according to the equation $$\begin{aligned}
\label{cs2}
\frac{dv_i}{dt}=\frac{1}{N}\sum_{j=1}^N\psi(|x_j-x_i|)(v_j-v_i),\end{aligned}$$ where $\psi:[0,\infty)\to[0,\infty)$ is a given function called the [*communication weight*]{} or the [*communication rate*]{}. The communication weight may be interpreted as perception of the particles. Indeed, we have $$\begin{aligned}
\frac{1}{N}\sum_{j=1}^N\psi(|x_j-x_i|)(v_j-v_i)=\frac{1}{N}\sum_{j=1}^N\psi(|x_j-x_i|) \left[\frac{\sum_{j=1}^N\psi(|x_j-x_i|)v_j}{\sum_{j=1}^N\psi(|x_j-x_i|)}-v_i\right],\end{aligned}$$ thus $v_i$ evolves towards $\frac{\sum_{j=1}^N\psi(|x_j-x_i|)v_j}{\sum_{j=1}^N\psi(|x_j-x_i|)}$, which is a convex combination of velocities $v_j$ and can be interpreted as the avarage velocity of the particles percieved by $i$-th particle. For instance if $\psi\equiv 1$ then it is exactly the avarage velocity of the particles and if $\psi\equiv 0$ then the particles move independently with constant velocity as if they did not see each other.

As the number of particles grows to infinity, i.e., $N\to\infty$, the discrete model is replaced by the following Vlasov-type equation: $$\begin{aligned}
\label{cscont}
\partial_tf+v\cdot\nabla f+{\rm div}_v(F(f)f)=0,\ \ x\in\r^d,\ v\in\r^d,\\
F(f)(x,v,t):=\int_{\r^{2d}}\psi(|y-x|)(w-v)f(y,w,t)dwdy\nonumber,\end{aligned}$$ where $f=f(x,v,t)$ is a density of particles that at the time $t$ have position $x$ and velocity $v$.

Smooth communication weight
---------------------------

The most well known communication weight is a bounded and smooth one given by $$\begin{aligned}
\label{cucu}
\psi_{cs}(s)=\frac{K}{(1+s^2)^\frac{\beta}{2}}, \ \ \ \beta\geq 0,\ \ \  K>0,\end{aligned}$$ or more generally $\psi_{cs}$– bounded and Lipschitz continuous. The C-S model with weight $\psi_{cs}$ was introduced in 2007 by Cucker and Smale in 