Competition between surface reconstruction and magnetic ordering plays an important role to determine geometries and electronic properties of semiconductor surfaces. Thus, it is expected that magnetic ordering is enhanced under certain conditions in which the surface reconstruction is suppressed. Here, we find that the magnetic ordering of the dangling-bond (DB) networks on the hydrogenated Si(111) surfaces is realized by controlling the DB network topology (or the deposition of H atoms) which suppresses flexibility of the surface reconstruction. The density functional calculations reveal that a triangular DB unit exhibits a high spin state and that two different types of hexagonal networks consisting of the DB unit exhibit ferrimagnetic ordering. The magnetic ordering on DB nanostructures on Si surfaces may be promising elements for the nanoscale Si based spintronic devices in future.