Explosive topological Kuramoto of node and link phases on a simplicial complex, with the Nodes-Links-Triangles variant
Phases on the nodes and links of a clique complex, coupled through its incidence by order parameters.
- Name:
topological_kuramoto - Field:
physics/synchronization - Formalism: ordinary differential equations
- References: Majhi, Perc, and Ghosh 2022; Ghorbanchian et al. 2021
Select this system
from skcomplex.systems import create_simulation, create_system, describe
system = create_system('topological_kuramoto')
entry = describe('topological_kuramoto')
trajectory = create_simulation('topological_kuramoto').prepare().run().trajectory_
Factory
@register_system(
field='physics/synchronization',
tags=('synchronization', 'oscillators', 'phase', 'simplicial complex', 'cell complex', 'explosive'),
references=('Majhi, Perc, and Ghosh 2022', 'Ghorbanchian et al. 2021'),
defaults={'until': 40.0, 'dt': 0.2},
observables=_topological_observables,
)
def topological_kuramoto(
n_nodes: int = 30,
*,
density: float = 0.4,
coupling: float = 2.0,
width: float = 1.0,
seed: int = 0,
model: str = 'NL',
) -> System:
"""Return phases on the nodes and links of a clique complex, coupled through its incidence by order parameters.
Node phases follow `theta' = omega + K R_links codifferential(sin(d theta))`. Link phases follow `phi' = omega + K
R_nodes d(sin(codifferential(phi))) - K codifferential(sin(d phi))`. With the signs of `codifferential`, these are
the Nodes-Links system of Ghorbanchian et al. (2021), Eq. 6 of the review of Majhi, Perc, and Ghosh (2022,
arXiv 2203.06601 §III), which is the source at hand. Here `R_nodes` is the order parameter of the node phases and
`R_links` that of the projection of the link phases on the nodes. The review takes frequencies of mean `Omega` and
standard deviation `1 / tau`, with `Omega = 2` and `tau = 1` for its Fig. 2, on complexes of 500 nodes. The mean 0,
the size, and the random graph are choices of this system.
The Nodes-Links-Triangles system, `model='NLT'`, Eq. 8 of the review, scales the coupling of the link phases through
the nodes by `R_nodes R_up` and through the triangles by `R_links`, where `R_up` is the order parameter of the
projection of the link phases on the triangles. Every projection then couples to another, so all three order
parameters jump together at one explosive transition, while in the Nodes-Links system `R_up` moves on its own.
Args:
n_nodes:
The number of nodes of the random graph.
density:
The probability of every edge of the random graph.
coupling:
The coupling `K`.
width:
The standard deviation of the natural frequencies of the nodes and the links.
seed:
The seed of the graph, the frequencies, and the initial phases.
model:
The coupling, `'NL'` for the Nodes-Links system or `'NLT'` for the Nodes-Links-Triangles system.
Returns:
system:
The system, with a group `triangles` of unit weights on the faces for the Nodes-Links-Triangles system,
which only gives the order parameter of the projection on the triangles.
Raises:
ValidationError:
The model is neither `'NL'` nor `'NLT'`.
Examples:
>>> from skcomplex.systems.physics.synchronization import topological_kuramoto
>>> [element.size_ for element in topological_kuramoto(6, density=0.8).prepare().elements]
[6, 13]
"""
if model not in MODELS:
error_msg = f'Unknown model {model!r}. Use one of {", ".join(MODELS)}.'
