4th-Order Adjacency Tensor

• 4th-Order Adjacency Tensor is a symmetric representation that captures complex 4-adic interactions in hypergraphs and hb-graphs.
• It utilizes null vertices and normalization techniques to standardize non-uniform edge sizes and incorporate vertex multiplicities.
• The tensor underpins advanced spectral analysis and scalable algorithms using methods like tensor-times-same-vector for efficient computation.

A 4th-order adjacency tensor is a symmetric tensorial representation that encodes the multiway structure of 4-adic interactions in hypergraphs, non-uniform hypergraphs, and generalized objects such as hyperbag-graphs (hb-graphs, i.e., hypergraphs with multiset edges). The 4th-order adjacency tensor construction is central for advancing spectral theory and higher-order network analysis, supporting both theoretical and computational approaches for systems with complex multiway relationships.

1. Notions of Adjacency in Higher-Order Structures

Adjacency in classical graphs is a binary relationship, naturally represented by a 2nd-order matrix. For hypergraphs—where edges can join any subset of the vertex set—adjacency must generalize to higher-order relations:

• k-adjacency: A tuple of $k$ (not necessarily distinct) vertices is k-adjacent if it is contained in a hyperedge of size at least $k$.
• e-adjacency: For each edge (hyperedge) $e$, all vertices within $e$ are e-adjacent; this captures the simultaneous co-membership structure central to hypergraph data.

For non-uniform hypergraphs, where edge sizes vary, pairwise matrices fail to capture the intrinsic higher-order connectivity; an order-$r$ tensor, with $r$ being the maximal hyperedge cardinality, is required for a lossless representation (Ouvrard et al., 2017, Ouvrard et al., 2018).

2. Construction of the 4th-Order Adjacency Tensor

Explicit constructions vary according to the type of structure—simple hypergraph, non-uniform hypergraph, or hb-graph. The construction outlined below follows the most general approach based on (Ouvrard et al., 2018), with cross-references to (Aksoy et al., 2023, Ouvrard et al., 2017), and (Ouvrard et al., 2018).

a) General Definition for hb-graphs (Natural Multisets)

Let $H = (V, E)$ with $V = \{v_1, \dots, v_n\}$ and each $e \in E$ a multiset, i.e., $m_e : V \rightarrow \mathbb{N}$ records vertex multiplicities. Set $k$0, the maximum multiset cardinality of any edge.

• For each $k$1 of size $k$2, introduce a "null" vertex $k$3 of multiplicity $k$4.
• The extended multiset $k$5 consists of the elements of $k$6, plus $k$7 copies of $k$8.
• The tensor $k$9 has dimension $e$0, symmetrized in all indices:

$e$1

where $e$2, and $e$3 is nonzero only when the indices match the extended multiset $e$4:

$e$5

b) Specialization to $e$6 (Order 4)

Set $e$7; introduce null vertices $e$8, $e$9, $e$0. For $e$1 of size $e$2:

• $e$3
• $e$4
• Nonzero values:

$e$5

if the index multiset is given by the multiplicities $e$6 for $e$7 and $e$8 entries equal to $e$9 (location of $r$0) (Ouvrard et al., 2018).

An entry is nonzero if and only if, for some $r$1 of size $r$2, $r$3 of the indices equal $r$4 for $r$5 and $r$6 of the indices equal $r$7.

c) Uniform Hypergraphs

If the hypergraph is 4-uniform (every edge has size exactly 4), no null vertices are needed, and the tensor reduces to $r$8:

$r$9

with full permutation symmetry in all indices (Pearson et al., 2012).

3. Key Properties

• Permutation symmetry: $r$0 is invariant under any permutation of indices.
• Degree normalization: The m-degree of vertex $r$1, denoted $r$2 (i.e., the sum of its multiplicities), is recovered by summing $r$3 over the last three indices:

$r$4

• Handling duplicates: Multiplicities $r$5 in multiset edges contribute factorial weights, such as $r$6 in the numerator.
• Edge encoding: The specific "null" vertex associated to edge size $r$7 ensures that edges of different cardinalities are uniquely identified in the tensor structure (Ouvrard et al., 2018, Ouvrard et al., 2017, Ouvrard et al., 2018).

