Spectral Methods in Brain Network Analysis
graph Laplacian, spectral clustering, brain networks, functional connectivity, Fiedler value, community detection, neuroscience
Graph Laplacian and Brain Networks
Given a brain connectivity matrix \(W \in \mathbb{R}^{n \times n}\) where \(w_{ij}\) represents the functional connectivity between regions \(i\) and \(j\), the normalized graph Laplacian is:
\[ \mathcal{L} = I - D^{-1/2} W D^{-1/2} \]
where \(D = \text{diag}(d_1, \ldots, d_n)\) is the degree matrix with \(d_i = \sum_j w_{ij}\).
Spectral Decomposition
The eigendecomposition of \(\mathcal{L}\) yields:
\[ \mathcal{L} = U \Lambda U^T = \sum_{k=0}^{n-1} \lambda_k \mathbf{u}_k \mathbf{u}_k^T \]
where \(0 = \lambda_0 \leq \lambda_1 \leq \cdots \leq \lambda_{n-1} \leq 2\).
Key Properties
- Number of zero eigenvalues = number of connected components
- The Fiedler value \(\lambda_1\) measures algebraic connectivity
- The spectral gap \(\lambda_1 - \lambda_0\) indicates community separation strength
Participation Ratio
The effective rank via participation ratio provides a model-free measure of spectral spread:
\[ \text{PR}(\boldsymbol{\lambda}) = \frac{\left(\sum_k \lambda_k\right)^2}{\sum_k \lambda_k^2} \]
This ranges from 1 (single dominant eigenvalue) to \(n\) (uniform spectrum), capturing the effective dimensionality of the network.
Connection to Statistical Physics
The graph Laplacian connects to the partition function of a Gaussian field on the network:
\[ Z(\beta) = \int \exp\left(-\frac{\beta}{2} \mathbf{x}^T \mathcal{L} \mathbf{x}\right) d\mathbf{x} = \prod_{k=1}^{n-1} \sqrt{\frac{2\pi}{\beta \lambda_k}} \]
The free energy \(F = -\frac{1}{\beta}\ln Z\) encodes thermodynamic properties of signal propagation on the network.