A Proof of the Convergence of the Hegselmann-Krause Dynamics on the Circle

the s-kinetic energy of the system, their proof strategy consists in showing that (1) the quadratic kinetic energy K2 is finite, (2) the influence graph is eventually constant, and (3) the 1-kinetic energy K1 is finite, which immediately implies the convergence of the sequence of position vectors (x(t))t∈N. To show the finiteness of K2, we present a simple proof in Section 2 which is based on a reduction of the HK dynamics on the circle to the HK dynamics on the line. Concerning the eventual stability of influence graphs, we are not able to understand the proof outlined in [4], and we give our proof of this key point in Section 3. For the third point, namely the finiteness of K1, Hegarty, Martinsson, and Wedin introduce the vector of position differences x(t), and show that the sequence (x(t))t∈N converges to some limit x ∗ ∞ such that ‖x∗(t) − x∞‖= O (e). The argument given in the first version [4] for the latter point is erroneous, and we fix the argument in Section 4. An alternative argument, based on the finiteness of the quadratic kinetic energy, is provided in [5] which still appeals to the vector of differences x(t). In fact, this argument can be directly applied to the position vector as we show in Section 6, which allows to circumvent the arguments of Sections 4 and 5 and prove convergence directly.