raise ValidationError(error_msg)
rng = np.random.default_rng(seed)
graph = nx.gnp_random_graph(n_nodes, density, seed=seed)
space = CellComplexSpace(NetworkSpace(graph), faces='cliques').prepare()
n_links = space.incidences_[0].shape[0]
nodes = GroupElement(
'nodes',
element=PhaseOscillatorElement,
space=space,
on='sites',
values={'theta': rng.uniform(-np.pi, np.pi, n_nodes), 'omega': rng.normal(0.0, width, n_nodes)},
)
links = GroupElement(
'links',
element=PhaseOscillatorElement,
space=space,
on='edges',
values={'theta': rng.uniform(-np.pi, np.pi, n_links), 'omega': rng.normal(0.0, width, n_links)},
visual=VisualSpec('cell', {'state': 'theta'}),
)
interactions = [
RateInteraction(
'node_coupling',
target='nodes',
state='theta',
sources=['links'],
rate=partial(_node_rate, coupling),
),
]
if model == 'NL':
link_coupling = RateInteraction(
'link_coupling',
target='links',
state='theta',
sources=['nodes'],
rate=partial(_link_rate, coupling),
)
return System(elements=[nodes, links], interactions=[*interactions, link_coupling])
triangles = GroupElement('triangles', element=LinkElement, space=space, on='hyperedges')
link_coupling = RateInteraction(
'link_coupling',
target='links',
state='theta',
sources=['nodes', 'triangles'],
rate=partial(_triangle_link_rate, coupling),
)
return System(elements=[nodes, links, triangles], interactions=[*interactions, link_coupling])
Components
Elements:
PhaseOscillatorElement, fromElement
Interactions:
RateInteraction
class PhaseOscillatorElement(Element):
"""Phase oscillator, a phase `theta` that turns at its natural frequency, `theta' = omega`.
Couplings to other oscillators add to the rate of the phase.
Args:
label:
The label of the oscillator or the group.
visual:
How the oscillator is drawn, or `None` for `default_visual_`.
theta:
The initial phase.
omega:
The natural frequency.
Attributes:
default_visual_:
The visual of the oscillator when it sets none.
Examples:
>>> from skcomplex.core.components.system import System
>>> from skcomplex.core.engine.runtime import Simulation
>>> from skcomplex.systems.physics.common.oscillators import PhaseOscillatorElement
>>> system = System(elements=[PhaseOscillatorElement('clock', omega=2.0)])
>>> float(Simulation(system, dt=0.5).prepare().run(until=1.0).trajectory_.values('clock.theta')[-1])
2.0
"""
def __init__(
self,
label: str,
visual: VisualSpec | None = None,
*,
theta: Annotated[float, StateSpec(role='phase')] = 0.0,
omega: Annotated[float, ParameterSpec()] = 1.0,
) -> None:
super().__init__(label, visual=visual)
self.theta = theta
self.omega = omega
def _get_laws(self, namespace: SimpleNamespace) -> list[BaseLaw]:
"""Return the turning of the phase at the natural frequency.
Args:
namespace:
The namespace of the symbols the laws can use.
Returns:
laws:
The rate of the phase.
"""
return [RateLaw(namespace.theta, namespace.omega)]
class RateInteraction(Interaction):
"""Rate `rate(ns)` and noise `noise(ns) dW` added to the highest time derivative of a state of a target.
On a state of order 1 the rate is its first derivative, and on a state of order `k` it adds to the `k`-th
derivative, so rates on one state superpose. With a divisor `d` the rate and the noise are divided by `d`, the name
of a state or parameter of the target, such as an inertia, a friction, or a capacitance, which backends can read.
A rate, a noise, or a divisor function takes the namespace of the interaction, with the namespaces `target` and
`sources` of the bound elements, the time `t`, and the symbols of the parameters a subclass declares. A number
stands for a constant, and a number on a vector state applies to every component. The noise is a Wiener process per
component and member by default, a process of a `WienerProcessSpec` to share or correlate it across states and
members, or a fractional Brownian motion of a Hurst index.
