Supra Adjacency Matrix Overview

• Supra adjacency matrix is a mathematical structure that aggregates all pairwise relationships from hyperedges and layers into a unified block matrix.
• It generalizes traditional adjacency matrices to represent complex interactions in oriented hypergraphs and multilayer networks.
• Its structured block format facilitates spectral analysis and combinatorial insights essential for studying interconnected network systems.

A supra adjacency matrix is a mathematical structure designed to aggregate and encode multiple layers or types of relationships in a single, comprehensive block matrix, most commonly arising in the study of multilayer, multiplex, or oriented hypergraph networks. The supra adjacency matrix generalizes the classical adjacency matrix of graphs to simultaneously represent all pairwise (2-adic) adjacencies induced by the full set of hyperedges, layers, or relations present in complex combinatorial objects.

1. Formal Definition and Generalization

In the context of hypergraphs and multilayer networks, a supra adjacency matrix collects all possible pairwise adjacencies simultaneously, regardless of edge or layer uniformity. For an oriented hypergraph $G=(V,E,\mathcal{I})$ with incidence orientation $\sigma$, the adjacency matrix $A(G)$ is defined entrywise by

$$a_{ij} = \sum_{e \in E} -\sigma(v_i, e)\sigma(v_j, e),$$

for each pair $v_i, v_j$ of vertices with at least one hyperedge in common. This construction:

• Accommodates arbitrary edge sizes (non-uniform hypergraphs).
• Sums all pairwise adjacencies induced by multiple hyperedges (multiplicity).
• Generalizes the signed adjacency matrix for signed graphs to the hypergraph context (Reff, 2015).

In the multilayer or multiplex scenario, layers correspond to distinct edge types or relations, and the supra adjacency matrix becomes a block matrix whose diagonal blocks represent intra-layer adjacencies and off-diagonal blocks encode inter-layer couplings [(Ouvrard et al., 2017), De Domenico et al., 2013].

2. Construction for Oriented Hypergraphs

Given an oriented hypergraph $G$, the construction of the supra adjacency matrix relies on the notion of adjacency signature

$$\operatorname{sgn}_e(v_i, v_j) = -\sigma(v_i,e)\sigma(v_j,e),$$

where $v_i, v_j \in V$, $e \in E$, and $\sigma$ encodes the incidence orientation. The supra adjacency matrix is then

$$A(G) = (a_{ij}), \quad a_{ij} = \sum_{e \in E} \operatorname{sgn}_e(v_i, v_j),$$

with $a_{ij}=0$ whenever $v_i$ and $v_j$ are not co-incident in any hyperedge. Notably,

• This matrix is always symmetric.
• It does not require edges of uniform cardinality.
• It aggregates adjacency contributions from arbitrarily overlapping edge sets.

This definition coincides with the adjacency matrix of the strict 2-section of the hypergraph (the oriented hypergraph with all 2-element edges derived from original hyperedges), reflecting all pairwise vertex adjacencies induced by the full set of hyperedges (Reff, 2015).

3. Supra Adjacency Matrix in Multilayer Networks

The concept extends naturally to multilayer network analysis: the supra adjacency matrix is constructed by combining adjacency matrices from all layers into a large block matrix:

• Diagonal blocks correspond to adjacency matrices within each layer.
• Off-diagonal blocks encode connections between vertices in different layers, including potential inter-layer coupling terms. This approach allows for the analysis of interdependencies and cross-layer interactions within a unified algebraic framework, facilitating spectral, algebraic, and combinatorial investigations (Ouvrard et al., 2017).

4. Algebraic Foundation and Array Framework

Matrix-based network representations are further generalized by associative array algebra, which unifies arrays, matrices, and graphs under the same formalism. The supra adjacency matrix can be constructed via the product of incidence arrays: $$\mathbf{A}(x,y) = \bigoplus_{k \in K} \mathbf{E}_\mathrm{out}(k,x) \otimes \mathbf{E}_\mathrm{in}(k,y),$$ where $\mathbf{E}_\mathrm{out}, \mathbf{E}_\mathrm{in}$ are incidence arrays and $(V, \oplus, \otimes)$ is a suitable algebraic structure. The resulting adjacency array (or matrix) is supra in the sense that it unifies all pairwise adjacencies across layers, directions, or edge types. Mathematical criteria guarantee that this construction accurately reflects the network's connectivity: the algebra $(V, \oplus, \otimes)$ must satisfy (i) no non-trivial additive inverses, (ii) the zero product property, and (iii) zero is an annihilator for $\otimes$ (Dibert et al., 2015). These criteria are necessary and sufficient for accurate array-based supra adjacency matrix construction.

5. Relationship to Hypergraph and Higher-Order Structures

In hypergraphs with arbitrary edge cardinalities, the supra adjacency matrix corresponds to the aggregation of all possible 2-adic adjacencies resulting from hyperedges, but does not capture higher-order ($k$-adic, $k>2$) relationships. For general hypergraphs, this limitation motivates the e-adjacency tensor representation, which genuinely encodes $n$-adic (multi-way) relations among vertex sets. The supra adjacency matrix thus occupies an intermediate structural role:

• It reflects all induced pairwise adjacencies.
• It coincides with the adjacency matrix of the strict 2-section and retains spectral properties relevant for many algebraic and combinatorial analyses (Ouvrard et al., 2017).
• It serves as a natural block matrix generalization for multilayer or multiplex network data.

6. Structural and Spectral Properties

The supra adjacency matrix, by construction, supports spectral analysis and algebraic characterizations of the network:

• For oriented hypergraphs, the supra adjacency matrix often coincides with adjacency matrices of related intersection graphs and the incidence dual under specific regularity conditions, particularly for linear hypergraphs (no repeated pairs in edges) (Reff, 2015).
• There exist matrix identities between the adjacency, Laplacian, and incidence matrices; for example, $L(G) = D(G) - A(G) = H(G)H(G)^\top$, and their analogues for the incidence dual.
• As a supra structure, it enables the analysis of inter-layer connectivity and multi-edge interactions, key for the study of network controllability, spectral clustering, and algebraic combinatorics in complex systems.

7. Summary Table: Supra Adjacency Matrix Contexts

Context	Supra Adjacency Matrix Role	Notes
Oriented hypergraphs	Sums all hyperedge-induced pairwise adjacencies	Generalizes classic adjacency, allows multiples
Multilayer/multiplex networks	Block-matrix stacking of all layer adjacencies	Intra- and inter-layer encoded in block structure
Associative array formalism	Product of incidence arrays yields supra adjacency	Algebraic criteria required for correctness

The supra adjacency matrix serves as a central structure in hypergraph theory, multilayer network analysis, and associative array algebra, enabling the comprehensive study of aggregated pairwise connectivity across diverse relational contexts. Its structural and spectral properties underlie much of the modern combinatorial analysis of complex systems.