Network of recurrent events for the Olami-Feder-Christensen model

We numerically study the dynamics of a discrete spring-block model introduced by Olami, Feder, and Christensen (OFC) to mimic earthquakes and investigate to what extent this simple model is able to reproduce the observed spatiotemporal clustering of seismicity. Following a recently proposed method to characterize such clustering by networks of recurrent events [J. Davidsen, P. Grassberger, and M. Paczuski, Geophys. Res. Lett. 33, L11304 (2006)], we find that for synthetic catalogs generated by the OFC model these networks have many nontrivial statistical properties. This includes characteristic degree distributions, very similar to what has been observed for real seismicity. There are, however, also significant differences between the OFC model and earthquake catalogs, indicating that this simple model is insufficient to account for certain aspects of the spatiotemporal clustering of seismicity.

Figure 1

(Color online) The average area $⟨a_r⟩$ of the smallest rectangle on the lattice containing a given avalanche of spatial extent $a$ for different values of the dissipation parameter $\alpha$. Only avalanches contained within the bulk, $L_{\text{bulk}}=800$, are considered (see text for details). The solid line corresponds to a linear increase. Inset: rescaled version indicating that the area scales linearly with $a$ independent of $\alpha$ for $300\le a\le 30000$ (area enclosed by straight lines).

Figure 2

(Color online) Mean degree $⟨k⟩$ vs catalog size $N$ for $s_{\text{th}}=300$ and different values of $\alpha$, for the regular and shuffled OFC catalogs. $N={10}^6$ corresponds to the full catalogs—see Table . The solid line is of the form $0.95\text{ln}\left(N\right)+\text{const}$.

Figure 3

(Color online) Schematic representation of two cascades of recurrences of size 4. In both cases, $d_4<d_3<d_2<d_1$. The cascade on the left is “open”—i.e., it is possible for a fifth recurrence to occur—since $d_4>0$. This is not the case for the cascade on the right. Since $d_4=0$, this cascade is “closed” and no distance will be able to break the record.

Figure 4

(Color online) In-degree and out-degree distributions for networks generated by the OFC model for $s_{\text{th}}=300$ and different $\alpha$’s (see Table for details). The solid (black) line is a Poisson distribution with the same $⟨k⟩$ as for $\alpha =0.20$.

Figure 5

(Color online) In-degree and out-degree distributions for networks generated from shuffled OFC catalogs with $s_{\text{th}}=300$ and different $\alpha$’s. The solid (black) line is a Poisson distribution with the same $⟨k⟩$ as for $\alpha =0.20$.

Figure 6

(Color online) Degree-degree correlations for $\alpha =0.2$ and $s_{\text{th}}=300$. The curves on the bottom (blue) are for the original catalog and on top (green) for the shuffled catalog. Note that the offset between the curves for the original OFC catalog and the shuffled version is simply due to different mean degrees.

Figure 7

(Color online) Single-vertex degree-degree correlations for $\alpha =0.2$ and $s_{\text{th}}=300$. The left panel shows the results for the original catalogs, and the right panel shows the results for the open cascade of recurrences only. The bottom curves (green) are for the unmodified catalogs, and the top ones (blue) are for the shuffled catalog.

Figure 8

(Color online) Distribution of the waiting times between events and their recurrences for different threshold sizes $s_{\text{th}}$, for $\alpha =0.2$. The solid line on the left is a power law with exponent $-0.9$ and on the right one with exponent $-1$. Inset: rescaled distributions with $N$ taken from Table .

Figure 9

(Color online) Distribution of the waiting times between events and their recurrences for $s_{\text{th}}=300$ and varying $\alpha$. Inset: rescaled distribution, zoomed in on around the characteristic time region.

Figure 10

(Color online) Distribution of waiting times between events and their recurrences for shuffled OFC catalogs with different threshold sizes $s_{\text{th}}$ and $\alpha =0.2$. The solid line on the left is a power law with exponent $-0.9$ and on the right one with exponent $-1$. Inset: rescaled distribution with $N$ taken from Table .

Figure 11

(Color online) Distribution of the waiting time ratios for subsequent recurrences, for $\alpha =0.2$ and $s_{\text{th}}=300$, for (a) the regular catalog, (b) only the “open” recurrences, (c) the shuffled catalog, and (d) only the “open” recurrences for the shuffled catalog. In all plots the solid line is a power law with inclination $-0.85$.

Figure 12

(Color online) Distribution of recurrence distances for $\alpha =0.2$ and different threshold values $s_{\text{th}}$. Inset: distributions rescaled according to Eq. (4) with $\delta =0.3$ and $\beta =1.5$. Recurrences of distance $l=0$ are not shown and were not used in the normalization.

Figure 13

(Color online) Distribution of recurrence distances for $\alpha =0.2$ and different threshold values $s_{\text{th}}$, but only considering “open” cascades of recurrences. Inset: distributions rescaled according to Eq. (4) with $\delta =0.3$ and $\beta =1.35$.

Figure 14

(Color online) Distribution of recurrence distances for the shuffled OFC catalog with $\alpha =0.2$ and different threshold values $s_{\text{th}}$. Inset: distributions rescaled according to the number of events (see Table ) as indicated in the legend.

Figure 15

(Color online) Distribution of recurrence distances for $\alpha =0.2$ and $s_{\text{th}}=300$, for an increasingly larger potion of the catalog $\Delta t$, rescaled as indicated in the legend, for (a) the regular catalog ($\gamma =0$—i.e., no scaling), (b) only the “open” recurrences $\left(\gamma =0.37\right)$, and (c) the shuffled catalog $\left(\gamma =0.18\right)$.

Figure 16

(Color online) Distribution of ratios between subsequent recurrence distances for $\alpha =0.2$ and $s_{\text{th}}=300$, for (a) the regular catalog, (b) only the “open” recurrences, (c) the shuffled catalog, and (d) only the “open” recurrences for the shuffled catalog. The solid line is a power law with exponent 1.