In my last post I treated a model for opinion dynamics. The specific feature of this model by Canuto, Fagnani and Tilli was that it is not based on a finite number of agents, each moving around in opinion space, but on the density of agents on the opinion space. On the one hand, this makes the “agents” somehow disappear in the model and especially, it is not possible to “track an individual agent” anymore (correct me, if I’m wrong). On the other hand, this “approximation by infinity” makes the analysis a bit easier.
In this post I’d like to collect similar equations for other related phenomena, namely flocking and synchronization.
1. Flocking dynamics
Some animals (as well as humans) build flocks and usually, this happens without a kind of leader. Flocks can be visually appealing, as you can see here or here. There are several mathematical models around to describe the leaderless formation of flocks.
One of the first flocking models is due to Vicsek at al. and a very popular one is more recent and due to Cucker and Smale: At time there are agents at positions with velocities and each agent moves according to its velocity and adapts its velocity in view of the velocities of its neighbors. The equations read like this:
In the second equation for the velocities, you recognize that the velocities of the th agent is a weighted mean of the velocities of the other agents and the weight for the th agent depends on their mutual distance.
For the continuous counterpart of this model, we need a density function of the phase space, that is a density function and describes “how many agents are at time at position with velocity ”. This is very similar to the famous Boltzmann equation from gas dynamics. The evolution of this density function according to the above model has been derived by Ha and Tadmor. With the operator
the model reads as
The term “” leads to the movement of the agent along the velocities, while the last term “” is the term which models the interaction of the velocities.
Another natural phenomenon where group behavior arises is that of synchronization. Different “oscillators” are mutually coupled and may evolve to a state in which all of them oscillate with the same frequency. This phenomenon is modelled by the Kuramoto equation: The state of the th oscillator at time is given by its phase (which is a real number but has to be considered modulo ). If an oscillator has natural frequency (or phase velocity), its phase evolves according to . Put differently, its phase satisfies the differential equation . The Kuromoto equation for coupled oscillators is
The coupling strength is given by and the coupling term says that an oscillator “slows down” if it sees some oscillators “behind him” and “speeds up” if it sees some “ahead of him”.
In the limit for we consider a distribution and described “how many oscillators at time have phase and natural frequency ”. Further, the distribution of natural frequencies is given by . Then, the equation for the evolution of the oscillators is
Finally, I’d like to mention another model for opinion formation, the model by Ben-Naim, Krapivsky and Redner: The density describes how many agents at time have the opinion and evolves according to
The model is derived from the Deffuant-Weisbuch model in which two randomly chosen agents meet in each step, and if their opinion is close enough to each other, they agree on their mean value.