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
\~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=1.00+-0.05. We also find that the
magnitude distribution of triggered earthquakes is independent of the
triggering earthquake magnitude m. When alpha=b, small earthquakes are roughly
as important to earthquake triggering as larger ones. We evaluate the fractal
correlation dimension of hypocenters D=2 using two relocated catalogs for
Southern California, and removing the effect of short-term clustering. Thus
D=2alpha as predicted by assuming that earthquake triggering is due to static
stress. The value D=2 implies that small earthquakes are as important as larger
ones for stress transfers between earthquakes.Comment: 14 pages, 7 eps figures, latex. In press in J. Geophys. Re