  We study static annihilation on complex networks, in which pairs of connected
particles annihilate at a constant rate during time. Through a mean-field
formalism, we compute the temporal evolution of the distribution of surviving
sites with an arbitrary number of connections. This general formalism, which is
exact for disordered networks, is applied to Kronecker, Erd\"os-R\'enyi (i.e.
Poisson) and scale-free networks. We compare our theoretical results with
extensive numerical simulations obtaining excellent agreement. Although the
mean-field approach applies in an exact way neither to ordered lattices nor to
small-world networks, it qualitatively describes the annihilation dynamics in
such structures. Our results indicate that the higher the connectivity of a
given network element, the faster it annihilates. This fact has dramatic
consequences in scale-free networks, for which, once the ``hubs'' have been
annihilated, the network disintegrates and only isolated sites are left.
