There is an interesting preprint today on arxiv by Sascha Kurz:
How long does it take to consensus in the Hegselmann-Krause model? http://arxiv.org/abs/1405.5757
Consider -particle system with total energy , and let be the range of interactions. The evolution of energy distribution is defined by the following algorithm:
- set the initial energies with .
- as time develops, interacts (i.e. exchanges energy) only with the particles with similar energies: let and . Then set .
- Suppose we have defined . Then let and . Then set .
A simple test example is for .
- : trivial system.
- : and . Then for all and all . All reach the same energy at time .
- : . Then , and . Now note that for all . So for all and all . Then reach the same energy at time .
The set plays an important role. One can check that and : the small groups can always reach the same energy. But the case is different:
Eventually this leads to two different clusters for and for with large energy difference . The two groups of particles never intersect with the other group and will stay in this status forever. So we set .
There is no general formula for . It is proved that , and conjectured that , and the worse scenario is given the initial data for .