Supra-Adjacency Matrix in Temporal Networks

• Supra-adjacency matrix is a block matrix that consolidates intra-layer adjacency matrices and inter-layer couplings, offering a unified representation of temporal networks.
• It allows closed-form spectral decompositions using the discrete Fourier transform, facilitating explicit computation of eigenvalues and eigenvectors.
• The model generalizes core graph-theoretic concepts, such as the Fiedler vector, enabling rigorous temporal community detection and network partitioning.

A supra-adjacency matrix is the fundamental mathematical object encoding the connectivity structure of a multilayer or temporal network by consolidating individual layer adjacency matrices and their inter-layer couplings into a single block matrix framework. This construction provides a systematic means to analyze the spectral, algebraic, and dynamical properties of temporal networks, particularly when the network structure evolves across ordered layers interconnected by specified coupling weights. The supra-adjacency matrix underlies the definition of the supra-Laplacian, and, in the context of the constant block Jacobi model, enables closed-form spectral decompositions and generalizations of core graph-theoretic concepts to the multilayer setting (Kuncheva et al., 2023).

1. Formal Construction of the Supra-Adjacency Matrix

Consider a temporal multilayer network comprising $T$ simple, undirected, connected layers with a common node set $V$ ($|V| = N$), where each layer at time $t$ is represented as $G^{t} = (V, A^{(t)})$, and $A^{(t)}$ is the binary $N \times N$ adjacency matrix of layer $t$. Neighbouring layers $t$ and $t\pm1$ are interconnected via diagonal inter-layer weight matrices $V$0. In the constant block Jacobi model, all interlayer weights are equal: $V$1.

The supra-adjacency matrix $V$2 of size $V$3 is defined as a block matrix: $V$4 where the off-diagonal $V$5 blocks couple a node’s copies in adjacent time layers, and $V$6. The block structure is periodic, reflecting the temporal topology's cyclical, time-stripe interlayer connections (Kuncheva et al., 2023).

In the specific case where all intra-layer adjacency matrices are identical ($V$7), and interlayer couplings are uniform, the supra-adjacency matrix is written as: $V$8 where $V$9 is the $|V| = N$0 circulant matrix with ones on the two off-diagonals and corners, corresponding to temporal coupling.

2. Supra-Degree Matrix and Supra-Laplacian

The degree structure at the supra-network level is captured by the supra-degree matrix $|V| = N$1, a block-diagonal matrix with: $|V| = N$2 For the constant-$|V| = N$3 case, this simplifies to $|V| = N$4, where $|V| = N$5 is the intra-layer degree.

The unnormalized supra-Laplacian is defined as: $|V| = N$6 and the normalized supra-Laplacian as: $|V| = N$7 These matrices generalize canonical spectral objects to the multilayer temporal setting, enabling the transfer of many graph-analytic techniques to multilayer data (Kuncheva et al., 2023).

3. Closed-Form Spectral Solution under the Constant Block Jacobi Model

If intra-layer structure is stationary ($|V| = N$8), and inter-layer weights are uniform, the supra-Laplacian $|V| = N$9 is a block-circulant Jacobi matrix. This permits simultaneous diagonalization via the discrete Fourier transform (DFT) applied in the layer (temporal) dimension.

Expressing a supra-Laplacian eigenvector blockwise, $t$0, the DFT: $t$1 block-diagonalizes the eigenproblem into $t$2 independent $t$3 eigenproblems: $t$4 with: $t$5 where $t$6 (Kuncheva et al., 2023). This analytic structure yields closed-form spectral representations and enables explicit computation of eigenvalues and eigenvectors by inverse DFT.

4. Perturbation Theory and Spectral Modes Near Zero

When $t$7 is small, the supra-Laplacian $t$8 is a perturbation of the block-diagonal matrix $t$9, whose zero-eigenspace is $G^{t} = (V, A^{(t)})$0-dimensional, spanned by layerwise constant vectors. Standard perturbation theory demonstrates that the $G^{t} = (V, A^{(t)})$1 eigenvalues of $G^{t} = (V, A^{(t)})$2 near zero originate from perturbing this invariant subspace. To first order,

$G^{t} = (V, A^{(t)})$3

where $G^{t} = (V, A^{(t)})$4 are vectors with support in only one layer. This coupling matrix possesses a Jacobi form; its eigenvalues are $G^{t} = (V, A^{(t)})$5.

The corresponding eigenvectors are DFT "Fourier modes" in the layer dimension, modulated by the intra-layer kernel vector (typically the principal eigenvector of each layer’s Laplacian). This implies that temporal dynamics and layer-to-layer structural correlations are encoded directly via the perturbative spectral structure (Kuncheva et al., 2023).

5. Generalization of the Fiedler Vector and Spectral Partitioning

In a single graph, the Fiedler vector (eigenvector of the Laplacian associated with its smallest nonzero eigenvalue) underpins optimal partitioning strategies. Supra-Laplacian analysis produces a spectrum of small eigenmodes indexed by temporal frequency $G^{t} = (V, A^{(t)})$6: the lowest nonzero eigenvalue, typically at $G^{t} = (V, A^{(t)})$7 or $G^{t} = (V, A^{(t)})$8, has an eigenvector of the form: $G^{t} = (V, A^{(t)})$9 where $A^{(t)}$0 is the Fiedler vector of the intra-layer Laplacian $A^{(t)}$1.

This results in a "time-stripe" or "wave-like" generalization of the Fiedler vector, encoding both within-layer and temporal consistency in community structure. The lowest-frequency mode ($A^{(t)}$2) naturally balances intra-layer cut cost with inter-layer coupling, and thus yields multilayer spectral partitions "consistent both in space (within each layer) and in time (across layers)" (Kuncheva et al., 2023).

6. Analytical Summary Table

Concept	Definition/Implementation	Key Features
Supra-Adjacency Matrix	Block matrix with intra-layer $A^{(t)}$3; off-diagonals $A^{(t)}$4	Encodes all layers and their coupling
Supra-Degree Matrix	Diagonal, entries $A^{(t)}$5	Aggregates within- and inter-layer degree
Supra-Laplacian	$A^{(t)}$6	Spectral basis for multilayer analysis
Closed-form diagonalization	DFT in layer index; reduces to $A^{(t)}$7 eigenproblems	Enables explicit spectral solutions
Fiedler vector generalization	Lowest nonzero mode: $A^{(t)}$8	Wave-like multilayer partition vector

7. Implications and Applications

The supra-adjacency matrix formalism, especially under the constant block Jacobi model, streamlines the spectral analysis of temporal and multilayer networks by furnishing analytically tractable eigendecompositions and perturbative approaches. This enables rigorous investigations into temporal community detection, spectral clustering, and the evolution of dynamical processes on networks. The explicit wave-like structure of multilayer Fiedler vectors provides a principled method for extracting time-consistent partitions and generalizes classical graph-theoretic methods to temporal domains (Kuncheva et al., 2023).