Join-a-club spins
Join-a-club spins, agents who join a club with their acquaintances while acquaintances follow the club.
- Name:
join_a_club - Field:
social/opinion_dynamics - Formalism: discrete map
- References: Thurner et al. 2018; Biely, Hanel, and Thurner 2009
Factory
@register_model(
field='social/opinion_dynamics',
tags=('opinion', 'spins', 'clubs', 'adaptive network', 'heat bath'),
references=('Thurner et al. 2018', 'Biely, Hanel, and Thurner 2009'),
defaults={'until': 150},
)
def join_a_club(
n_agents: int = 200,
*,
mean_degree: float = 6.0,
beta: float = 0.5,
appeal: float = 0.0,
rewiring: float = 0.5,
random_state: int | np.random.Generator | None = 0,
) -> System:
"""Return join-a-club spins, agents who join a club with their acquaintances while acquaintances follow the club.
Agents of an Erdős-Rényi graph are in or out of a club, a spin, and choose by a heat bath with the inverse
temperature `beta` in the field of their acquaintances and of the appeal of the club. Acquaintances across the
club boundary dissolve and reform among agents on the same side. Without coupling the choices stay random and
many acquaintances cross the boundary, and with strong coupling the members and the outsiders separate into
communities with few acquaintances between them.
Args:
n_agents:
The number of agents.
mean_degree:
The mean number of acquaintances.
beta:
The inverse temperature of the heat bath.
appeal:
The field of the club on every agent.
rewiring:
The probability that a reconsidered acquaintance across the club boundary reforms on one side.
random_state:
The seed or generator of the graph and of the uniform random choices.
Returns:
system:
The model, with the groups `voters` and `ties` and the interaction `separation`.
Examples:
>>> from skcomplex.models.social.opinion_dynamics import join_a_club
>>> join_a_club(30).elements[0].element.__name__
'ClubMemberElement'
"""
rng = np.random.default_rng(random_state)
space = NetworkSpace(nx.gnp_random_graph(n_agents, mean_degree / (n_agents - 1), seed=rng))
opinions = (rng.random(n_agents) < FIRST_OPINION_SHARE).astype(float)
choice = MapInteraction('choice', target='voters', state='opinion', function=_choose_club)
return _coevolving(ClubMemberElement, space, opinions, choice, beta=beta, appeal=appeal, rewiring=rewiring)
Components
class ClubMemberElement(Element):
"""Agent that is in a club (spin 1) or out of it (spin 0), aligned by a heat bath with the agents it knows.
Every step an agent reconsiders with probability `deciding` and then joins with probability
`1 / (1 + exp(-2 β h))`, where the field `h` counts the acquaintances in the club minus those out of it, plus the
appeal of the club. The interaction `SeparationInteraction` dissolves acquaintances between a member and an outsider
and reforms them towards an agent of the same side.
Args:
label:
The label of the agent or group.
visual:
How the agent is drawn, or `None` for `default_visual_`.
opinion:
1 for a member of the club and 0 for an outsider.
degree:
The number of acquaintances, read at the start of every step.
beta:
The inverse temperature of the heat bath.
appeal:
The field of the club on every agent, positive for an attractive club.
deciding:
The probability that an agent reconsiders its choice in a step.
activity:
The probability that an acquaintance is reconsidered in a step.
rewiring:
The probability that a reconsidered acquaintance across the club boundary dissolves and reforms.
Examples:
>>> from skcomplex.models.social.opinion_dynamics import ClubMemberElement
>>> ClubMemberElement('agents', beta=2.0).beta
2.0
"""
def __init__(
self,
label: str,
visual: VisualSpec | None = OPINION_VISUAL,
*,
opinion: Annotated[float, StateSpec(domain=RealSet.closed_open(0, oo))] = 0.0,
degree: Annotated[float, StateSpec(domain=RealSet.closed_open(0, oo))] = 0.0,
beta: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 1.0,
appeal: Annotated[float, ParameterSpec()] = 0.0,
deciding: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.1,
activity: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.5,
rewiring: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.5,
) -> None:
super().__init__(label, visual=visual)
self.opinion = opinion
self.degree = degree
self.beta = beta
self.appeal = appeal
self.deciding = deciding
self.activity = activity
self.rewiring = rewiring
class TieElement(Element):
"""Edge of a coevolving network, which holds no state of its own.
Args:
label:
The label of the edge group.
Examples:
>>> from skcomplex.models.social.opinion_dynamics import TieElement
>>> TieElement('ties').label
'ties'
"""
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())]
class SeparationInteraction(TopologyInteraction):
"""Rewiring of discordant edges by one end, which keeps the edge and moves its other end to a like-minded node.
It is a `TopologyInteraction` that rewires the edges of the voters. The end `keep` acts on an edge with
probability `activity / degree` per step, and when it holds the opinion `opinion` and the other end does not, it
rewires the edge with probability `rewiring`, drawing a new end among the voters of the opinion `opinion`.
Args:
label:
The label of the interaction.
voters:
The path of the voters on the nodes.
keep:
The end of an edge that acts and keeps the edge, `'source'` or `'target'`.
opinion:
The opinion of the end that acts, 0 or 1.
Examples:
>>> from skcomplex.models.social.opinion_dynamics import SeparationInteraction
>>> SeparationInteraction('separation', voters='voters', keep='target', opinion=0).check().action
'rewire'
"""
def __init__(
self,
label: str,
*,
voters: Annotated[str, RoleSpec()],
keep: str = 'source',
opinion: float = 1.0,
) -> None:
super().__init__(
label,
target=voters,
action='rewire',
probability=self._get_chance,
toward=self._get_weight,
keep=keep,
)
self.voters = voters
self.keep = keep
self.opinion = opinion
def _get_weight(self, node: SimpleNamespace) -> object:
"""Return the weight of a candidate node as a new end, 1 when it holds the opinion of the acting end."""
return node.opinion if self.opinion == 1 else 1 - node.opinion
def _get_chance(self, namespace: SimpleNamespace) -> object:
"""Return the probability that the acting end rewires the edge, when the ends hold different opinions."""
actor, other = (namespace.source, namespace.target)[:: 1 if self.keep == 'source' else -1]
chance = actor.activity * actor.rewiring / (actor.degree + SMALLEST_DEGREE)
if self.opinion == 1:
return chance * actor.opinion * (1 - other.opinion)
return chance * (1 - actor.opinion) * other.opinion
Equations
voters.opinion[i]: opinion_voters(t + 1) == -(active_voters_topology - 1)*opinion_voters(t) + active_voters_topology*where(uniform_d_choice < deciding_voters, where(uniform_u_choice < (1/(e^(-2*(_neighbours_1 + appeal_voters)*beta_voters) + 1)), 1, 0), opinion_voters(t))
voters.degree[i]: degree_voters(t + 1) == _neighbours_0*active_voters_topology - (active_voters_topology - 1)*degree_voters(t)