Two-network Kuramoto-Sakaguchi model under tempered stable Lévy noise

Abstract

We examine a model of two interacting populations of phase oscillators labelled 'Blue' and 'Red'. To this we apply tempered stable Lévy noise, a generalisation of Gaussian noise where the heaviness of the tails parametrised by a power law exponent α can be controlled by a tempering parameter λ . This system models competitive dynamics, where each population seeks both internal phase synchronisation and a phase advantage with respect to the other population, subject to exogenous stochastic shocks. We study the system from an analytic and numerical point of view to understand how the phase lag values and the shape of the noise distribution can lead to steady or noisy behaviour. Comparing the analytic and numerical studies shows that the bulk behaviour of the system can be effectively described by dynamics in the presence of tilted ratchet potentials. Generally, changes in α away from the Gaussian noise limit, 1 < α < 2, disrupts the locking between Blue and Red, while increasing λ acts to restore it. However we observe that with further decreases of α to small values, α 1, with λ = 0, locking between Blue and Red may be restored. This is seen analytically in a restoration of metastability through the ratchet mechanism, and numerically in transitions between periodic and noisy regions in a fitness landscape using a measure of noise. This non-monotonic transition back to an ordered regime is surprising for a linear variation of a parameter such as the power law exponent and provides a novel mechanism for guiding the collective behaviour of such a complex competitive dynamical system.

Adversarial decision strategies in multiple network phased oscillators: The Blue-Green-Red Kuramoto-Sakaguchi model

“…For numerical exploration of the BGR model, we construct graphs of size |B| = |G| = |R| = 21, given explicitly in Figure 2. This extends the example followed in previous Bluevs-Red studies in [30,42,43]. As shown on the left side of Figure 2, the Blue population forms a hierarchy stemming from a single root, followed by a series of four branches two layers deep.…”

“…The Blue and Red tactical networks interact with each other, attempting to stay ahead in the phase of their adversary's tactical nodes. In the absence of a Green network, the adversarial dynamics between Blue and Red networks has been explored in [30,41,42,43].…”

Emergent behavior in an adversarial synchronization and swarming model

“…Using a similar parametrization to the works in Refs. [38][39][40] to differentiate between blue and red, we specify the elements of the matrices α and K as…”

“…Recently it has been applied to understanding organizational decision making [31] in the command and control context where the phase represents the actor's position in the "Observe-Orient-Decide-Act" (OODA) loop [32,33]. Extending these ideas to multi-networks offers a way to model competitive dynamics [34][35][36][37] and, in the specific case of two populations, this has been referred to as the "blue vs. red model" [38][39][40]. In parallel, (spatial dynamic) swarming has been extensively studied since Vicsek [1], Cucker, and Smale [8] showed that, through a simple continuous-time model, complicated flocking behavior can emerge.…”