Partial convergence of heterogeneous Hegselmann-Krause opinion dynamics

In opinion dynamics, the convergence of the heterogeneous Hegselmann-Krause (HK) dynamics has always been an open problem for years which looks forward to any essential progress. In this short note, we prove a partial convergence conclusion of the general heterogeneous HK dynamics. That is, there must be some agents who will reach static states in finite time, while the other opinions have to evolve between them with a minimum distance if all the opinions does not reach consensus. And this result leads to the convergence of several special cases of heterogeneous HK dynamics, including when the minimum confidence bound is large enough, the initial opinion difference is small enough, and so on.