Complex agents with internal states and standardization
Complex agents whose interactions make them standardized, all showing one characteristic, and ordered.
- Name:
standardized_agents - Field:
social/opinion_dynamics - Formalism: agent-based model
- References: Bertin 2021; Bertin and Jensen 2019
Factory
@register_model(
field='social/opinion_dynamics',
tags=('agents', 'standardization', 'phase transition', 'large deviations', 'mean field'),
references=('Bertin 2021', 'Bertin and Jensen 2019'),
defaults={'until': 1000},
observables=lambda system: [
ObservableSpec('q', group='agents', variable='spin', reduce=_standardization),
ObservableSpec('m', group='agents', variable='spin', reduce='mean'),
],
)
def standardized_agents(
agents: int = 100,
*,
K: float = 10.0,
J: float = 1.0,
T: float = 0.5,
H: int = 50,
standardized: float = 1.0,
random_state: int | np.random.Generator | None = 0,
) -> System:
"""Return complex agents whose interactions make them standardized, all showing one characteristic, and ordered.
Below the temperature `T_S = K / (2 ln(H / 2))` the most probable share of agents with the characteristic jumps
from `q ~ 0` to `q ~ 1`, a discontinuous transition, and below `T_M = J` the standardized agents order, with a
magnetization `m` of either sign. Both states are long-lived on either side of `T_S` in large groups, so the
initial share `standardized` selects the branch that the run follows.
Args:
agents:
The number of agents `N`.
K:
The coupling of the characteristics `|S_i|`, which standardizes the agents.
J:
The coupling of the spins `S_i`, which orders the standardized agents.
T:
The temperature, the randomness of the choices.
H:
The number of configurations without the characteristic.
standardized:
The initial share of agents with the characteristic, each with a random spin. The others start in random
configurations without it.
random_state:
The seed or generator of the initial configurations.
Returns:
system:
The model, with the group `agents`, the share `q` of agents with the characteristic, and the magnetization
`m`, the mean spin.
Examples:
>>> from skcomplex.models.social.opinion_dynamics import standardized_agents
>>> standardized_agents(10, standardized=0.0).elements[0].values['spin'].tolist()
[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
"""
rng = np.random.default_rng(random_state)
marked = rng.random(agents) < standardized
configurations = np.where(marked, rng.integers(MARKED, size=agents), rng.integers(MARKED, H + MARKED, size=agents))
configurations = configurations.astype(float)
return System(
elements=[
GroupElement(
'agents',
element=ComplexAgentElement,
size=agents,
values={'configuration': configurations, 'spin': StandardizationInteraction._spins_of(configurations)},
),
],
interactions=[
StandardizationInteraction('standardization', target='agents', K=K, J=J, H=H, intensity=1 / T, phase=PHASE),
MapInteraction(
'spin',
target='agents',
state='spin',
function=_spin_of,
schedule='member',
form='agent',
phase=PHASE,
),
],
time=DiscreteTime(schedule=PhasesScheduleSpec([PhaseSpec(PHASE, activation='visits', groups=('agents',))])),
)
Components
class ComplexAgentElement(Element):
"""Agent with many internal configurations, of which two show a characteristic with the spin `1` or `-1`.
Configuration 0 has the spin `S = 1`, configuration 1 the spin `S = -1`, and the configurations from 2 to `H + 1`
the spin 0. In `standardized_agents`, a `MapInteraction` sets the spin after every change of configuration.
Args:
label:
The label of the agent or group.
visual:
How the agent is drawn, or `None` for `default_visual_`.
configuration:
The internal configuration, from 0 to `H + 1`.
spin:
The spin of the configuration.
Examples:
>>> from skcomplex.core.elements import GroupElement
>>> from skcomplex.models.social.opinion_dynamics import ComplexAgentElement
>>> GroupElement(
... 'agents', element=ComplexAgentElement, size=10, values={'configuration': 5.0, 'spin': 0.0}
... ).size
10
"""
def __init__(
self,
label: str,
visual: VisualSpec | None = None,
*,
configuration: Annotated[float, StateSpec(domain=NN)] = 2.0,
spin: Annotated[float, StateSpec(domain=ZZ)] = 0.0,
) -> None:
super().__init__(label, visual=visual)
self.configuration = configuration
self.spin = spin
@classmethod
def _get_view(cls) -> type[ComplexAgentView]:
"""Return the class of the views of the agents, whose attributes are the states and parameters."""
return ComplexAgentView
class StandardizationInteraction(ChoiceInteraction):
"""Changes of configuration of random agents, accepted with the logistic probability of their gain in utility.
