  Opinion dynamics is crucial for unraveling the complexities of human
interaction in the information age. How to speed up consensus without
disturbing the fate of the system is key for opinion dynamics. We propose a
voter model on adaptive networks, which resembles the coevolutionary process
between opinions and social relationships. We prove the existence of a
one-dimensional stable manifold for the system, which facilitates us to study
both the fate of the system and the consensus time it takes. Surprisingly, we
find the adjustment of social relations speeds up consensus but does not affect
the fate of the system. For echo-chamber-like networks which consist of two
homogeneous subnetworks connected by few sparse links, a small probability of
adaptive edge dynamics is sufficient to accelerate consensus formation, which
is counterintuitive. If the network structure makes consensus much slower than
that of the regular networks, minor random rewiring makes a discontinuous drop
in consensus time. Our work opens up an avenue for speeding up consensus
without disturbing the fate of the system. It can be insightful for crowd
control.
