We estimate the relative importance of small and large earthquakes for static stress changes and for earthquake triggering, assuming that earthquakes are triggered by static stress changes and that earthquakes are located on a fractal network of dimension D. This model predicts that both the number of events triggered by an earthquake of magnitude m and the stress change induced by this earthquake at the location of other earthquakes increase with m as similar to 10(Dm/2). The stronger the spatial clustering, the larger the influence of small earthquakes on stress changes at the location of a future event as well as earthquake triggering. If earthquake magnitudes follow the Gutenberg-Richter law with b > D/2, small earthquakes collectively dominate stress transfer and earthquake triggering because their greater frequency overcomes their smaller individual triggering potential. Using a southern California catalog, we observe that the rate of seismicity triggered by an earthquake of magnitude m increases with m as 10(alpha m), where alpha = 1.05 +/- 0.05. We also find that the magnitude distribution of triggered earthquakes is independent of the triggering earthquake's magnitude m. When alpha approximate to b, small earthquakes are roughly as important to earthquake triggering as larger ones. We evaluate the fractal correlation dimension D of hypocenters using two relocated catalogs for southern California. The value of D measured for distances 0.1 < r < 5 km is D = 1.54 for the Shearer et al. catalog and D = 1.73 for the Hauksson et al. catalog. The value of D reflects both the structure of the fault network and the nature of earthquake interactions. By considering only those earthquake pairs with interevent times larger than 1000 days, we can largely remove the effects of short-term clustering. Then D approximate to 2, close to the value D = 2 alpha = 2.1 predicted by assuming that earthquake triggering is due to static stress. The value D approximate to 2b implies that small earthquakes are as important as larger ones for stress transfers between earthquakes and that considering stress changes induced by small earthquakes should improve models of earthquake interactions.
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