Analysis of the Heterogeneous Vectorial Network Model of Collective Motion

We analyze the vectorial network model, a stochastic protocol that describes collective motion of groups of agents, randomly mixing in a planar space. Motivated by biological and technical applications, we focus on a heterogeneous form of the model, where agents have different propensity to interact with others. By linearizing the dynamics about a synchronous state and leveraging an eigenvalue perturbation argument, we establish a closed-form expression for the mean-square convergence rate to the synchronous state in the absence of additive noise. These closed-form findings are extended to study the effect of added noise on the agents' coordination, captured by the polarization of the group. Our results reveal that heterogeneity has a detrimental effect on both the convergence rate and the polarization, which is nonlinearly moderated by the average number of connections in the group. Numerical simulations are provided to support our theoretical findings.