  The $q$-voter model with independence is generalized to signed random graphs
and studied by means of Monte Carlo simulations and theoretically using the
mean field approximation and different forms of the pair approximation. In the
signed network with quenched disorder, positive and negative signs associated
randomly with the links correspond to reinforcing and antagonistic
interactions, promoting, respectively, the same or opposite orientations of
two-state spins representing agents' opinions; otherwise, the opinions are
called mismatched. With probability $1-p$, the agents change their opinions if
the opinions of all members of a randomly selected $q$-neighborhood are
mismatched, and with probability $p$, they choose an opinion randomly. The
model on networks with finite mean degree $\langle k \rangle$ and fixed
fraction of the antagonistic interactions $r$ exhibits ferromagnetic transition
with varying the independence parameter $p$, which can be first- or
second-order, depending on $q$ and $r$, and disappears for large $r$. Besides,
numerical evidence is provided for the occurrence of the spin-glass-like
transition for large $r$. The order and critical lines for the ferromagnetic
transition on the $p$ vs. $r$ phase diagram obtained in Monte Carlo simulations
are reproduced qualitatively by the mean field approximation. Within the range
of applicability of the pair approximation, for the model with $\langle k
\rangle$ finite but $\langle k \rangle \gg q$, predictions of the homogeneous
pair approximation concerning the ferromagnetic transition show much better
quantitative agreement with numerical results for small $r$ but fail for larger
$r$. A more advanced signed homogeneous pair approximation is formulated which
distinguishes between classes of active links with a given sign connecting
nodes occupied by agents with mismatched opinions...
