Quantitative convergence guarantees for the mean-field dispersion process

Figure 1.  Illustration of the dispersion dynamics on a complete graph with $ N = 5 $ nodes/sites and $ M = 9 $ particles. Particles which share a common site will be "active" and move across sites

Figure 2.  Stackplots of the agent-based simulation with $ \mu = 0.8 $ and $ \mu = 2 $ during the first $ 10 $ units of time. At a given time $ t $, the width of $ n $-th layer represents the number of sites hosting $ n $ particles, making a total of $ N = 1000 $ sites. For $ \mu = 0.8 $, we initially put all particles at one single site; the distribution of particles converges to a Bernoulli distribution and no site hosts two or more particles (left). For $ \mu = 2 $, we put two particles in each site initially; each site hosts at least one particle and the distribution of particles stabilizes around a zero-truncated Poisson distribution after a few units of time (right)

Figure 3.  Distribution of particles for the dispersion model with $ N = 1000 $ agents after $ 2000 $ units of time, using two different values of $ \mu $. For $ \mu = 0.8 $, the final distribution coincides exactly with the Bernoulli distribution with mean $ 0.8 $, where we put all the particles into a single site initially. For $ \mu = 2 $, the terminal distribution is well-approximated by a zero-truncated Poisson distribution (4), where we put $ X_i(0) = \mu $ for all $ 1\le i\le N $ initially

Figure 4.  Stackplot of the numerical solution to the truncated ODE system $ \{p _n (t)\} _{n = 0} ^{100} $ with $ \mu = 0.8 $ and $ \mu = 2 $ during the first 10 units of time. At a given time $ t $, the width of $ n $-th layer represents $ p _n (t) $ which sum up to $ 1 $. For $ \mu = 0.8 $ the distibution of particles converges to a Bernoulli distribution with mean $ \mu $ (left). For $ \mu = 2 $, the distribution converges to the zero-truncated Poisson distribution with mean $ \mu $ (right). The overlayed yellow lines represent the corresponding agent-based simulation results

Figure 5.  Schematic illustration of the Fokker–Planck type system of nonlinear ODEs (10) as a jump process with loss and gain

Figure 6.  Evolution of the energy $ \mathcal E [ {\bf p} (t)] $ over $ 0\le t \le 3 $ with $ \mu = 0.8 $ and $ \mu = 1 $. It can be seen from the picture that the energy decays exponentially for $ \mu = 0.8 $ with rate $ C{\text{e}} ^{-0.4 t} $. For $ \mu = 1 $, the decay is slower

Figure 7.  Evolution of the $ \ell ^1 $ error $ \left\lVert {\bf p} (t) - \mkern 1.5mu\overline{\mkern-1.5mu {\bf p} \mkern-1.5mu}\mkern 1.5mu \right\rVert _{\ell ^1} $ over time for different values of $ \nu $. It can be seen that larger values of $ \nu $ (or $ \mu $) leads to faster convergence, although such improvement in terms of the convergence rate saturates when $ \nu $ becomes large enough