KuramotoState#

class graph_tool.dynamics.KuramotoState(g, omega=1, w=1, sigma=0, t0=0, s=None)[source]#

Bases: ContinuousStateBase

The Kuramoto model.

Parameters:

gGraph: Graph to be used for the dynamics
omegaVertexPropertyMap or float (optional, default: 1): Intrinsic frequencies for each node. If a scalar is given, it will be used for all nodes.
wEdgePropertyMap or float (optional, default: 1): Coupling strength of each edge. If a scalar is given, it will be used for all edges.
sigmaVertexPropertyMap or float (optional, default: 0): Stochastic noise magnitude for each node. If a scalar is given, it will be used for all nodes.
sVertexPropertyMap (optional, default: None): Initial global state. If not provided, a random state will be chosen.

Notes

This implements Kuramoto’s model for synchronization [kuramoto_self-entrainment_1975] [rodrigues_kuramoto_2016].

Each node has an angle \(\theta_i\), which evolves in time obeying the differential equation:

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

where \(\xi_i(t)\) is a Gaussian noise with zero mean and unit variance.

References

[kuramoto_self-entrainment_1975]

Y. Kuramoto, “Self-entrainment of a population of coupled non-linear oscillators”, International Symposium on Mathematical Problems in Theoretical Physics. Lecture Notes in Physics, vol 39. Springer, Berlin, Heidelberg (1975), DOI: 10.1007/BFb0013365 [sci-hub, @tor]

[rodrigues_kuramoto_2016]

Francisco A. Rodrigues, Thomas K. DM.Peron, Peng Ji, Jürgen Kurth, “The Kuramoto model in complex networks”, Physics Reports 610 1-98 (2016) DOI: 10.1016/j.physrep.2015.10.008 [sci-hub, @tor], arXiv: 1511.07139

Examples

>>> g = gt.collection.data["karate"]
>>> omega = g.new_vp("double", np.random.normal(0, 1, g.num_vertices())) 
>>> state = gt.KuramotoState(g, omega=omega, w=1.5)
>>> thetas = []
>>> ts = linspace(0, 40, 1000)
>>> for t in ts:
...     ret = state.solve(t, first_step=0.0001)
...     thetas.append(state.get_state().fa % (2 * pi))

>>> figure(figsize=(6, 4))
<...>
>>> for v in g.vertices():
...    plot(ts, [t[int(v)] - t.mean() for t in thetas])
[...]
>>> xlabel(r"Time")
Text(...)
>>> ylabel(r"$\theta_i - \left<\theta\right>$")
Text(...)
>>> tight_layout()
>>> savefig("karate-kuramoto.svg")

../_images/karate-kuramoto.svg — Kuramoto oscillator dynamics on the Karate Club network.#

Methods

`copy`()	Return a copy of the state.
`get_diff`(dt)	Returns the current time derivative for all the nodes.
`get_r_phi`()	Returns the phase coherence \(r\) and average phase \(\phi\), defined as
`get_state`()	Returns the internal `VertexPropertyMap` with the current state.
`solve`(t, args, *kwargs)	Integrate the system up to time `t`.
`solve_euler`(t[, dt])	Integrate the system up o time `t` using a simple Euler's method with step size `dt`.

copy()#: Return a copy of the state.

get_diff(dt)#: Returns the current time derivative for all the nodes. The parameter dt is the time interval in consideration, which is used only if the ODE has a stochastic component.

Parallel implementation.

If enabled during compilation, this algorithm will run in parallel using OpenMP. See the parallel algorithms section for information about how to control several aspects of parallelization.

get_r_phi()[source]#: Returns the phase coherence \(r\) and average phase \(\phi\), defined as

\[re^{i\phi} = \frac{1}{N}\sum_j e^{i\theta_j}.\]

get_state()#: Returns the internal VertexPropertyMap with the current state.

solve(t, *args, **kwargs)#: Integrate the system up to time t. The remaining parameters are passed to scipy.integrate.solve_ivp(). This solver is not suitable for stochastic ODEs.

solve_euler(t, dt=0.001)#: Integrate the system up o time t using a simple Euler’s method with step size dt. This solver is suitable for stochastic ODEs.

KuramotoState

Contents

KuramotoState#