  We investigate the conditions required for general spin systems with
frustration and disorder to display self-organized criticality, a property
which so far has been established only for the fully-connected infinite-range
Sherrington-Kirkpatrick Ising spin-glass model [Phys. Rev. Lett. 83, 1034
(1999)]. Here we study both avalanche and magnetization jump distributions
triggered by an external magnetic field, as well as internal field
distributions in the short-range Edwards-Anderson Ising spin glass for various
space dimensions between 2 and 8, as well as the fixed-connectivity mean-field
Viana-Bray model. Our numerical results, obtained on systems of unprecedented
size, demonstrate that self-organized criticality is recovered only in the
strict limit of a diverging number of neighbors, and is not a generic property
of spin-glass models in finite space dimensions.
