Spectral: bridge weight and heat diffusion

Question

How does a single bridge weight affect the spectrum and the measured decay of graph heat?

Mathematical background

The fixed graph consists of two 4×4 grids joined by one weighted edge. Internal weights are 1 and total initial heat is 1 at a fixed left-grid node.

For the unnormalized Laplacian, a connected graph’s second eigenvalue gives an upper bound on decay toward the global mean.

Read the mathematical derivations.

Hypothesis or expected result

Theoretical expectations

A zero bridge disconnects the graph, producing two independent component means and two zero eigenvalues.

For positive bridge weights, measured deviation is expected to stay within the spectral decay bound, subject to numerical tolerance. The slowest possible bound need not be attained.

Parameters and assumptions

• Only the bridge weight changes: 0,0.01,0.05,0.2,1. Node IDs, internal edges and the source node are fixed.
• Observation times run from 0 to 200 at intervals of 0.1. This interval samples the modal solution; it is not an integration step.
• The comparison threshold is 10% of the initial distance from the uniform mean.

Each case’s complete parameter object appears below and in its JSON file. Display rounding is used only in this article; the saved data retain full numeric precision.

Method and numerical settings

Compute the eigensystem once per bridge setting with the shared kernel, and evaluate the heat solution at all observation times using those modes.

Save the complete graph, spectrum, field and norm curves, theoretical bound, and threshold observations. Do not rerun eigendecomposition separately for each time.

Methods reported by the executed kernel

• 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.

Software identifier

0.1.0+0e6017e6a747.dirty.02665ba9098c

Git revision

0e6017e6a747d3c610b956612b4125e232283001

Uncommitted worktree state

Present when the batch was generated

Source SHA-256

02665ba9098c7b8fe2a4b728fbd2b3749b1724fe9999f830d8fa90cb5d8b209b

Runtime

v26.5.1

Results from an actual run

This saved batch contains 5 completed calculations. The table and downloads below refer to those actual runs.

The largest recorded absolute diagnostic residual was 9.15329e-13. 5 of 5 residual-bearing cases were within their own reported numerical tolerance.

Across 5 recorded readings, Second eigenvalue λ₂ ranged from -3.75699e-16 to 0.06190975.

Across 5 recorded readings, Selected eigenvalue ranged from -3.75699e-16 to 0.06190975.

Across 5 recorded readings, Total heat mass ranged from 1 to 1.

Across 5 recorded readings, Eigensystem residual ranged from 1.14475e-15 to 9.15329e-13.

Bridge weight 0: λ₂ = -3.75699e-16; distance from the global mean at t = 200 was 0.1767767; connected bound Not applicable; first observed 10% crossing: Not observed by t = 200.

Bridge weight 0.01: λ₂ = 0.001238157; distance from the global mean at t = 200 was 0.1383664; connected bound 0.7683532; first observed 10% crossing: Not observed by t = 200.

Bridge weight 0.05: λ₂ = 0.005963459; distance from the global mean at t = 200 was 0.05431949; connected bound 0.2986252; first observed 10% crossing: 100.4.

Bridge weight 0.2: λ₂ = 0.02091855; distance from the global mean at t = 200 was 0.002814583; connected bound 0.01500181; first observed 10% crossing: 30.1.

Bridge weight 1: λ₂ = 0.06190975; distance from the global mean at t = 200 was 8.36409e-7; connected bound 4.12756e-6; first observed 10% crossing: 11.5.

Download summary CSVDownload complete summary JSON

Lines connect the recorded cases to aid comparison. Intermediate settings are not additional experiments.

Interpretation

The bridge comparisons above are readings from the saved runs, including the cases where no crossing was observed. Near-zero eigenvalues are reported with their numerical sign; connectivity is decided from edge weights.

Use component counts from the positive-weight edge data when interpreting the zero-weight case.

Changing one bridge modifies a bottleneck. Scaling every weight would instead rescale the whole time axis; that is not the operation studied here.

Limitations and possible errors

• Repeated eigenvalues define an eigenspace rather than a unique displayed basis. A mode can change sign without changing the mathematics.
• A crossing outside the observation window is not recorded as a finite time. Disconnected cases need not approach the global uniform mean.

Warnings recorded during these runs

• The graph has 2 components. Each retains its own initial mean; the connected global decay bound does not apply.
• The selected eigenvalue has 2 numerically indistinguishable modes. Its eigenspace is meaningful; the displayed orthonormal basis is not unique.
• Raw heat contains a minimum value of -5.065392549852277e-16. Negative values within floating-point error are retained in data; colors use the nonnegative heat scale.
• Raw heat contains a minimum value of -8.803721640582296e-16. Negative values within floating-point error are retained in data; colors use the nonnegative heat scale.
• Raw heat contains a minimum value of -2.982987121202818e-14. Negative values within floating-point error are retained in data; colors use the nonnegative heat scale.
• Raw heat contains a minimum value of -1.4137996329210978e-14. Negative values within floating-point error are retained in data; colors use the nonnegative heat scale.
• Raw heat contains a minimum value of -7.511352650979575e-16. Negative values within floating-point error are retained in data; colors use the nonnegative heat scale.

Reproduce this study

1. Open Spectral with the bridged-grid preset, 32 nodes, source node 0 and the saved bridge weight.
2. Read JSON for the full eigensystem outputs and CSV for the recorded evolution. Compare times without changing the graph layout or mathematical inputs.

Open spectral

{
  "preset": "bridge",
  "size": 32,
  "bridgeWeight": 0,
  "source": 0,
  "time": 200,
  "mode": 1,
  "field": "heat"
}

{
  "preset": "bridge",
  "size": 32,
  "bridgeWeight": 0.01,
  "source": 0,
  "time": 200,
  "mode": 1,
  "field": "heat"
}

{
  "preset": "bridge",
  "size": 32,
  "bridgeWeight": 0.05,
  "source": 0,
  "time": 200,
  "mode": 1,
  "field": "heat"
}

{
  "preset": "bridge",
  "size": 32,
  "bridgeWeight": 0.2,
  "source": 0,
  "time": 200,
  "mode": 1,
  "field": "heat"
}

{
  "preset": "bridge",
  "size": 32,
  "bridgeWeight": 1,
  "source": 0,
  "time": 200,
  "mode": 1,
  "field": "heat"
}

Each run’s download address is available in the summary JSON. The printed article preserves the methods, measurements and limitations.

References and review status

Project attribution: BIPH Math Society. Exposition and implementation were prepared with tool assistance.

Review status: Not reviewed. Independent teacher or member review has not yet been recorded.

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.