The different localisation properties of the eigenmodes of the Laplacian and adjacency matrix of 2D random geometric graphs

Abstract

We compare the spectrum and the localisation properties of the eigenmodes of the Laplacian and the adjacency matrix of 2D random geometric graphs, using numerical diagonalization of these matrices for different system sizes and connectivities. For sufficiently large ensembles of systems, we evaluate the spectrum, the probability distribution of the participation ratio and the relation between participation ratios and eigenvalues. While all eigenmodes of the adjacency matrix are localised for sufficiently large system sizes, the Laplacian matrix always leads to a small proportion of system-spanning modes due to a conservation law, and therefore to power-law tails in the probability distribution of the participation ratio and its relation to the eigenvalues. By disentangling the effects of finite system size, of mean degree, of component size distribution, and of network motifs, we provide a thorough understanding of the data.

Figures (6)

• Figure 1: Left: Component size distribution of systems of size $N=10000$ and different $z$ (1000 realisations). The dashed line indicates a power law with the Fisher exponent $187/91$. The deviation of the points from 1 at the system size although there is only a single component for $z=16,32$ is due to the normalization. Right: The average of all maximum components of an ensemble of 1000 for $N=625$ and 100 for $N=2500,10000$ dependent on $z$. With increasing system size the curve becomes steeper in proximity of the critical connectivity radius $4.51218\leq z_c \leq 4.51228$balisterContinuumPercolationSteps2005 (grey dashed line).
• Figure 2: In the top row, the first two eigenvectors associated with the eigenvalues at the band edges of the AM (($N=4000,~z=32$) are shown. Red indicates a high, blue a low, and white an amplitude of zero. For negative (positive) eigenvalues, neighbouring entries of the eigenvector tend to have the same (the opposite) sign. In the bottom row, eigenvectors close to $E=1$ and at $E=1$ are shown. At $E=1$, the eigenvector is fully localised on the orbits, while for $E$ close to 1 the participation ratio is large.
• Figure 3: Eigenvectors of the four smallest eigenvalues greater than zero for the LM. Top row: $N=4000,~z=32$; bottom row: $N=4000,~z=6$. Red indicates positive, blue negative, and white zero amplitude. In both cases the sinusoidal form of the eigenmodes is apparent but for $z=6$ weakly-connected substructures are visible.
• Figure 4: (a) Density of states (DOS) of the (negative) adjacency matrix of a regular lattice with 4, 8, and 24 neighbours. The DOS of the regular 2D lattice ($z=4$) has been calculated analytically in economou2006, showing that there is a logarithmic divergence in the centre of the band and the band itself extends over the range of $8$. (b) DOS of the AM on an RGG (c) Discrete spectrum of the LM. (d) Continuous spectrum of the LM. The last three graphs were evaluated for an ensemble of 1000 systems and $N=10^4$.
• Figure 5: Distribution of the participation ratio for the continuous part of the spectrum. On the left-hand side $z$ is varied for constant $N=10^4$. On the right-hand side $z=4$ is plotted in blue, $z=6$ in green, and $z=8$ in red for different system sizes. (a)-(d) show the AM with participation ratios belonging to positive eigenvalues in the top row and those arising from eigenvectors with negative eigenvalues in the second row. (e) and (f) show the LM, with the asymptotically expected power law of exponent $-1.5$ indicated by a dashed line in (f). The ensemble size is $10^3$ apart from $N=4\times 10^4$ where it is $10$.