{
  "handle": "SpectralEntropy",
  "ref": "SpectralEntropy#d557",
  "stub": "d557",
  "semaId": "sema:SpectralEntropy#mh:SHA-256:d557f8c3d693a0ee6c24c656efb113cf0fab9dabae8c60f5ba1ae076dbc6b79a",
  "layer": "Infrastructure",
  "category": "Primitives",
  "gloss": "Entropy of a graph Laplacian's eigenvalues: the diversity of global flow and relaxation modes",
  "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 {{shannon_entropy}} 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 {{degree_entropy}}.",
  "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"
  ],
  "failureModes": [
    "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"
  ],
  "ring": 1,
  "tier": 1,
  "aliases": {
    "shannon_entropy": "ShannonEntropy",
    "degree_entropy": "DegreeEntropy"
  },
  "buildsOn": [
    {
      "handle": "ShannonEntropy",
      "type": "references",
      "alias": "shannon_entropy",
      "url": "/knowledge-graph/ShannonEntropy/",
      "json": "/knowledge-graph/ShannonEntropy.json"
    },
    {
      "handle": "DegreeEntropy",
      "type": "composes_with",
      "alias": "degree_entropy",
      "url": "/knowledge-graph/DegreeEntropy/",
      "json": "/knowledge-graph/DegreeEntropy.json"
    }
  ],
  "usedBy": [
    {
      "handle": "InteractionSequenceEntropySignature",
      "type": "references",
      "alias": "spectral_entropy",
      "url": "/knowledge-graph/InteractionSequenceEntropySignature/",
      "json": "/knowledge-graph/InteractionSequenceEntropySignature.json"
    },
    {
      "handle": "SecondOrderInteractionNetwork",
      "type": "composes_with",
      "alias": "spectral_entropy",
      "url": "/knowledge-graph/SecondOrderInteractionNetwork/",
      "json": "/knowledge-graph/SecondOrderInteractionNetwork.json"
    }
  ],
  "sources": {
    "research": [],
    "works": [
      {
        "kind": "thesis",
        "label": "Kevin Do, PhD thesis on ant collective behaviour (NCSU, 2026)",
        "note": "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",
        "url": ""
      }
    ],
    "homePage": false
  },
  "url": "/knowledge-graph/SpectralEntropy/",
  "graphUrl": "/knowledge-graph#SpectralEntropy",
  "directory": "/knowledge-graph/patterns.json"
}