Do not symmetrize if direction is physically meaningful.
5) How do self-loops affect the Laplacian?
Self-loops increase the degree and modify diagonal terms. Depending on your physical model, they may represent self-coupling or measurement bias. Include them only when they have a justified meaning.
6) What does the Fiedler value tell me?
The second-smallest eigenvalue measures algebraic connectivity. Larger values generally imply stronger global connectivity and faster diffusion, while small values can indicate bottlenecks and community structure.
7) Why do directed graphs produce complex behavior?
Directed Laplacians are generally not symmetric, so eigenvalues can be complex and interpretations change. For spectral clustering or energy-like models, many workflows instead use symmetrized or normalized undirected forms.