Numerical models in which grid-cells self-organize, so that the shear stress in each cell equals a predetermined threshold value, are remarkably successful in producing the fundamental structure of river networks. Such models are a reasonable approximation of networks characterized by near-constant dimensionless shear stress (&tgr;* &agr; bed shear stress/bed grain size), with each cell's threshold value being proportional to median-bed grain size. Previous work has shown that this is the case for channels cut into alluvium, and that the characteristic narrow range of &tgr;* is different for gravel- and sand-bed channels. Channels in the Italian Apennines that arc slowly incising into weak bedrock, but which are covered by a veneer of gravel for most of the time, are shown to be characterized by the same &tgr;* as alluvial gravel-bed channels. Thus threshold models capture the fundamental behavior of many river networks, even in areas of long-term bedrock incision. ¿ 2000 American Geophysical Union