  We study zero-temperature Glauber dynamics for Ising-like spin variable
models in quenched random networks with random zero-magnetization initial
conditions. In particular, we focus on the absorbing states of finite systems.
While it has quite often been observed that Glauber dynamics lets the system be
stuck into an absorbing state distinct from its ground state in the
thermodynamic limit, very little is known about the likelihood of each
absorbing state. In order to explore the variety of absorbing states, we
investigate the probability distribution profile of the active link density
after saturation as the system size $N$ and $<k >$ vary. As a result, we find
that the distribution of absorbing states can be split into two self-averaging
peaks whose positions are determined by $<k>$, one slightly above the ground
state and the other farther away. Moreover, we suggest that the latter peak
accounts for a non-vanishing portion of samples when $N$ goes to infinity while
$<k >$ stays fixed. Finally, we discuss the possible implications of our
results on opinion dynamics models.
