Self-organized branching process
The self-organized branching process of Zapperi, Lauritsen, and Stanley, a mean-field sandpile.
- Name:
self_organized_branching - Field:
physics/criticality - Formalism: discrete map
- References: Zapperi, Lauritsen, and Stanley 1995; Jensen 2023
Select this system
from skcomplex.systems import create_simulation, create_system, describe
system = create_system('self_organized_branching')
entry = describe('self_organized_branching')
trajectory = create_simulation('self_organized_branching').prepare().run().trajectory_
Factory
@register_system(
field='physics/criticality',
tags=('branching process', 'avalanches', 'mean field', 'self-organized criticality', 'stochastic'),
references=('Zapperi, Lauritsen, and Stanley 1995', 'Jensen 2023'),
defaults={'until': 30000, 'dt': 1},
)
def self_organized_branching(p: float = 0.3, *, generations: int = 10) -> System:
"""Return the self-organized branching process of Zapperi, Lauritsen, and Stanley, a mean-field sandpile.
An avalanche starts from one active site at the top of a binary tree of `generations` generations. Every active
site relaxes with the probability `p` into two active sites of the next generation, and stays still otherwise, so
every generation has `2 Binomial(z, p)` active sites. The `sigma` active sites that reach the last generation leave
the system as dissipation. After every avalanche one unit of energy has entered and `sigma` have left, so the
probability changes by `(1 - sigma) / N` with `N = 2^(generations + 1) - 1` sites. A supercritical `p` makes large
dissipating avalanches that lower it, and a subcritical one lets energy accumulate, so `p` organizes itself to the
critical value `p_c = 1 / 2`, where the branching ratio `2 p` is 1. The process advances one generation per step,
so a step is a time of the avalanche and avalanches follow each other without pause.
Args:
p:
The initial branching probability.
generations:
The number of generations of the tree, at least 1.
Returns:
system:
The system, with the probability `tree.p`, the active sites `tree.z`, and the generation `tree.g`.
Examples:
>>> from skcomplex.systems.physics.criticality import self_organized_branching
>>> self_organized_branching(0.4).elements[0].p
0.4
"""
return System(
elements=[BranchingTreeElement('tree', p=p)],
interactions=[
MapInteraction('branching', target='tree', state='z', function=partial(_branch, generations)),
MapInteraction('generation', target='tree', state='g', function=partial(_advance, generations)),
MapInteraction('tuning', target='tree', state='p', function=partial(_tune, generations)),
],
time=DiscreteTime(),
)
Components
Elements:
BranchingTreeElement, fromElement
Interactions:
MapInteraction
class BranchingTreeElement(Element):
"""Self-organized branching process: avalanches on a binary tree whose branching probability the avalanches tune.
The probability `p` that an active site relaxes into two active sites of the next generation, the number `z` of
active sites of the current generation of the avalanche, and its generation `g`. Map interactions advance the
avalanche by one generation per step.
Args:
label:
The label of the tree.
visual:
How the tree is drawn, or `None` for `default_visual_`.
p:
The initial branching probability.
z:
The initial number of active sites.
g:
The initial generation.
Attributes:
default_visual_:
The visual of the tree when it sets none.
Examples:
>>> from skcomplex.systems.physics.criticality import BranchingTreeElement
>>> BranchingTreeElement('tree').p
0.3
"""
def __init__(
self,
label: str,
visual: VisualSpec | None = None,
*,
p: Annotated[float, StateSpec(domain=RealSet.closed(0, 1))] = 0.3,
z: Annotated[float, StateSpec(domain=NN)] = 1,
g: Annotated[float, StateSpec(domain=NN)] = 0,
) -> None:
super().__init__(label, visual=visual)
self.p = p
self.z = z
self.g = g
class MapInteraction(Interaction):
"""Map of a state of discrete time, whose next value is a function, or the state plus it.
In the mode `'replace'` the next value of the state is `f`, and in the mode `'increment'` it is the state plus `f`.
Under the `'synchronous'` schedule every member of a bound group updates at once, from the states at the start of
the step. Under the `'sequential'` schedule the members update one after the other, in their order, and under the
`'random-sequential'` schedule in a new random order every step, each member reading the values the members before
it set. Under the `'member'` schedule of the agent form, every member updates in its own activation, together with
the rules of its element and the other member interactions, in the order the system lists them.
