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Spectral

Calculate graph Laplacian eigenmodes and use them to follow heat diffusion through a weighted network.

Computing new parameters. The previous result is shown until ready.
Visualization controls

Two 4 × 4 grids joined by one bridge of weight 0.15; internal weights are 1. Heat at t = 1.

Edge 0–1, weight 1Edge 0–4, weight 1Edge 1–2, weight 1Edge 1–5, weight 1Edge 2–3, weight 1Edge 2–6, weight 1Edge 3–7, weight 1Edge 4–5, weight 1Edge 4–8, weight 1Edge 5–6, weight 1Edge 5–9, weight 1Edge 6–7, weight 1Edge 6–10, weight 1Edge 7–11, weight 1Edge 8–9, weight 1Edge 8–12, weight 1Edge 9–10, weight 1Edge 9–13, weight 1Edge 10–11, weight 1Edge 10–14, weight 1Edge 11–15, weight 1Edge 12–13, weight 1Edge 13–14, weight 1Edge 14–15, weight 1Edge 16–17, weight 1Edge 16–20, weight 1Edge 17–18, weight 1Edge 17–21, weight 1Edge 18–19, weight 1Edge 18–22, weight 1Edge 19–23, weight 1Edge 20–21, weight 1Edge 20–24, weight 1Edge 21–22, weight 1Edge 21–25, weight 1Edge 22–23, weight 1Edge 22–26, weight 1Edge 23–27, weight 1Edge 24–25, weight 1Edge 24–28, weight 1Edge 25–26, weight 1Edge 25–29, weight 1Edge 26–27, weight 1Edge 26–30, weight 1Edge 27–31, weight 1Edge 28–29, weight 1Edge 29–30, weight 1Edge 30–31, weight 1Edge 7–20, weight 0.150Node 0, heat 0.27438269692240813, component 0Node 1, heat 0.16172844539253867, component 0Node 2, heat 0.06471549939957634, component 0Node 3, heat 0.022904575840684883, component 0Node 4, heat 0.1617302854778009, component 0Node 5, heat 0.09532327316737013, component 0Node 6, heat 0.03810251488982123, component 0Node 7, heat 0.013175746395832655, component 0Node 8, heat 0.0647310546286561, component 0Node 9, heat 0.038152713803454995, component 0Node 10, heat 0.015257580549334926, component 0Node 11, heat 0.005361193650723401, component 0Node 12, heat 0.02297013367444052, component 0Node 13, heat 0.013538926055861556, component 0Node 14, heat 0.005416321415019271, component 0Node 15, heat 0.0019139592090610608, component 0Node 16, heat 0.0000656527211568111, component 0Node 17, heat 0.000015902623588341484, component 0Node 18, heat 0.0000027258438988858482, component 0Node 19, heat 4.240505636631313e-7, component 0Node 20, heat 0.0003637233635044195, component 0Node 21, heat 0.00005247594146290944, component 0Node 22, heat 0.0000068318943186858245, component 0Node 23, heat 8.857586127737606e-7, component 0Node 24, heat 0.00005785360818069862, component 0Node 25, heat 0.00001360083025453359, component 0Node 26, heat 0.0000022770278380286813, component 0Node 27, heat 3.4739452849232763e-7, component 0Node 28, heat 0.000009014034756571985, component 0Node 29, heat 0.0000027258438958757972, component 0Node 30, heat 5.437034608414915e-7, component 0Node 31, heat 9.488739730971511e-8, component 0

Playback advances observation time at 2 model units per second while calculations keep pace. It pauses when this view is offscreen or the page is hidden.

00.274383Heat

Color scale adapts to this field; use exact values to compare times or modes. A dashed ring marks the initial source. Use the Node ID control to inspect closely spaced nodes. Changing layout leaves every eigenvalue and heat value unchanged.

Node 0

Heat
0.2743826969
Weighted degree Lᵢᵢ
2
Component
0
Long-time component mean
0.03125
Computed spectrum · selected mode 1
λ0 = -1.1861182178982618e-16λ1 = 0.016366137947723867λ2 = 0.5857864376269049λ3 = 0.5857864376269054λ4 = 0.5857864376269057λ5 = 0.6205725292006484λ6 = 1.1715728752538106λ7 = 1.1805234657410628λ8 = 1.9999999999999984λ9 = 2.0000000000000004λ10 = 2.000000000000004λ11 = 2.0340796952796762λ12 = 2.585786437626902λ13 = 2.5857864376269033λ14 = 2.585786437626905λ15 = 2.6223653675771574λ16 = 3.414213562373094λ17 = 3.414213562373095λ18 = 3.414213562373095λ19 = 3.4493875319497285λ20 = 3.9999999999999987λ21 = 4.000000000000001λ22 = 4.000000000000002λ23 = 4.000000000000002λ24 = 4.000000000000002λ25 = 4.082074714071167λ26 = 5.41421356237309λ27 = 5.4142135623730905λ28 = 5.414213562373091λ29 = 5.45589380635107λ30 = 6.828427124746187λ31 = 6.83873675188175940λMode index: 0 through 31

Eigenvectors may change sign. The displayed sign convention makes the largest-magnitude entry positive.

