Localized and delocalized modes on random geometric graphs in one dimension.

Area of Science:

• Graph theory
• Network science
• Mathematical physics

Background:

• Eigenmode localization is crucial for understanding wave phenomena in disordered systems.
• Random geometric graphs offer a tractable model for complex network structures.
• Spectral properties of matrices (Laplace, adjacency) reveal network topology and dynamics.

Purpose of the Study:

• To extensively investigate the localization properties of eigenmodes in 1D random geometric graphs.
• To evaluate the density of states and participation ratio distribution.
• To understand the influence of system size, component size, mean degree, network motifs, and degeneracy on localization.

Main Methods:

• Analysis of Laplace and adjacency matrices for 1D random geometric graphs.
• Evaluation of the density of states and participation ratio.
• Systematic disentanglement of various graph parameters (size, degree, motifs, degeneracy).
• Comparison with ordered graphs and tight-binding models.

Main Results:

• Detailed characterization of eigenmode localization in 1D random geometric graphs.
• Quantification of the participation ratio and its relationship with eigenvalues.
• Identification of key factors (system size, mean degree, etc.) affecting localization behavior.
• Comparative analysis highlighting differences and similarities with ordered networks and tight-binding models.

Conclusions:

• A comprehensive understanding of eigenmode localization in 1D random geometric graphs is achieved.
• The study provides insights into how network structure influences spectral properties.
• Findings contribute to the broader understanding of disordered systems and network dynamics.