Dynamical maximum entropy approach to flocking

We derive a new method to infer from data the out-of-equilibrium alignment dynamics of collectively moving animal groups, by considering the maximum entropy model distribution consistent with temporal and spatial correlations of flight direction. When bird neighborhoods evolve rapidly, this dynamical inference correctly learns the parameters of the model, while a static one relying only on the spatial correlations fails. When neighbors change slowly and the detailed balance is satisfied, we recover the static procedure. We demonstrate the validity of the method on simulated data. The approach is applicable to other systems of active matter.

Figure 1

Comparison between dynamical and static inference. Data were generated using Voronoi neighborhood. The inference was performed by using either a Voronoi rule or a nearest-neighbor (NN) topological rule, parametrized by the number $n_c$ of interacting neighbors. Main panel: The inferred number of interacting neighbors $n_c^{*}$ is shown as a function of the mixing rate $\mu$. Circles: dynamical inference; triangles: static inference. The dashed line marks the real average value $n_V=6$. Static inference badly overestimates the number of interacting neighbors at large mixing, while dynamical inference does a much better job. Inset: Dynamical normalized log-likelihood $-{\mathcal{L}}_t/N$ as a function of $n_c$ for the NN topological rule (circles). The maximum of this function gives the NN value of $n_c^{*}$ reported in the main panel. The Voronoi likelihood (dashed line) is larger than the NN one, revealing that Voronoi was the actual generating rule. Data are for high mixing.

Figure 2

Upper panel: Comparison of the normalized log-likelihood for the nearest-neighbor and metric rules, as a function of $n_c$. For the metric case, for increasing values of $r_c$, the empirical $n_c=(1/N){\sum }_i{n}_i$ is shown. The dashed line corresponds to the log-likelihood calculated with the Voronoi rule. Lower panel: Inferred interaction range $n_c^{*}$ for the nearest-neighbor and metric cases, as a function of the mixing parameter $\mu$.