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 {t} there are agents at positions {x_i(t)} with velocities {v_i(t)} 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:

\displaystyle  \dot x_i(t) = v_i(t),\qquad \dot v_i(t) = \lambda\sum_{j=1}^N a(|x_j(t)-x_i(t)|)(v_j(t)-v_i(t)).

In the second equation for the velocities, you recognize that the velocities of the {i}th agent is a weighted mean of the velocities of the other agents and the weight {a} for the {j}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 {f:{\mathbb R}^3\times{\mathbb R}^3\times{\mathbb R}\rightarrow{\mathbb R}} and {f(x,v,t)} describes “how many agents are at time {t} at position {x} with velocity {v}”. 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

\displaystyle  Q(f)(x,v,t) = \int\int a(|x-y|)(w-v)f(y,w,t)dwdy

the model reads as

\displaystyle  \partial_t f(x,v,t) + v\cdot\nabla_x f(x,v,t) + \lambda\nabla_v \Big(Q(f)(x,v,t)f(x,v,t))\Big) = 0.

The term “{v\cdot\nabla_x f(x,v,t)}” leads to the movement of the agent along the velocities, while the last term “{\lambda\nabla_v\Big(Q(f)(x,v,t)f(x,v,t))\Big)}” is the term which models the interaction of the velocities.

2. Synchronization

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 {i}th oscillator at time {t} is given by its phase {\theta_i(t)} (which is a real number but has to be considered modulo {2\pi}). If an oscillator has natural frequency {\omega_i} (or phase velocity), its phase evolves according to {\theta_i(t) = \theta_i(0) + t\omega_i}. Put differently, its phase satisfies the differential equation {\dot\theta_i = \omega_i}. The Kuromoto equation for {N} coupled oscillators is

\displaystyle  \dot\theta_i(t) = \omega_i + \tfrac{K}{N}\sum\sin(\theta_j(t)-\theta_i(t)).

The coupling strength is given by {K>0} 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 {N\rightarrow\infty} we consider a distribution {\rho:{\mathbb R}\times{\mathbb R}\times {\mathbb R}\rightarrow{\mathbb R}} and {\rho(\theta,\omega,t)} described “how many oscillators at time {t} have phase {\theta} and natural frequency {\omega}”. Further, the distribution of natural frequencies is given by {g(\omega)}. Then, the equation for the evolution of the oscillators is

\displaystyle  \partial_t\rho(\theta,\omega,t) + \partial_\theta\Big[\rho(\theta,\omega,t)\Big(\omega + K \int_0^{2\pi}\int_{\mathbb R} \sin(\theta'-\theta)\rho(\theta',\omega',t)g(w')dw'd\theta'\Big)\Big].

For more information, see the Wikipedia page or this article by Strogatz.

3. Consensus

Finally, I’d like to mention another model for opinion formation, the model by Ben-Naim, Krapivsky and Redner: The density {P(x,t)} describes how many agents at time {t} have the opinion {x} and evolves according to

\displaystyle  \partial_t P(x,t) = \iint_{|x_1-x_2|\leq1} P(x_1,t)P(x_2,t)\big[\delta(x - \tfrac{x_1+x_2}{2}) - \delta(x-x_1)\big]dx_1dx_2.

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.

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