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 $k = r_m(H) = \max_{e \in E} \#_m e$, the maximum multiset cardinality of any edge.

• For each $e$ of size $s = \#_m e \leq k$, introduce a "null" vertex $N_s$ of multiplicity $k-s$.
• The extended multiset $\hat{e}$ consists of the elements of $e$, plus $k-s$ copies of $N_s$.
• The tensor $A = (A_{i_1...i_k})$ has dimension $n + (k-1)$, symmetrized in all indices:

$$A_{i_1\cdots i_k} = \sum_{e\in E} c_e\,\Delta^{(e)}_{i_1\cdots i_k}$$

where $c_e = k/s$, and $\Delta^{(e)}$ is nonzero only when the indices match the extended multiset $\hat{e}$:

$$\Delta^{(e)}_{i_1...i_k} = \begin{cases} \dfrac{\prod_{v \in e^*} m_e(v)!\,(k-s)!}{(k-1)!}, & \text{if the multiset of } (i_1,...,i_k) \text{ matches } \hat{e} \ 0, & \text{otherwise.} \end{cases}$$

b) Specialization to $k=4$ (Order 4)

Set $k=4$; introduce null vertices $N_1$, $N_2$, $N_3$. For $e$ of size $s$:

• $c_e = 4/s$
• $m_e(N_s) = 4-s$
• Nonzero values:

$$A_{i_1 i_2 i_3 i_4} = \sum_{e \in E}\frac{4}{s}\frac{\prod_{v \in e^*} m_e(v)!\;(4-s)!}{6}$$

if the index multiset is given by the multiplicities $m_e(v_j)$ for $j=1..n$ and $(4-s)$ entries equal to $n+s$ (location of $N_s$) (Ouvrard et al., 2018).

An entry is nonzero if and only if, for some $e$ of size $s$, $m_e(v_j)$ of the indices equal $j$ for $1\leq j\leq n$ and $4-s$ of the indices equal $n+s$.

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 $A \in \mathbb{R}^{n \times n \times n \times n}$:

$$A_{i_1 i_2 i_3 i_4} = \begin{cases} 1/6 &\text{if } \{i_1,i_2,i_3,i_4\}\in E \ 0 &\text{otherwise} \end{cases}$$

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

3. Key Properties

• Permutation symmetry: $A_{i_1 i_2 i_3 i_4}$ is invariant under any permutation of indices.
• Degree normalization: The m-degree of vertex $i$, denoted $\deg_m(v_i)$ (i.e., the sum of its multiplicities), is recovered by summing $A_{i\, j_2\, j_3\, j_4}$ over the last three indices:

$$\sum_{j_2,j_3,j_4} A_{i\, j_2\, j_3\, j_4} = \deg_m(v_i)$$

• Handling duplicates: Multiplicities $m_e(v)$ in multiset edges contribute factorial weights, such as $m_e(v)!$ in the numerator.
• Edge encoding: The specific "null" vertex associated to edge size $s$ 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 $V=\{a,b,c\}$ ($n=3$) and edges: $e_1=\{a^2, b^1\}$ ($s=3$), $e_2=\{b^1, c^2\}$ ($s=3$), $e_3=\{a^1, b^2, c^1\}$ ($s=4$). The tensor $A$ (of dimension 6) includes, for instance:

• For $e_1$: $c_{e_1}=4/3$, $m_{e_1}(N_3)=1$. Each of the 12 tuples matching two $a$, one $b$, and one $N_3$ is assigned $A$-value $4/9$.
• For $e_3$ ($s=4$): pattern $\{a, b, b, c\}$, with each tuple receiving $A$-value $1/3$.

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

For $V=\{1,2,3,4\}$ and $E=\{\{1,2,3,4\}\}$: Each permutation of $(1,2,3,4)$ indexes yields $A_{i_1 i_2 i_3 i_4}=1/6$; 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 $n+3$ (for order 4), assigning $1/6$ to each valid pattern obtained by augmenting an edge of size $j$ with $4-j$ 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:

$$(A x^3)_i = \lambda x_i^3, \quad x\in\mathbb{R}^n\setminus\{0\}$$

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

• Z-eigenvalues:

$A x^3 = \mu x, \quad \|x\|_2=1$

The largest Z-eigenvalue $\rho_Z(A)>0$ has an associated nonnegative unit-norm eigenvector; strict positivity is guaranteed under additional "nice" connectivity (Pearson et al., 2012).

• E-eigenvalues:

$A x^3 = \nu x, \quad x_1^2+...+x_n^2 = 1$

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

A Gershgorin-type bound applies for $A$ symmetric and degree-normalized:

$|\lambda| \leq \max(\Delta, \Delta^*) + 4$

where $\Delta$ is the maximum vertex m-degree, and $\Delta^*$ is the maximum m-degree over null indices (Ouvrard et al., 2018). In the 4-uniform case, this reduces to $|\lambda| \leq \Delta$ (Ouvrard et al., 2017).

6. Computational and Algorithmic Aspects

Direct formation of the $n^4$-size adjacency tensor is typically avoided in practice for large $n$. Recent developments, e.g., tensor-times-same-vector (TTSV) methods, achieve efficient computation ($O(\sum_{e \in E}|e|)$ time), central for centrality and clustering algorithms:

• For vector $x\in\mathbb{R}^n$, $y = \mathcal{A} \times_2 x \times_3 x \times_4 x$ is computed without explicit tensor storage (Aksoy et al., 2023).
• The tensor is formally $n^4$-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.