Laplacian matrix · local view
Rows and columns 0–7 of the 32 × 32 matrix L. JSON includes the full matrix.
Node01234567
02-100-1000
1-13-100-100
20-13-100-10
300-12000-1
4-10003-100
50-100-14-10
600-100-14-1
7000-100-13.15

Diagonal entries are weighted degrees; off-diagonal entries are negative edge weights. Zero means no edge. The diagonal includes neighbors outside this displayed block.

Node 0 neighbors: 1 (weight 1), 4 (weight 1).

Node heatpositiveRange 9.489e-8 to 0.27438Color scale adapts to this run; inspect numeric values to compare runs.

Measurements

Nodes
32
Positive-weight edges
49
Connected components
1
Second eigenvalue λ₂
0.016366138
Selected eigenvalue
0.016366138
Selected mode index
1
Total heat mass
1
Distance from global mean
0.3441288
Distance from component means
0.3441288
Connected decay bound
0.9682737
10% threshold first observed
37.9First sampled crossing; observation interval 0.1. The graph and source are fixed.
Eigensystem residual
7.600e-13Frobenius norm of LQ − QΛ divided by max(1, ‖L‖F).
Orthogonality residual
3.060e-14
Maximum mass residual
4.396e-14
Raw heat contains a minimum value of -8.965050923848139e-15. Negative values within floating-point error are retained in data; colors use the nonnegative heat scale.

Guided exploration

How does one bridge edge change the timescale of diffusion?

Laplacian eigenvalue spectrum

Vertical: Eigenvalue λ

-1.186e-1601.717.753.4215.55.1323.36.8431Mode index (zero-based)
Computed eigenvalues
Heat decay over a fixed observation window

Vertical: Euclidean distance

0.0069300.251500.4961000.741500.984200Observation time
Distance from global meanDistance from component meansConnected spectral upper bound

Experiment

Save the current parameters and full-precision results, or reproduce the documented study. Importing an experiment reruns its calculation. Display quality and camera position do not affect the exported mathematics.

Open the spectral study →
Method details

Unnormalized L = D − W; double-precision cyclic Jacobi (maximum 100 sweeps), followed by the modal matrix exponential. Observation sampling 0.1 on [0,200] is not an integration step. Node layout is excluded from L. Eigenvector signs and repeated-eigenvalue bases are not unique.

Model version: spectral-1.0.0. Software: 0.1.0+uncommitted.dirty.f24df3120282. Numerical tolerance: 1e-10.

Unrounded residual: 7.600110870080371e-13

Classification: numerical experiment. Mathematical claims require the stated assumptions and derivations.

Reasoning

Read the definitions, follow the arguments and examine the conditions of each result.

Background: Vectors and matrices, Eigenvalues, Exponential decay. Start with the model for a brief introduction, follow the guided exploration for a closer look, or work through the arguments and experiment for a detailed study.

What you can learn
  • Construct the unnormalized Laplacian from graph weights.
  • Express heat flow in an orthonormal eigenbasis.
  • Separate graph connectivity from display layout and numerical eigenvector choices.

Reading step 1 of 5: Construct the Laplacian

Step 1 · Derivation · Selected

Construct the Laplacian

The graph contains a finite set of nodes and undirected edges with nonnegative weights. The matrix W stores those weights symmetrically, and D places each node’s weighted degree on the diagonal. Their difference L = D − W is the unnormalized Laplacian. Node positions in the drawing do not enter this construction.

Expand vᵀLv and collect the two appearances of each edge. The result is one term wᵢⱼ(vᵢ − vⱼ)² per undirected edge. Every term is nonnegative, so L is positive semidefinite. This calculation also explains why counting each edge twice would give an incorrect energy formula.

vTLv=∑{i,j}∈Ewij(vi−vj)2≥0v^{\mathsf T}Lv=\sum_{\{i,j\}\in E}w_{ij}(v_i-v_j)^2\ge0

Theorem. The graph Laplacian is positive semidefinite.

