Spectral Coarse-Graining Theory

Spectral Coarse-Graining (SCG) reduces a complex network while preserving diffusion dynamics.

Given adjacency matrix \(A\), the normalized Laplacian is:

\[L = D^{-1/2}(D - A)D^{-1/2}\]

Key spectral properties:

• Algebraic connectivity \(\lambda_2\): network robustness

• Spectral gap \(\Delta = \lambda_{k+1} - \lambda_k\): cluster boundary

• Spectral radius \(\rho\): contagion speed

SCG risk score:

\[\text{SCG Risk} = 1 - \frac{\lambda_2}{\rho}\]

Ranges from 0 (robust) to 1 (fragile).