Heterophilious Dynamics Enhances Consensus

Abstract

We review a general class of models for self-organized dynamics based on alignment. The dynamics of such systems is governed solely by interactions among individuals or "agents", with the tendency to adjust to their 'environmental averages'. This, in turn, leads to the formation of clusters, e.g., colonies of ants, flocks of birds, parties of people, rendezvous in mobile networks, etc. A natural question which arises in this context is to understand when and how clusters emerge through the self-alignment of agents, and what type of "rules of engagement" influence the formation of such clusters. Of particular interest to us are cases in which the self-organized behavior tends to concentrate into one cluster, reflecting a consensus of opinions, flocking of birds, fish or cells, rendezvous of mobile agents, and in general, concentration of other traits intrinsic to the dynamics. Many standard models for self-organized dynamics in social, biological and physical science assume that the intensity of alignment increases as agents get closer, reflecting a common tendency to align with those who think or act alike. Moreover, "Similarity breeds connection," reflects our intuition that increasing the intensity of alignment as the difference of positions decreases, is more likely to lead to a consensus. We argue here that the converse is true: when the dynamics is driven by local interactions, it is more likely to approach a consensus when the interactions among agents increase as a function of their difference in position. Heterophily -the tendency to bond more with those who are different rather than with those who are similar, plays a decisive rôle in the process of clustering. We point out that the number of clusters in heterophilious dynamics decreases as the heterophily dependence among agents increases. In particular, sufficiently strong heterophilious interactions enhance consensus.Date: October 15, 2018. 1991 Mathematics Subject Classification. 92D25,74A25,76N10.

Critical thresholds in flocking hydrodynamics with non-local alignment

“…The expected long-time behaviour of these agents is to self-organize into finitely many clusters, and in particular, depending on the properties of the interaction kernel, a(·, ·), to flock into one such cluster; consult the recent reviews [2,3]. The goal of this paper is to study the flocking phenomenon of the corresponding hydrodynamic description (1.1), or flocking hydrodynamics for short.…”

Clustering and asymptotic behavior in opinion formation

“…This dynamics of this discrete dynamics has been studied in [7,19]. In [6], it has been conjectured that the dynamics converges toward a stationary state.…”

“…The only remaining question is how fast can one prove convergence in that case. But this case was extensively studied in [19] and we recall one of the main result in this paper …”

“…Without loss of generality, one can assume that φ(0) = 1. Several studies have been conducted to study numerically the long time behavior of the consensus model [5,7,19]. It has been observed that the dynamics generate concentration of opinions (also called clusters) and that, after a transient period, the configuration stabilizes.…”

“…When the interaction function φ is globally supported (i.e. φ(r) > 0 for all r ≥ 0) meaning that every agent interacts with each other, the dynamics is easily understood: the system converges to a so-called consensus [19]. In other words, there exists one opinion x ∞ such that…”