  The Hegselmann-Krause system (HK system for short) is one of the most popular
models for the dynamics of opinion formation in multiagent systems. Agents are
modeled as points in opinion space, and at every time step, each agent moves to
the mass center of all the agents within unit distance. The rate of convergence
of HK systems has been the subject of several recent works. In this work, we
investigate two natural variations of the HK system and their effect on the
dynamics. In the first variation, we only allow pairs of agents who are friends
in an underlying social network to communicate with each other. In the second
variation, agents may not move exactly to the mass center but somewhere close
to it. The dynamics of both variants are qualitatively very different from that
of the classical HK system. Nevertheless, we prove that both these systems
converge in polynomial number of non-trivial steps, regardless of the social
network in the first variant and noise patterns in the second variant.
