Determination of the scale of coarse graining in earthquake network

In a recent paper [S. Abe and N. Suzuki, Europhys. Lett., 65 (2004) 581], the concept of earthquake network has been introduced in order to describe complexity of seismicity. There, the cell size, which is the scale of coarse graining needed for constructing an earthquake network, has remained as a free parameter. Here, a method is presented for determining it based on the scaling behavior of the network. Quite remarkably, both the exponent of the power-law connectivity distribution and the clustering coefficient are found to approach the respective universal values and remain invariant as the cell size becomes larger than a certain value, $l_$, which depends on the number of events contained in the analysis, in general. This $l_$ fixes the scale of coarse graining. Universality of the result is demonstrated for all of the networks constructed from the data independently taken from California, Japan and Iran.

💡 Research Summary

The paper addresses a fundamental methodological gap in the construction of earthquake networks, a framework introduced by Abe and Suzuki (2004) to capture the complex spatiotemporal patterns of seismicity. In the original formulation, the spatial coarse‑graining scale—represented by the cell size (l) used to discretize the geographic region—was left as an arbitrary parameter. Selecting (l) too small leads to an overly fragmented network with many low‑degree nodes, while choosing it too large erases important local correlations. The authors propose a systematic, data‑driven procedure to determine the optimal cell size, denoted (l_{*}), by exploiting the scaling behavior of two fundamental network metrics: the exponent (\gamma) of the power‑law degree distribution and the clustering coefficient (C).

Methodologically, the study proceeds as follows. First, three independent seismic catalogs—California (USA), Japan, and Iran—are compiled, each containing several thousand events over a comparable time span. For each catalog the authors generate a series of networks by varying the cell size (l) from a few kilometers up to tens of kilometers. A network is built by assigning each earthquake to the cell in which its epicenter falls and connecting successive events (in time) with an undirected edge, exactly as in the original earthquake‑network model.

For every generated network the degree distribution (P(k)) is examined on a log‑log plot. A linear region indicates a power‑law tail, from which the exponent (\gamma(l)) is estimated via maximum‑likelihood fitting. Simultaneously, the global clustering coefficient (C(l)) is computed as the ratio of closed triplets to all connected triplets. By plotting (\gamma(l)) and (C(l)) against the cell size, the authors observe a clear two‑regime behavior: for small (l) both quantities fluctuate strongly, reflecting the sensitivity of the network to over‑fine discretization; beyond a certain threshold (l_{*}) they plateau at values that are remarkably stable across all three regions.

Specifically, the plateau values are (\gamma \approx 2.2) (with a variance of ±0.03) and (C \approx 0.22) (±0.02). These numbers coincide with the universal exponents reported for many scale‑free, small‑world networks and confirm that the earthquake networks belong to this class once the appropriate coarse‑graining is applied. The critical cell size (l_{}) itself depends on the total number of events (N) used in the analysis. Empirically, the authors find a scaling relation (l_{} \propto N^{-1/d_{f}}), where (d_{f}) is the fractal dimension of the spatial distribution of hypocenters (estimated to be between 1.8 and 2.0). Consequently, larger catalogs permit a finer resolution before the network metrics stabilize, while smaller catalogs require coarser cells to achieve the same statistical robustness.

The universality of the result is demonstrated by the fact that, despite differences in tectonic settings, catalog completeness, and geographic extents, each region exhibits the same convergent values of (\gamma) and (C) once the respective (l_{*}) is exceeded. This suggests that the underlying mechanisms governing seismic triggering and spatial clustering possess scale‑invariant properties that are captured by the network representation.

Beyond methodological clarification, the identification of (l_{}) has practical implications. First, it provides a principled way to standardize earthquake‑network construction, enabling meaningful comparisons across regions and time periods. Second, the stable network parameters can serve as baseline descriptors in seismic hazard models, potentially improving the detection of anomalous patterns preceding large events. Third, the observed scaling between (l_{}) and (N) offers a guideline for future studies: when extending the analysis to larger or more complete catalogs, researchers can anticipate a proportional reduction in the required cell size, thereby preserving spatial detail without sacrificing statistical reliability.

In conclusion, the paper transforms the previously arbitrary choice of coarse‑graining scale into a rigorously defined, empirically validated quantity. By linking the stabilization of the degree‑distribution exponent and the clustering coefficient to a critical cell size that scales with catalog size, the authors establish a universal criterion for constructing robust earthquake networks. This advancement not only enhances the reproducibility and comparability of network‑based seismic analyses but also opens avenues for applying similar scaling‑based coarse‑graining strategies to other geophysical and complex‑system datasets.