Dimensionality reduction and spectral properties of multilayer networks

Area of Science:

• Network Science
• Graph Theory
• Complex Systems Analysis

Background:

• Real-world systems often comprise multiple interconnected layers, a departure from traditional single-edge network models.
• Characterizing the structure and dynamics of these multilayer networks is a significant challenge in network science.
• Spectral properties of networks offer a powerful lens for understanding their behavior.

Purpose of the Study:

• To investigate the spectral properties of undirected multilayer networks.
• To explore the relationship between multilayer network spectra and their simplified representations (quotients).
• To understand the dynamical implications of using these simplified network models.

Main Methods:

• Application of graph quotient framework to multilayer networks.
• Analysis of eigenvalue interlacing for adjacency and Laplacian matrices.
• Comparison of spectra between multilayer networks and their layer and aggregate network quotients.

Main Results:

• Established relationships between the eigenvalue spectra of multilayer networks and their natural graph quotients.
• Demonstrated how spectral properties of simplified network representations reflect those of the full multilayer network.
• Quantified the dynamical implications of using the network of layers or the aggregate network for analysis.

Conclusions:

• Graph quotients provide valuable insights into the spectral and dynamical properties of multilayer networks.
• Simplified network representations can be effectively used to study complex system dynamics.
• This work aids in understanding dynamical processes governed by spectral network properties.