Conditions and assumptions

  • Finite undirected graph; nonnegative weights; no self-loops.
  • A finite undirected graph.
  • Nonnegative edge weights and no self-loops.
  • Each undirected edge appears once in the sum.

Sources: [M8] Spectral and Algebraic Graph Theory

Step 2 · Derivation

Connect the kernel to components

A vector has zero Laplacian energy exactly when its endpoint values agree across every positive-weight edge. It is therefore constant on each connected component. Each component supplies an independent constant vector, so the dimension of the kernel equals the number of components.

The implementation determines components from the edge data rather than from a rounded eigenvalue. A small positive bridge still connects the graph; a zero bridge removes that connection. A connected graph has one zero eigenvalue and a positive second eigenvalue. A disconnected graph has a separate conserved mean in every component, so global uniformity is generally unavailable.

Theorem. The multiplicity of the zero eigenvalue equals the number of connected components.

Conditions and assumptions

  • Only positive-weight edges count as connections.
  • L = D − W is the unnormalized Laplacian.

Sources: [M8] Spectral and Algebraic Graph Theory

Step 3 · Derivation

Solve the graph heat equation

Because L is symmetric, choose an orthonormal basis of eigenvectors φₖ with eigenvalues λₖ. Expand the initial node values in that basis. Substituting a modal expansion into du/dt = −Lu gives one scalar equation aₖ′ = −λₖaₖ for each coefficient.

Each coefficient is therefore its initial value multiplied by exp(−λₖt). Summing the modes yields the displayed field. The zero modes remain unchanged. Since every row of L sums to zero and L is symmetric, the derivative of the sum of node values is zero. The heat experiment has no external source after its initial condition is set.

u(t)=∑k=1ne−λkt⟨u(0),ϕk⟩ϕku(t)=\sum_{k=1}^{n}e^{-\lambda_kt}\langle u(0),\phi_k\rangle\phi_k

Derivation. Each Laplacian eigenmode decays at the rate given by its eigenvalue.

Conditions and assumptions

  • du/dt = −Lu with fixed L.
  • The eigenvectors form an orthonormal basis.
  • No external sources or sinks are added.

Sources: [M8] Spectral and Algebraic Graph Theory

Step 4 · Derivation

Bound decay toward the mean

On a connected graph, subtract the uniform mean ū1. The remaining vector has no component in the constant eigenvector. Its squared norm is the sum of the squared nonconstant coefficients multiplied by exp(−2λₖt). Every relevant eigenvalue is at least λ₂, giving the stated upper bound after taking a square root.

The bound describes the slowest available decay rate; a particular initial field need not attain it. Changing one bridge alters a bottleneck while leaving the internal edge weights fixed. Scaling every weight instead rescales all eigenvalues together and simply changes the model’s time unit. The study keeps those operations distinct.

∥u(t)−uˉ1∥2≤e−λ2t∥u(0)−uˉ1∥2\|u(t)-\bar u\mathbf 1\|_2\le e^{-\lambda_2t}\|u(0)-\bar u\mathbf 1\|_2

Theorem. On a connected graph, the second eigenvalue bounds decay toward the uniform mean.

Conditions and assumptions

  • A connected graph with at least two nodes.
  • The graph is connected and n ≥ 2.
  • Eigenvalues are sorted in nondecreasing order.
  • The bound need not be attained by the selected initial state.

Sources: [M8] Spectral and Algebraic Graph Theory

Step 5 · Numerical experiment

Interpret computed modes

A normalized eigenvector and its negative describe the same mode. At a repeated eigenvalue, many orthonormal bases describe the same eigenspace. A solver’s sign flip or basis change is not a sudden change in the graph. The display identifies near-repeated modes and uses a signed scale for eigenvectors, separate from the heat scale.

The eigensolver is checked through residuals and orthogonality, then against independent small-graph answers. With one unit-weight edge, the eigenvalues are 0 and 2 and heat from (1,0) becomes ((1 + exp(−2t))/2, (1 − exp(−2t))/2). Saved bridge studies include weights, spectra, norms and actual threshold observations.

Sources: [M8] Spectral and Algebraic Graph Theory

References and review status

Project attribution: BIPH Math Society. The exposition was prepared with tool assistance.

Review status: Not reviewed. Independent review by a teacher or society member has not yet been recorded. Numerical checks do not replace mathematical review.

  1. [M8] Spectral and Algebraic Graph Theory

    Daniel A. Spielman

    Incomplete draft dated 2 April 2025. Sections 1.3, 1.5.4 and 3.1 and Chapter 5 support the Laplacian, positive semidefiniteness, connectivity and example spectra. The heat equation solution and decay estimate are derived in the exhibit from this spectral framework. Source checked 16 September 2026.

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