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Math: multilayer operators

Let L = A₁ × A₂ × … × A_k be the cartesian product of aspects, where each aspect A_i has a finite set of elementary layers. A layer index is a tuple α ∈ L.

Supra incidence and adjacency

  • Build the set of present vertex‑layers V×M = {(v, α) : v has presence in layer α}.
  • Define supra incidence B̃ ∈ R^{|V×M| × |E|} by mapping each original endpoint (v, α) of edge e to a row in B̃ with the same sign convention as the monolayer case.

From B̃, obtain: - Supra Laplacian: L̃ = B̃ W B̃ᵀ (undirected or symmetrized case). - Directed supra adjacency: Â = B̃⁺ W (B̃⁻)ᵀ.

Tensor representation

Alternatively, use a 4‑index tensor view 𝒜[u, α, v, β] giving the weight of arcs from (u, α) to (v, β).

  • Fold to supra adjacency by indexing (u,α) and (v,β) pairs.
  • Unfold from supra adjacency back to tensor by inverting the index mapping.

Edge types in multilayer

  • Intra‑layer: endpoints share the same layer tuple α.
  • Inter‑layer: endpoints lie in different layer tuples.
  • Coupling: endpoint is the same vertex across two layers (v, α) ↔ (v, β).

These categories determine which blocks of  are populated and how presence constraints apply.