Args:
label:
The label of the interaction, unique among the parts of its system.
target:
The path of the element or group whose state the rate changes.
state:
The name of the state of the target that the rate changes.
rate:
A function of the namespace that returns the rate, or a number.
sources:
The paths of the elements or groups the rate and the noise read but do not change.
noise:
A function of the namespace that returns the amplitude of the noise, a number, or `None` for no noise.
divisor:
The name of the state or parameter of the target that divides the rate and the noise, a function of the
namespace of the target that returns the divisor, such as a moment of inertia `m l^2`, or `None`.
process:
The Wiener processes that drive the noise, or `None` for one independent process per component and member.
index:
The process of `process` that drives the noise, or `None` for every process, one per component of a vector
state.
convention:
The stochastic calculus of the noise, `'ito'`, `'stratonovich'`, or `'anti-ito'`.
hurst:
The Hurst index in `(0, 1)` of a fractional Brownian motion that drives the noise, a number or a function
of the namespace that returns a parameter, or `None` for a Wiener process.
conserved:
Whether the noise is the divergence of a random flux on the links of the lattice of a field, so it keeps the
total of the state.
Raises:
ValidationError:
The state is not a name, a function is neither callable nor a number, the convention is unknown, the Hurst
index is outside `(0, 1)`, or the noise options cannot combine.
Examples:
>>> from skcomplex.core.components.elements import CompartmentElement
>>> from skcomplex.core.components.interactions import RateInteraction
>>> from skcomplex.core.components.system import System
>>> from skcomplex.core.engine.runtime import Simulation
>>> decay = RateInteraction('decay', target='stock', state='n', rate=lambda s: -0.5 * s.target.n)
>>> system = System(elements=[CompartmentElement('stock', n=8.0)], interactions=[decay])
>>> trajectory = Simulation(system, dt=0.01).prepare().run(until=2.0).trajectory_
>>> round(float(trajectory.values('stock.n')[-1]), 3)
2.943
"""
law_classes: ClassVar[tuple[type[BaseLaw], ...]] = (RateLaw, NoiseLaw, FractionalNoiseLaw)
def __init__(
self,
label: str,
*,
target: Annotated[str, RoleSpec(elements=(ElementKind.ANY,), states=('state', 'divisor'))],
state: str = 'x',
rate: Callable[[SimpleNamespace], object] | float = 0.0,
sources: Annotated[Sequence[str], RoleSpec(variadic=True, elements=(ElementKind.ANY,))] = (),
noise: Callable[[SimpleNamespace], object] | float | None = None,
divisor: str | Callable[[SimpleNamespace], object] | None = None,
process: Annotated[WienerProcessSpec | None, RoleSpec(noise=True)] = None,
index: int | None = None,
convention: str = 'ito',
hurst: Callable[[SimpleNamespace], object] | float | None = None,
conserved: bool = False,
) -> None:
super().__init__(label)
self.target = target
self.state = state
self.rate = rate
self.sources = sources
self.noise = noise
self.divisor = divisor
self.process = process
self.index = index
self.convention = convention
self.hurst = hurst
self.conserved = conserved
def _check_functions(self) -> None:
"""Raise an error for a state that is not a name, or a function that is neither callable nor a number."""
if not isinstance(self.state, str) or not self.state.isidentifier():
error_msg = (
f'{type(self).__name__} {self.label!r} changes the state {self.state!r}, which is not a name of a '
'state.'
)
raise ValidationError(error_msg)
for name in ('rate', 'noise'):
value = getattr(self, name)
if not (value is None or callable(value) or isinstance(value, int | float)):
error_msg = f'The {name} of {self.label!r} is {value!r}. Give a function of the namespace or a number.'
raise ValidationError(error_msg)
if not (
self.divisor is None
or callable(self.divisor)
or (isinstance(self.divisor, str) and self.divisor.isidentifier())
):
error_msg = (
f'The divisor of {self.label!r} is {self.divisor!r}. Give the name of a state or parameter, or a '
'function of the namespace of the target.'
