Eigenvector localization in hypergraphs: Pairwise versus higher-order links

Localization behaviors of Laplacian eigenvectors of complex networks furnish an explanation to various dynamical phenomena of the corresponding complex systems. We numerically examine roles of higher-order and pairwise links in driving eigenvector localization of hypergraphs Laplacians. We find that pairwise interactions can engender localization of eigenvectors corresponding to small eigenvalues for some cases, whereas higher-order interactions, even being much much less than the pairwise links, keep steering localization of the eigenvectors corresponding to larger eigenvalues for all the cases considered here. These results will be advantageous to comprehend dynamical phenomena, such as diffusion, and random walks on a range of real-world complex systems having higher-order interactions in better manner.

Figure 1

Schematic of the hypergraph model used here with $N=10, M^h=1$, and $M^p=1$ for two different realizations. (a) $One$ pairwise link $E^p=\{1,8\}$ and $one$ hyperedge $E^h=\{1,3,7\}$ are added into the ring lattice. (b) $One$ pairwise link $E^p=\{3,7\}$ and $one$ hyperedge $E^h=\{3,7,10\}$ are added into the ring lattice. The pairwise links are solid lines (green) and the hyperedge with the dashed line (sky-blue) enclosing the involved nodes.

Figure 2

Laplacian eigenvalues ${\lambda }_i$ (red) and node degree $k_i$ (blue) against index $i$ arranged in an increasing order for the $\gamma$ values. The size of the hypergraphs $N=2000$ and $M^h=500$ remain fixed for all $\gamma$ values with 40 random realizations.

Figure 3

The relative deviations of eigenvalues from the higher-order degrees, ${\delta }_i^h$ (blue) and pairwise degrees, ${\delta }_i^p$ (red) against index $i$. The hypergraph parameters, $N=2000$ and $M^h=500$, remain fixed for all $\gamma$ values with 40 random realizations.

Figure 4

Average IPR $\left[Y_{{\mathit{x}}_j}\left(\lambda \right)\right], k^h\left(\lambda \right)$ ($\circ$), $k^p\left(\lambda \right)$ ($\square$), ${\widehat{k}}^h\left(\lambda \right)$ ($---$), ${\widehat{k}}^p\left(\lambda \right)$ ($---$) against $\lambda$ for various $\gamma <1$'s. The corresponding higher-order and pairwise degree distribution are also plotted in last two rows. $---$ and $-·-·-$ depict, respectively, $\langle k^h\rangle$ and $\langle k^p\rangle$ on the $y$ axis. The size of the hypergraphs $N=2000$ and $M^h=500$ remain fixed for all $\gamma$ values with 40 random realizations for the bottom $three$ rows. Red ($-$), green ($-$), black ($-$), blue ($-$), magenta ($-$) are used for $N=500,1000,2000,4000$, and 8000, respectively, in the upper panels.

Figure 5

Average IPR $\left[Y_{{\mathit{x}}_j}\left(\lambda \right)\right], k^h\left(\lambda \right)$ ($\circ$), $k^p\left(\lambda \right)$ ($\square$), ${\widehat{k}}^h\left(\lambda \right)$ ($---$), and ${\widehat{k}}^p\left(\lambda \right)$ ($---$) against $\lambda$ for various $\gamma >1$'s. The corresponding higher-order and pairwise degree distributions are also plotted in last two rows. $---$ and $-·-·-$ are at $\langle k^h\rangle$ and $\langle k^p\rangle$ on the $y$ axis. The size of the hypergraph, $N=2000$ and $M^h=500$ remain fixed for all $\gamma$ values with 40 random realizations for bottom $three$ rows. Red ($-$), green ($-$), black ($-$), blue ($-$), and magenta ($-$) are used for $N=500,1000,2000,4000$, and 8000, respectively, in the upper panels.

Figure 6

Plot of $\alpha$ against $\lambda$ for various $\gamma$ values. (a) $\gamma =0.25$, (b) $\gamma =0.50$, (c) $\gamma =0.75$, (d) $\gamma =1.25$, (e) $\gamma =3$, and (f) $\gamma =5$.

Figure 7

$\langle \left|x_i\right|\rangle$ (black), $\langle k_i^h\rangle$ (red), and $\langle k_i^p\rangle$ (blue) against index $i$ for $\lambda$ belonging to the delocalized region. (a) and (b) $\lambda \approx 8.5$, (c) and (d) $\lambda \approx 14$, (e) and (f) $\lambda \approx 12.5$, (g) and (h) $\lambda \approx 23.5$, (i) and (j) $\lambda \approx 18$, and (k) and (l) $\lambda \approx 26$. The size of the hypergraph, $N=2000$, and $M^h=500$ remains fixed for all $\gamma$ values with 40 random realizations.

Figure 8

$\langle \left|x_i\right|\rangle$ (black), $\langle k_i^h\rangle$ (red), and $\langle k_i^p\rangle$ (blue) against index $i$ for $\lambda$ belonging to the localized region. (a) and (b) $\lambda \approx 2$, (c) and (d) $\lambda \approx 33$, (e) and (f) $\lambda \approx 3$, (g) and (h) $\lambda \approx 36$, (i) and (j) $\lambda \approx 5$, and (k) and (l) $\lambda \approx 46$. The size of the hypergraphs $N=2000$, and $M^h=500$ remain fixed for all $\gamma$ values with 40 random realizations.

Figure 9

Average IPR $\left[Y_{{\mathit{x}}_j}\left(\lambda \right)\right]$ ($\circ$), $k^h\left(\lambda \right)$ ($\circ$), and $k^p\left(\lambda \right)$ ($\square$) against $\lambda$ for various $\gamma <1$'s of the conventional Laplacian. The corresponding higher-order and pairwise degree distributions are also plotted in the last two rows. The size of the hypergraphs $N=2000$ and $M^h=500$ remain fixed for all $\gamma$ values with 40 random realizations.

Figure 10

Average IPR $\left[Y_{{\mathit{x}}_j}\left(\lambda \right)\right]$ ($\circ$), $k^h\left(\lambda \right)$ ($\circ$), and $k^p\left(\lambda \right)$ ($\square$), for various $\gamma >1$'s of the conventional Laplacian. The corresponding higher-order and pairwise degree distributions are also plotted in last two rows. The size of the hypergraphs $N=2000$ and $M^h=500$ remain fixed for all $\gamma$ values with 40 random realizations.