Scalar model of flocking dynamics on complex social networks

Abstract

We investigate the effects of long-range social interactions in flocking dynamics by studying the dynamics of a scalar model of collective motion embedded in a complex network representing a pattern of social interactions, as observed in several social species. In this scalar model we find a phenomenology analogous to that observed in the classic Vicsek model: In networks with low heterogeneity, a phase transition separates an ordered from a disordered phase. At high levels of heterogeneity, instead, the transition is suppressed and the system is always ordered. This observation is backed up analytically by the solution of a modified scalar model within an heterogeneous mean-field approximation. Our work extends the understanding of the effects of social interactions in flocking dynamics and opens the path to the analytical study of more complex topologies of social ties.

Flocking dynamics mediated by weighted social networks

“…The same particular role of the degree exponent was recovered analytically in Ref. [32] using the scalar version of the Vicsek model proposed by Czirók, Barabási and Vicsek, the CBV model [33], in which velocity is a real number, instead of a vector in a two-dimensional space. These results are relevant for the understanding of the collective motion in social animals, as they indicate that the flocking phase is more robust against noise effects in the case of a highly heterogeneous pattern of social contacts.…”

“…Individual velocities are finally updated by this modulated local average velocity with the addition of a noise term. For a binary network, the update rule can be defined as [32]…”

“…where ξ i is a uniform random number in the interval [−1/2, 1/2] and η ∈ [0, ∞) gauges the strength of the external noise. For simplicity, the modulating function G(U ) is chosen to be the sign function, taking value G(U ) = +1 when U ≥ 0 and G(U ) = −1 otherwise [32]. We notice that, in this prescription, we do not consider the interaction of the velocity of a node with itself.…”

“…The CBV model in weighted networks can be tackled numerically applying the HMF approximation developed in Ref. [32] (see also [46]). We start by rewriting the update dynamics in terms of the dual velocities u * i as…”

Phase transitions on a class of generalized Vicsek-like models of collective motion

“…In order to gain some understanding of the applications of mean-field theory in our class of flocking dynamics models, we consider the case in which interactions are mediated by a static complex network [21]. The relevance of this case is based on the recent claims that a network structure, representing the patterns of social interactions [31], can be relevant in understanding the flocking behavior of social animals [15,16,[32][33][34], which might have a stronger tendency to follow individuals with which they have stronger social ties [34]. Previous works have shown that the scalar Vicsek model with networked interactions display a second-order transition [16,27], whose properties depend on the degree of heterogeneity of the network.…”