In the `'symbolic'` form, `function(namespace)` returns a symbolic expression, or a vector of them for a vector
state, where `namespace.target` holds the states and parameters of the target, `namespace.sources` the namespaces
of the sources, and `namespace.t` the time, and may hold draws such as `normal('noise')`. In the `'agent'` form,
the target is an agent group and `function(agent, system)` returns the value as a number, or as an array of the
shape of a vector state, so it can read the agent, its neighbours, and the history and storage of the system view.
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 map sets.
state:
The name of the state of the target.
function:
The function `f(namespace)` in the symbolic form, or `f(agent, system)` in the agent form.
sources:
The paths of the elements, or members of groups such as `cells[0]`, that the function reads.
mode:
`'replace'` or `'increment'`.
schedule:
`'synchronous'`, `'sequential'`, `'random-sequential'`, or `'member'` for the agent form.
form:
`'symbolic'` for update laws of the states of the target, or `'agent'` for a rule over an agent group.
phase:
The phase of the step the agent form runs in, or `None` for the default phase.
Raises:
ValidationError:
The mode, the schedule, or the form is unknown, the symbolic form has the schedule `'member'`, the
function is not callable, or the agent form returns a value that does not match the state, raised by
`run` of the simulation.
Examples:
>>> from skcomplex.core.components.elements import Element
>>> from skcomplex.core.components.interactions import MapInteraction
>>> from skcomplex.core.components.spaces import DiscreteTime
>>> from skcomplex.core.components.system import System
>>> from skcomplex.core.engine.runtime import Simulation
>>> from skcomplex.core.specs import StateSpec
>>> from typing import Annotated, SupportsFloat
>>> class ValueElement(Element):
... def __init__(self, label, x: Annotated[float, StateSpec()] = 0.5):
... super().__init__(label)
... self.x = x
>>> logistic = MapInteraction(
... 'map', target='value', state='x', function=lambda s: 3 * s.target.x * (1 - s.target.x)
... )
>>> system = System(elements=[ValueElement('value')], interactions=[logistic], time=DiscreteTime())
>>> Simulation(system).prepare().run(until=2).trajectory_.values('value.x').tolist()
[0.5, 0.75, 0.5625]
>>> agents = MapInteraction('map', target='walkers', state='x', function=lambda a, m: a.x + 1, form='agent')
>>> type(agents.prepare().run(SimpleNamespace()).laws_[0]).__name__
'RuleLaw'
>>> from skcomplex.core.components.elements import GroupElement
>>> class WeightsElement(Element):
... def __init__(self, label, visual=None, *, w: Annotated[object, StateSpec(vector=True)] = 0.0):
... super().__init__(label, visual=visual)
... self.w = w
>>> group = GroupElement('weights', element=WeightsElement, size=1, values={'w': [1.0, 2.0]})
>>> doubling = MapInteraction(
... 'double', target='weights', state='w', function=lambda a, m: [2 * x for x in a.w], form='agent'
... )
>>> trajectory = Simulation(System(elements=[group], interactions=[doubling])).prepare().run(until=2)
>>> trajectory.trajectory_.agents('weights', 2).values['w'].tolist()
[[4.0, 8.0]]
"""
law_classes: ClassVar[tuple[type[BaseLaw], ...]] = (RuleLaw, UpdateLaw)
def __init__(
self,
label: str,
*,
target: Annotated[str, RoleSpec(elements=(ElementKind.ANY,), states=('state',))],
state: str,
function: Callable[..., object],
sources: Annotated[Sequence[str], RoleSpec(variadic=True, elements=(ElementKind.ANY,))] = (),
mode: str = 'replace',
schedule: str = 'synchronous',
form: str = 'symbolic',
phase: str | None = None,
) -> None:
super().__init__(label)
self.target = target
self.state = state
self.function = function
self.sources = sources
self.mode = mode
self.schedule = schedule
self.form = form
self.phase = phase
def _check(self) -> None:
"""Raise an error for an unknown mode, schedule, or form, or a function that cannot be called.
Raises:
ValidationError:
The mode, the schedule, or the form is unknown, the symbolic form has the schedule `'member'`, or the
function is not callable.
