Effects of Heterogeneous Social Interactions on Flocking Dynamics

Abstract

Social relationships characterize the interactions that occur within social species and may have an important impact on collective animal motion. Here, we consider a variation of the standard Vicsek model for collective motion in which interactions are mediated by an empirically motivated scale-free topology that represents a heterogeneous pattern of social contacts. We observe that the degree of order of the model is strongly affected by network heterogeneity: more heterogeneous networks show a more resilient ordered state, while less heterogeneity leads to a more fragile ordered state that can be destroyed by sufficient external noise. Our results challenge the previously accepted equivalence between the static Vicsek model and the equilibrium XY model on the network of connections, and point towards a possible equivalence with models exhibiting a different symmetry.

Scalar model of flocking dynamics on complex social networks

“…To avoid this problem, in the following we will consider a sufficiently large minimum degree. As in the case of the vectorial Vicsek model on networks, the position of particles plays no role and we do not keep track of them [17]. The order parameter of the model, φ(η) is defined in terms of the time average of the average velocity…”

“…(23) to self-consistently solve for y in Eq. (17), considering that in the steady state ρ k = ψ k . Since y = Q − 1/2, we have, from Eq.…”

“…This simplification, however, comes at the cost of disregarding the possible effects of social interactions, that can cause individuals to follow preferentially those others with which they have strong social ties [12]. The effects of such social ties, represented in terms of non-metric pairwise interactions encoded in a complex network [13], have been considered in the framework of the Vicsek model in different works [14][15][16][17]. In particular, Ref.…”

“…In particular, Ref. [17] studied the effects of a heterogeneous complex topology, as represented by a scale-free degree distribution (defined as the probability that a node is connected to k others, or has degree k) with a powerlaw form [18], P (k) ∼ k −γ , on the order-disorder transition experimented by the model. In this work, it was reported that for γ > 5/2, a standard transition is observed for a finite value of η, while for γ < 5/2, the transition is suppressed in the thermodynamic limit of infinite network size, being the system in the ordered state for all physical values of the noise parameter η 1 .…”

“…In the standard Vicsek model, the range of physical values of η is restricted to a finite interval that can be taken to be 0 ≤ η ≤ 1[5]. In Ref [17]. it was shown that ηc → 1 in the thermodynamic limit for γ < 5/2.…”