  In this work we present novel results to the problem of the Hegselmann-Krause
dynamics in networks obtained by an extensive study of the behavior of the
standard order parameter sensitive to the onset of consensus: the normalized
size of the giant cluster. This order parameter reveals the non trivial effect
of the network topology on the steady states of the dynamics, overlooked by
previous works, which concentrated on the onset of \textit{unanimity}, and
allows to detect regions of polarization between the fragmented and the
consensus phases. While the previous results on unanimity are confirmed, the
consensus threshold shifts in the opposite direction compared to the threshold
for unanimity. A detailed finite size scaling analysis shows that, in general,
consensus is easier to obtain in networks than in mixed populations. At a
difference with previous studies, we show that the network topology is relevant
beyond the finitness of the average degree with increasing system size. In
particular, in pure random networks (either uniform random graphs or scale free
networks), the consensus threshold seems to vanish in the thermodynamic limit.
A detailed analysis of the time evolution of the dynamics reveals the role of
\textit{bridges} in the network, which allow for the interaction between agents
belonging to clusters of very different opinions, after several repeated
interaction steps. These bridges are at the origin of the shift of the
confidence threshold to lower values in networks as compared to lattices or the
mixed population.