4. Concrete Examples

a) hb-graph Example (Ouvrard et al., 2018)

Given $r$8 ($r$9) and edges: $H = (V, E)$0 ($H = (V, E)$1), $H = (V, E)$2 ($H = (V, E)$3), $H = (V, E)$4 ($H = (V, E)$5). The tensor $H = (V, E)$6 (of dimension 6) includes, for instance:

• For $H = (V, E)$7: $H = (V, E)$8, $H = (V, E)$9. Each of the 12 tuples matching two $V = \{v_1, \dots, v_n\}$0, one $V = \{v_1, \dots, v_n\}$1, and one $V = \{v_1, \dots, v_n\}$2 is assigned $V = \{v_1, \dots, v_n\}$3-value $V = \{v_1, \dots, v_n\}$4.
• For $V = \{v_1, \dots, v_n\}$5 ($V = \{v_1, \dots, v_n\}$6): pattern $V = \{v_1, \dots, v_n\}$7, with each tuple receiving $V = \{v_1, \dots, v_n\}$8-value $V = \{v_1, \dots, v_n\}$9.

b) Uniform Hypergraph Example (Pearson et al., 2012)

For $e \in E$0 and $e \in E$1: Each permutation of $e \in E$2 indexes yields $e \in E$3; all other entries vanish.

c) General Hypergraph via Polynomial Homogeneization

The construction via homogeneous polynomials or the hypergraph uniformisation process leads to a tensor of dimension $e \in E$4 (for order 4), assigning $e \in E$5 to each valid pattern obtained by augmenting an edge of size $e \in E$6 with $e \in E$7 auxiliary indices; all other entries are zero (Ouvrard et al., 2017, Ouvrard et al., 2018).

5. Spectral Theory

The spectral theory of the 4th-order adjacency tensor underpins higher-order generalizations of eigenvalues and centralities:

• H-eigenvalues:

$e \in E$8

The set of real H-eigenvalues is finite; if the hypergraph is connected, the spectral radius $e \in E$9 is attained at a unique positive eigenvector (up to scaling) (Pearson et al., 2012).

• Z-eigenvalues:

$m_e : V \rightarrow \mathbb{N}$0

The largest Z-eigenvalue $m_e : V \rightarrow \mathbb{N}$1 has an associated nonnegative unit-norm eigenvector; strict positivity is guaranteed under additional "nice" connectivity (Pearson et al., 2012).

• E-eigenvalues:

$m_e : V \rightarrow \mathbb{N}$2

For 4-partite 4-graphs, the E-spectrum is symmetric about zero.

A Gershgorin-type bound applies for $m_e : V \rightarrow \mathbb{N}$3 symmetric and degree-normalized:

$m_e : V \rightarrow \mathbb{N}$4

where $m_e : V \rightarrow \mathbb{N}$5 is the maximum vertex m-degree, and $m_e : V \rightarrow \mathbb{N}$6 is the maximum m-degree over null indices (Ouvrard et al., 2018). In the 4-uniform case, this reduces to $m_e : V \rightarrow \mathbb{N}$7 (Ouvrard et al., 2017).

6. Computational and Algorithmic Aspects

Direct formation of the $m_e : V \rightarrow \mathbb{N}$8-size adjacency tensor is typically avoided in practice for large $m_e : V \rightarrow \mathbb{N}$9. Recent developments, e.g., tensor-times-same-vector (TTSV) methods, achieve efficient computation ($k$00 time), central for centrality and clustering algorithms:

• For vector $k$01, $k$02 is computed without explicit tensor storage (Aksoy et al., 2023).
• The tensor is formally $k$03-dimensional but only nonzero on patterns prescribed by the edge structure and the normalization scheme.

TTSV algorithms enable the use of 4th-order adjacency tensors in scalable hypergraph data analysis, extracting higher-order structure inaccessible to matrix-based approaches (Aksoy et al., 2023).

7. Connections and Generalizations

The 4th-order adjacency tensor is encapsulated by multiple frameworks:

• Hypergraph uniformization and polynomial homogeneization (Ouvrard et al., 2017, Ouvrard et al., 2018): Orders non-uniform edges by introducing auxiliary vertices, producing a homogeneous symmetric tensor encoding all hyperedge layers in a single structure.
• hb-graphs and multisets (Ouvrard et al., 2018): Accommodates multiedges with multiplicity, requiring careful normalization and null-vertex bookkeeping.
• Stirling number–based weighting (Aksoy et al., 2023): Equidistributes edge-weight over all index patterns representing full coverage of edge vertices, crucial for algorithms exploiting multilinear spectral theory.

All approaches converge on the core requirement: a fully symmetric, degree-normalized, combinatorially faithful encoding of 4-adic adjacency, serving as a canonical representation for higher-order spectral, algebraic, and algorithmic analyses in hypergraph-based models.