The Model

This page is a compact reference for the equations pyGD integrates. The full development — the information-theoretic reading, the viability measure, and the semantic threshold — is in Sowinski, Frank & Ghoshal, Phys. Rev. Research 6, 043188 (2024)[1].

The environment

The environment is a population of \(N\) Kuramoto phase oscillators[2] on the nodes of a graph with adjacency matrix \(A\). Each oscillator turns at its own natural frequency \(\omega_i\) and is pulled toward its neighbors,

\[\frac{d\theta_i}{dt} = \omega_i + \sigma \sum_{j} A_{ij}\,\sin(\theta_j - \theta_i),\]

with coupling \(\sigma\). The frequencies are standardized to zero mean and unit variance, and time is measured in units of the root-mean-square frequency. Kuramoto steps this with an explicit Euler update of timestep dt.

Synchronization is read off the global order parameter

\[Z = r\,e^{i\phi} = \frac{1}{N}\sum_j e^{i\theta_j},\]

whose magnitude \(r\) measures phase coherence — near 0 for scattered phases, near 1 for a locked population. Above a critical coupling \(\sigma_c\) an extensive fraction of the oscillators locks and \(r\) becomes nonzero[3].

The agent

The Yokai occupies one node at a time. At node \(i\) it estimates the local mean-field angle \(\hat\phi_i\) from the neighbors’ phases and kicks its own phase to oppose the local alignment,

\[\theta_i \mapsto \theta_i + \alpha\,\kappa_i, \qquad \kappa_i = \operatorname{sign}\big(\sin(\theta_i - \hat\phi_i)\big),\]

then hops to a uniformly chosen neighbor. Its strength \(\alpha\) and hop rate \(\beta\) enter all observables only through the product \(\alpha\beta\).

The coarse-grained limit

Averaging the agent’s rare, degree-biased visits into a continuous drift plus a fluctuation turns the driven environment into a stochastic Kuramoto equation,

\[\frac{d\theta_i}{dt} = \omega_i + \sigma \sum_j A_{ij}\,\sin(\theta_j - \theta_i) + \alpha\beta\,\frac{k_i}{\langle k \rangle}\,\kappa_i + \sqrt{\alpha\beta\,\frac{k_i}{\langle k \rangle}}\;\frac{dW_i}{dt},\]

where \(k_i\) is the degree of node \(i\) and \(W_i\) a Wiener process. The second added term is a degree-weighted shift of the natural frequency; the third is a heat bath whose temperature scales with degree, so that hubs fluctuate more than leaves. This is what KuramotoCG integrates, with ab \(= \alpha\beta\).

The continuum model reproduces the agent-based order parameter closely but not exactly — it slightly underestimates the synchronization — and pyGD keeps the two as independent classes for that reason. The derivation of the drift and diffusion coefficients is given in Appendix B of the paper.

References