Probability of Consensus of Hegselmann-Krause Dynamics

The original Hegselmann-Krause (HK) model comprises a set of $n$ agents characterized by their opinion, a number in $[0,1]$. Agent $i$ updates its opinion $x_i$ via taking the average opinion of its neighbors whose opinion differs by at most $\epsilon$ from $x_i$. In the article, the opinion space is extended to $\mathbf{R^d}.$ The main result is to derive bounds for the probability of consensus. In general, we have a positive lower bound for the probability of consensus and demonstrate a lower bound for the probability of consensus on a unit cube. In particular for one dimensional case, we derive an upper bound and a better lower bound for the probability of consensus and demonstrate them on a unit interval.