)
raise ValidationError(error_msg)
def _check_noise(self) -> None:
"""Raise an error for an unknown convention, a Hurst index outside `(0, 1)`, or noise options that conflict."""
if self.convention not in CONVENTIONS:
error_msg = f'Unknown noise convention {self.convention!r}. Use one of {", ".join(CONVENTIONS)}.'
raise ValidationError(error_msg)
if not (
self.hurst is None or callable(self.hurst) or (isinstance(self.hurst, int | float) and 0 < self.hurst < 1)
):
error_msg = f'The Hurst index of {self.label!r} is {self.hurst!r}. Give a number in (0, 1) or a function.'
raise ValidationError(error_msg)
if self.index is not None and self.process is None:
error_msg = f'{self.label!r} picks the process {self.index!r} but binds no processes. Give `process`.'
raise ValidationError(error_msg)
options = [
name
for name, given in (('hurst', self.hurst), ('process', self.process), ('conserved', self.conserved))
if given
]
if len(options) > 1:
error_msg = f'The noise of {self.label!r} takes {" and ".join(options)}. Give at most one of them.'
raise ValidationError(error_msg)
def _check(self) -> None:
"""Raise an error for a state, functions, or noise options the interaction cannot take.
Raises:
ValidationError:
The type declares states, the state is not a name, a function is neither callable nor a number, the
convention is unknown, the Hurst index is outside `(0, 1)`, or the noise options cannot combine.
"""
super()._check()
self._check_functions()
self._check_noise()
def _evaluate(self, function: Callable[[SimpleNamespace], object] | float, namespace: SimpleNamespace) -> object:
"""Return the value of a function of the namespace, or a number as it is."""
return function(namespace) if callable(function) else function
def _broadcast(self, value: object, state: object) -> object:
"""Return a value with one component per component of a vector state, repeating a number."""
if isinstance(state, FreeModuleElement) and not isinstance(value, FreeModuleElement | list | tuple):
return vector(SR, [value] * len(state))
if isinstance(value, list | tuple):
return vector(SR, value)
return value
def _divide(self, value: object, namespace: SimpleNamespace) -> object:
"""Return a value divided by the divisor of the target, or the value without a divisor."""
if self.divisor is None:
return value
divisor = (
self.divisor(self._get_member(namespace, 'target'))
if callable(self.divisor)
else getattr(self._get_member(namespace, 'target'), self.divisor)
)
return operator.truediv(value, divisor)
def _get_rate(self, namespace: SimpleNamespace) -> object:
"""Return the rate before the divisor, the hook a subclass overrides to derive it from its parameters."""
return self._evaluate(self.rate, namespace)
def _get_noise(self, namespace: SimpleNamespace) -> object:
"""Return the noise amplitude before the divisor, the hook a subclass overrides, or `None` for no noise."""
return None if self.noise is None else self._evaluate(self.noise, namespace)
def _get_noise_law(self, state: object, amplitude: object, namespace: SimpleNamespace) -> BasePart:
"""Return the noise law of an amplitude, by the process, the convention, and the Hurst index."""
if self.hurst is not None:
hurst = self._evaluate(self.hurst, namespace)
return FractionalNoiseLaw(state, amplitude, hurst=hurst, convention=self.convention)
process = None
if self.process is not None:
handle = self._get_member(namespace, 'process').dW
process = handle if self.index is None else handle[self.index]
return NoiseLaw(state, amplitude, convention=self.convention, process=process, conserved=self.conserved)
def _get_laws(self, namespace: SimpleNamespace) -> list[BasePart]:
"""Return the rate law of the state and its noise law, both divided by the divisor.
Args:
namespace:
The namespace of the symbols the laws can use, with the namespaces of the target and the sources.
Returns:
laws:
The rate law, and the noise law when the interaction has noise.
"""
state = getattr(self._get_member(namespace, 'target'), self.state)
rate = self._divide(self._broadcast(self._get_rate(namespace), state), namespace)
laws: list[BasePart] = [RateLaw(state, rate)]
noise = self._get_noise(namespace)
if noise is not None:
laws.append(self._get_noise_law(state, self._divide(noise, namespace), namespace))
return laws
Equations
nodes.theta[i]: diff(theta_nodes(t), t) == 2.0*sqrt(_reduced_0^2 + _reduced_1^2)*_neighbours_1 + omega_nodes
links.theta[i]: diff(theta_links(t), t) == 2.0*sqrt(_reduced_2^2 + _reduced_3^2)*_neighbours_5 - 2.0*_neighbours_4 + omega_links