  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 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 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.