The utility of agent `i` is `u_i = K / N sum_j |S_i| |S_j| + J / N sum_j S_i S_j` over the other agents. An agent
proposes a configuration drawn uniformly from the `H + 2` configurations, its own included, and takes it with the
probability `1 / (1 + exp(-beta (u_i' - u_i)))` of the rule `'fermi'`, so the agents sample `P ~ exp(beta E)`. The
utilities read the counts of the configurations of the group, so every proposal sees the changes accepted before
it. In `standardized_agents`, a phase of visits draws `N` agents with replacement every unit of time.
Args:
label:
The label of the interaction.
target:
The path of the group of agents.
K:
The coupling of the characteristics `|S_i|`, which standardizes the agents.
J:
The coupling of the spins `S_i`, which orders the standardized agents.
H:
The number of configurations without the characteristic.
intensity:
The intensity of choice `beta`, the inverse of the temperature.
activity:
The probability that a visited agent proposes a configuration.
phase:
The phase of the step the choice runs in, or `None` for the default phase.
Examples:
>>> from skcomplex.models.social.opinion_dynamics import StandardizationInteraction
>>> StandardizationInteraction('standardization', target='agents', H=100).check().options[-1]
101.0
"""
def __init__(
self,
label: str,
*,
target: Annotated[str, RoleSpec()],
K: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 10.0,
J: Annotated[float, ParameterSpec()] = 1.0,
H: int = 50,
intensity: float = 2.0,
activity: float = 1.0,
phase: str | None = None,
) -> None:
super().__init__(
label,
target=target,
state='configuration',
options=tuple(float(option) for option in range(H + MARKED)),
utility=self._get_utility,
rule='fermi',
intensity=intensity,
schedule='member',
activity=activity,
form='agent',
phase=phase,
proposal='all',
)
self.K = K
self.J = J
self.H = H
def _get_utility(self, agent: ComplexAgentView, model: ModelViewProtocol, option: float) -> float:
"""Return the utility of a configuration for an agent, from the counts of the configurations of the others."""
counts = model.composition('target', 'configuration')
up, down = counts.get(0.0, 0), counts.get(1.0, 0)
spin, new = float(StandardizationInteraction._spins_of(np.array(agent.configuration))), float(
StandardizationInteraction._spins_of(np.array(option)),
)
standardizing = model.parameter('K') * abs(new) * (up + down - abs(spin))
ordering = model.parameter('J') * new * (up - down - spin)
return (standardizing + ordering) / sum(counts.values())
@staticmethod
def _spins_of(configurations: Array) -> Array:
"""Return the spin of each configuration: 1 for configuration 0, -1 for configuration 1, and 0 otherwise."""
return np.select([configurations == 0, configurations == 1], [1.0, -1.0], 0.0)
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, model)` 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 model 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, model)` 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.elements import Element
>>> from skcomplex.core.interactions import MapInteraction
>>> from skcomplex.core.spaces import DiscreteTime
>>> from skcomplex.core.system import System
>>> from skcomplex.engine.runtime import Simulation
>>> from skcomplex.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).check().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.check().prepare().run(SimpleNamespace()).laws_[0]).__name__
'RuleLaw'
>>> from skcomplex.core.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])).check().prepare().run(until=2)
>>> trajectory.trajectory_.agents('weights', 2).values['w'].tolist()
[[4.0, 8.0]]
"""
def __init__(
self,
label: str,
*,
target: Annotated[str, RoleSpec()],
state: str,
function: Callable[..., object],
sources: Annotated[Sequence[str], RoleSpec(variadic=True)] = (),
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, model: ModelViewProtocol) -> 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, model), 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, model: ModelViewProtocol) -> 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, model)
found = self.function(agent, model)
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, model: ModelViewProtocol) -> None:
"""Set the state of one member to its next value."""
if agent.alive:
self._set_value(agent, self._get_value(agent, model))
def update(self, model: ModelViewProtocol) -> None:
"""Set the state of every agent of the target to its next value, by the schedule.
Args:
model:
The model view of the interaction.
"""
self._update_agents(model, 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())]