Dimensionality reduction and spectral properties of multilayer networks

Network representations are useful for describing the structure of a large variety of complex systems. Although most studies of real-world networks suppose that nodes are connected by only a single type of edge, most natural and engineered systems include multiple subsystems and layers of connectivity. This new paradigm has attracted a great deal of attention and one fundamental challenge is to characterize multilayer networks both structurally and dynamically. One way to address this question is to study the spectral properties of such networks. Here, we apply the framework of graph quotients, which occurs naturally in this context, and the associated eigenvalue interlacing results, to the adjacency and Laplacian matrices of undirected multilayer networks. Specifically, we describe relationships between the eigenvalue spectra of multilayer networks and their two most natural quotients, the network of layers and the aggregate network, and show the dynamical implications of working with either of the two simplified representations. Our work thus contributes in particular to the study of dynamical processes whose critical properties are determined by the spectral properties of the underlying network.

💡 Research Summary

The paper addresses the challenge of characterizing multilayer networks—systems in which nodes are connected through several distinct types of edges or layers—by focusing on their spectral properties. The authors employ the concept of graph quotients, which arise naturally when one partitions a graph’s vertex set, and apply eigenvalue interlacing theorems to both adjacency and Laplacian matrices of undirected multilayer networks.

Two natural quotients are defined: (i) the “network of layers,” obtained by collapsing each layer into a single meta‑node and weighting edges by the average inter‑layer connectivity, and (ii) the “aggregate network,” formed by projecting all layers onto a single‑layer graph. For each quotient the authors construct appropriate adjacency matrices and, for the Laplacian, a specially defined quotient Laplacian that discards self‑loops while preserving average degree information.

The central theoretical results are:

1. Interlacing of adjacency spectra – If the multilayer network has n vertices and the quotient has m vertices (m < n), then the m eigenvalues of the quotient interlace those of the original network: λ_i ≤ μ_i ≤ λ_{i+n−m} for i = 1,…,m, where λ’s are the original eigenvalues and μ’s those of the quotient.

2. Interlacing of Laplacian spectra – Using the defined quotient Laplacian, the same interlacing inequality holds for Laplacian eigenvalues.

3. Equitable and almost‑equitable partitions – When the partition of nodes into layers is equitable (each node in a layer has identical numbers of connections to any other layer, including within its own layer), the quotient’s eigenvalues are not only interlaced but actually form a subset of the original spectrum. Moreover, eigenvectors can be “lifted”: each eigenvector of the quotient can be expanded to an eigenvector of the full network by repeating its components across all nodes in the corresponding layer. For almost‑equitable partitions (regular inter‑layer connections but possibly irregular intra‑layer connections) a similar lifting holds for the Laplacian spectrum.

4. Subnetworks – The authors also discuss induced subgraphs and general subgraphs. For induced subgraphs, adjacency eigenvalues interlace the original ones; for Laplacians, a one‑sided interlacing inequality holds (μ_i ≤ λ_{i+n−m}).

These spectral relationships have direct implications for dynamical processes on multilayer networks. Many dynamical thresholds—such as epidemic spreading thresholds, synchronization onset, or percolation critical points—are determined by specific eigenvalues (e.g., the second smallest Laplacian eigenvalue for synchronization, the largest adjacency eigenvalue for epidemic spreading). Because the quotients preserve these eigenvalues within known bounds, one can safely use the reduced representations to estimate critical parameters while dramatically reducing computational cost. In the case of equitable or almost‑equitable layer partitions, the reduced models give exact critical values.

The paper also quantifies the information loss incurred by dimensionality reduction: the interlacing inequalities provide explicit upper and lower bounds on how much the spectrum can shift when moving from the full multilayer network to a quotient. This allows practitioners to assess the trade‑off between model simplicity and accuracy.

Overall, the work situates multilayer network analysis within a rigorous algebraic framework, extending classical graph‑theoretic tools to modern complex systems. By linking structural reduction (quotients) with spectral interlacing and lifting, it offers a principled method for simplifying multilayer networks without sacrificing essential dynamical information, and it clarifies when such simplifications are exact versus approximate. Future research directions include exploring how to handle highly heterogeneous, non‑equitable layer structures and developing refined bounds for those cases.