Manifold Learning and Dimensionality Reduction in Collective Motion

We study mathematical aspects of the behaviors of animal groups, both to empirically reveal underlying simplicity, and to model possible mechanisms of them. Social groups consisting of multi-agents perform spatially coherent behaviors to self-defense against predation or to attain dietary benefits. As positions of agents in a group at a time can be represented as a point in a higher-dimensional space, all such points describing the behavior create a hidden manifold in the higher-dimensional space. Dimensional Reduction (DR) methods revealing the underlying manifold sometimes misinterpret the topology of the embedding. In fact, multi-agent systems may undergo abrupt changes defined as phase transitions, due to natural perturbations. A phase transition splits the underlying manifold into two sub-manifolds with distinct dimensionalities and often immobilizes DR methods. However, behavioral coherence is costly, as it requires agents to change their activities such as eating and resting in a communal time rather than their ideal time. Segregation of a group into subgroups may reduce the cost if subgroups are homogeneous, but it may also create small subgroups that are then vulnerable to predation.;Here we first construct an alternative DR approach that demands a two-dimensional embedding which topologically summarizes the higher-dimensional data. Then, we device a method to detect phase transitions in a multi-agent system. We also adopt a distinct metric that is capable of computing changes in a multi-agent group, and characterizing alternation of manifolds. Finally, we model an efficient segregation of a group by optimizing a cost function.