"""
super()._check()
self._check_variant('mode', self.mode, MODES, (BaseSpec,))
for name, value, variants in (
('schedule', self.schedule, SCHEDULES),
('form', self.form, FORMS),
):
if value not in variants:
error_msg = f'The {name} of map {self.label!r} is {value!r}. Use one of {", ".join(variants)}.'
raise ValidationError(error_msg)
if self.schedule == 'member' and self.form != 'agent':
error_msg = (
f'The map {self.label!r} updates every member in its activation, which only the agent form does.'
)
raise ValidationError(error_msg)
if not callable(self.function):
error_msg = f'The function of map {self.label!r} must be callable, but it is {self.function!r}.'
raise ValidationError(error_msg)
def _apply_mode(self, current: Value, value: Value) -> Value:
"""Return the next value of a state from its current value and the value of the function, by the mode."""
if self.mode == 'increment':
return current + value
if self.mode == 'replace':
return value
raise self._get_variant_error(self.mode, 'mode', '_apply_mode')
def _get_vector(self, agent: AgentViewProtocol, system: SystemViewProtocol) -> Value:
"""Return the next value of a vector state of an agent."""
current = np.asarray(getattr(agent, self.state), dtype=np.float64)
value = np.asarray(self.function(agent, system), dtype=np.float64)
if value.shape != current.shape:
error_msg = (
f'The function of map {self.label!r} returns the shape {value.shape} for the state {self.state!r} of '
f'the shape {current.shape}. Return an array of the shape of the state.'
)
raise ValidationError(error_msg)
return self._apply_mode(current, value)
def _get_value(self, agent: AgentViewProtocol, system: SystemViewProtocol) -> Value:
"""Return the next value of the state of an agent, a number or the array of a vector state."""
if np.ndim(getattr(agent, self.state)) > 0:
return self._get_vector(agent, system)
found = self.function(agent, system)
if not isinstance(found, SupportsFloat):
error_msg = f'The function of map {self.label!r} returns {found!r} for an agent. Return a number.'
raise ValidationError(error_msg)
value = float(found)
return self._apply_mode(float(getattr(agent, self.state)), value)
def _set_value(self, agent: AgentViewProtocol, value: Value) -> None:
"""Set the state of an agent to its next value."""
setattr(agent, self.state, value)
def _update_member(self, agent: AgentViewProtocol, system: SystemViewProtocol) -> None:
"""Set the state of one member to its next value."""
if agent.alive:
self._set_value(agent, self._get_value(agent, system))
def update(self, system: SystemViewProtocol) -> None:
"""Set the state of every agent of the target to its next value, by the schedule.
Args:
system:
The system view of the interaction.
"""
self._update_agents(system, self.schedule, self._get_value, self._set_value)
def _get_order(self) -> object:
"""Return the order of the members of the symbolic form: none, their own order, or a uniform draw."""
if self.schedule == 'synchronous':
return None
if self.schedule == 'sequential':
return SR(0)
return uniform(re.sub(r'\W', '_', f'order_{self.label}'))
def _get_laws(self, namespace: SimpleNamespace) -> list[BasePart]:
"""Return the update of the state to its next value, or the rule of the agent form.
Args:
namespace:
The namespace of the symbols the laws can use, with `namespace.target` and `namespace.sources`.
Returns:
laws:
The update law of the state, or the rule that runs the map of the agents.
"""
if self.form == 'agent' and self.schedule == 'member':
return [RuleLaw(self._update_member, phase=self.phase, members=self._get_role('target'))]
if self.form == 'agent':
return [RuleLaw(self.update, phase=self.phase)]
value = getattr(self._get_member(namespace, 'target'), self.state)
mapped = cast(Value, self.function(namespace))
next_value = self._apply_mode(value, mapped)
return [UpdateLaw(value, next_value, increment=self.mode == 'increment', order=self._get_order())]
Equations
tree.p: p_tree(t + 1) == where(either(g_tree(t) >= 10, z_tree(t) == 0), p_tree(t) - 1/2047*where(g_tree(t) >= 10, z_tree(t), 0) + 1/2047, p_tree(t))
tree.z: z_tree(t + 1) == where(either(g_tree(t) >= 10, z_tree(t) == 0), 1, 2*draw_children_branching)
tree.g: g_tree(t + 1) == where(either(g_tree(t) >= 10, z_tree(t) == 0), 0, g_tree(t) + 1)