The Dynamics of Multidimensional Secession: Fixed Points and Ideological Condensation

Abstract

We explore a generalized Seceder Model with variable size selection groups and higher dimensional genotypes, uncovering its well-defined mean-field limiting behavior. Mapping to a discrete, deterministic version, we pin down the upper critical size of the multiplet selection group, characterize all relevant dynamically stable fixed points, and provide a complete analytical description of its self-similar hierarchy of multiple branch solutions.Dynamical phenomena in which an initially homogeneous population of weakly interacting individual agents can disperse, aggregate and form clusters arises in many different physical, biological, and sociological contexts. Condensation and droplet formation [1] is, of course, a well-known example in physics; galaxy formation and clustering [2], another. In traffic patterns [3], the realspace jams that plague highway driving are, for some, a daily reminder of such intrinsic tendencies in correlated systems far-from-equilibrium. The formation of swarms and herds in zoology [4], or the flocking of birds [5,6], provide additional illustrations. In these cases, particularly, joining the group yields advantages over standing out alone, be it by better exploration of food resources, protection from predators, or easing the aerodynamic flow in flight. Nevertheless, sometimes, as in fashion trends and similar social (or even financial) settings, standing apart from the crowd can also be a seed for the formation of new groups, splitting off the mainstream, though maybe becoming the mainstream themselves later on. In these instances, steady-state multiple groups can be the norm. Such matters are manifest in recent, though now classic implementations of Arthur's variant of the El Farol Bar problem [7], as for example, discussed by Zhang and Challet [8], where a multitude of competing agents, armed with limited memory strategies, compete via statistical Sisyphian dynamics to be in the minority group. Interestingly, with stochasticity introduced to the decision-making process, Johnson and coworkers [9] uncovered a tendency towards self-organized segregation within such evolutionary minority games. Subsequently, Hod & Nakar [10] discovered a dynamical phase transition in this setting, between 2-group segregation and single group clustering, driven by the economic cost-benefit ratio defined in the model. In biological systems, clustering can appear on multiple scales [11], with aggregation a consequence of dynamic correlations, whether they be hidden or explicit. Even so, the complexity that arises, for example, in statistical models of evolution [12], resulting in the formation of species is not, per se, self-evident via the direct interplay of mutation and selection-the system can dynamically bring itself to a critical state. The Seceder Model [13] was introduced, initially, to demonstrate that an interative mechanism favoring individuality cannot only create distinct groups, but also yields a rich diversity of cluster-forming dynamics. The essential tack was to give a small advantage to i...

Diverging tendencies in multidimensional secession

“…Lastly, with the replicator equations, we see why the greatest degree of population fragmentation occurs for the case of one ideological dimension where three groups are the norm though 4, 5, 6 groups are not uncommon. By contrast, in d ≥ 3, the population self-organizes to a coarsened trio of three tightly-knit opinion clusters, effectively collapsing to a 2D ideological plane [1].…”

“…This paper discusses recent research [1] on the Seceder Model [2], an intriguing far-from-equilibrium stochastic model of opinion dynamics in which the competing tendencies of conformity and dissent cause an evolving population to fragment, disperse, and coalesce, forming distinct groups characterized by free and frequent interchange of individuals. We imagine a population that is ultimately economic, sociological, or political in character, with interactions based upon shared investment strategies, cultural opinions, or electoral inclinations.…”

“…For the case at hand, invoking the constraint x 2 = 1/2 − x 1 and demandingẋ 1 = 0 leads to the cubic equation ). For additional details regarding the self-similar hierarchical gapped spectrum intrinsic to triplet selection, see [1]. Finally, the replicator formalism provides some hidden extra benefits, among them an explanation for the critical critical selection multiplet size m c = 4.…”

“…Finally, the replicator formalism provides some hidden extra benefits, among them an explanation for the critical critical selection multiplet size m c = 4. Assuming arbitrary m, one performs a stability analysis of the 2-branch FP ( 1 2 , 0, 1 2 ), focussing on the Seceder term arising from the multinomial expansion [1]; in this fashion, we discoverẋ 2 = 2 m 1 1 2…”

Multi-agent complex systems and many-body physics

“…Within these simulations, N decision-making particles or agents (e.g. commuters, traders, computer programs [2][3][4]) repeatedly compete with each other for some limited global resource (e.g. road space, best buy/sell price, processing time) using sets of rules which may differ between agents and may change in time.…”