Graph Laplacian and Signal Energy
Connect a graph’s matrix to signal differences, connected components and zero-energy patterns.
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Graph Laplacian and Signal Energy
A graph can be represented by its Laplacian matrix L = D - A, where D is the degree matrix and A is the adjacency matrix. At vertex i, (Lx)i = Σj w_ij(xi-xj), the weighted sum of differences between its signal and each neighbor. The associated quadratic form computes the total energy of the signal across all edges.
Worked example: Path graph with weight 3
Consider a Path graph with 4 vertices and edge weight w = 3. The input signal vector is x = [2, -1, 1, 0].
- The Laplacian matrix is L = [[3, -3, 0, 0], [-3, 6, -3, 0], [0, -3, 6, -3], [0, 0, -3, 3]].
- Multiplying L by the signal vector gives Lx = [9, -15, 9, -3].
- The signal energy is the sum of squared differences across edges multiplied by the edge weight: 3 * ( (2 - (-1))2 + (-1 - 1)2 + (1 - 0)2 ) = 3 * (9 + 4 + 1) = 42.
- The sorted eigenvalues of this Laplacian are [0, 1.757359312881, 6, 10.242640687119].
| Edge (u, v) | Signal Difference (xu - xv) | Energy w(xu - xv)2 |
|---|---|---|
| (0, 1) | 2 - (-1) = 3 | 3 * 9 = 27 |
| (1, 2) | -1 - 1 = -2 | 3 * 4 = 12 |
| (2, 3) | 1 - 0 = 1 | 3 * 1 = 3 |
The total energy exactly matches the quadratic form xTLx = 42. Since there is 1 connected component with positive edge weights, there is exactly one zero eigenvalue.
Investigate Graph Connectivity and Zero Energy
Open Graph Laplacian and Signal Energy to experiment with different graph structures.
- Select the "Two pairs" graph and set the Edge weight to 2. Set the signals to [1, 1, -1, -1].
- Observe that the signal energy drops to 0, even though the signal vector is not all zeros.
- This happens because the graph has two disconnected components, allowing each component to hold a constant but different signal value without contributing to edge differences.
- Verify that the number of zero eigenvalues (the nullity of L) matches the number of positive-weight connected components.
- Change the Edge weight to 0 and observe that all edges become inactive, the graph splits into 4 components, and the Laplacian becomes a zero matrix with 4 zero eigenvalues.
Limitations of this specific model
This module uses unnormalized Laplacians for simple undirected graphs with a single common edge weight. It does not compute spectral clustering assignments or provide normalized Laplacian forms. The visualizations only show 4-node canonical layouts rather than general graph drawing, and matrix eigenvalues are analytic rather than numerically factored.
References
The primary reference detailing the unnormalized Laplacian matrix definition used in this module:
- NetworkX Laplacian Matrix Documentation - Official documentation detailing the definition of the unnormalized Laplacian matrix L = D - A for weighted graphs.
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Graph Laplacian and Signal Energy module.
Try Graph Laplacian and Signal Energy on Riano →