SpectralEntropy#d557

Mechanism

Build the graph Laplacian L. For an undirected graph L = D - A, with D the diagonal degree matrix and A the adjacency matrix. For a directed graph use the incidence matrix B (B_ie = +1 if edge e leaves node i, -1 if it enters, 0 otherwise) and L = B B^T. Take its eigenvalues lambda_i, normalise them to p_i = lambda_i / sum_j lambda_j, and compute H_s = -sum_i p_i log2 p_i, the ShannonEntropy of the spectrum. Because the Laplacian governs diffusion, synchronisation and relaxation on the graph, H_s describes global structure: a spread-out spectrum (high H_s) means many distinct flow modes, a concentrated spectrum (low H_s) means regular, modular or tightly structured topology. It complements local measures such as DegreeEntropy.

Invariants

• Uses the whole eigenvalue spectrum, so it reflects global rather than local structure
• Laplacian eigenvalues are non-negative, so the normalised spectrum is a valid distribution
• The zero eigenvalues (one per connected component) contribute nothing

Failure modes

• Density domination: when the spectrum mostly tracks node or edge count, H_s scales almost perfectly with density and carries little else, so an excellent data collapse can be a trivial one
• Comparing spectra of graphs with different Laplacian definitions (combinatorial, normalised, incidence-based)
• Eigendecomposition cost on very large graphs

Where this came from

• Thesis Kevin Do, PhD thesis on ant collective behaviour (NCSU, 2026), Ch. 1.2.1 Eq. 1.2; Ch. 1.3.3 Eqs. 1.8-1.9; Ch. 3.2.1 Eqs. 3.22-3.25; Ch. 4.2.3

Builds on

• ShannonEntropyReferences
• DegreeEntropyComposes with

Used by

• InteractionSequenceEntropySignatureReferences
• SecondOrderInteractionNetworkComposes with

Identity

The name is the hash of the definition. Change a word and it becomes a different pattern.

Sema ID

sema:SpectralEntropy#mh:SHA-256:d557f8c3d693a0ee6c24c656efb113cf0fab9dabae8c60f5ba1ae076dbc6b79a

Reference

SpectralEntropy#d557

Ring

1

Tier

1