Cascades of failures in interdependent networks
The cascade of failures of Buldyrev et al. between two networks whose nodes depend on each other.
- Name:
interdependent_network_cascade - Field:
mathematics/networks - Formalism: discrete map
- References: Buldyrev et al. 2010; Artime et al. 2025
Select this system
from skcomplex.systems import create_simulation, create_system, describe
system = create_system('interdependent_network_cascade')
entry = describe('interdependent_network_cascade')
trajectory = create_simulation('interdependent_network_cascade').prepare().run().trajectory_
Factory
@register_system(
field='mathematics/networks',
tags=('network', 'interdependent networks', 'cascade', 'percolation', 'multilayer'),
references=('Buldyrev et al. 2010', 'Artime et al. 2025'),
defaults={'until': 40},
)
def interdependent_network_cascade(
n_nodes: int = 2000,
*,
mean_degree: float = 4.0,
p: float = 0.8,
random_state: int | np.random.Generator | None = 0,
) -> System:
"""Return the cascade of failures of Buldyrev et al. between two networks whose nodes depend on each other.
Two Erdős-Rényi networks of the mean degree `mean_degree` share their nodes, every node of one depending on the
node of the same index in the other, so a node is one pair of dependent nodes. An attack leaves every node
functional with probability `p`. At even steps the nodes outside the giant component of the functional nodes of
the first network fail, and at odd steps those outside that of the second network, until the cascade stops at the
mutual giant component (Buldyrev et al. 2010, as Artime et al. 2025, section 5, Eqs. 9 to 11, review it). Its
share `S` solves `S = p (1 - exp(-k S))²`, which has a solution above 0 only for `p` above `2.4554 / k`, where the
networks collapse at once in a discontinuous transition.
Args:
n_nodes:
The number of nodes of every network.
mean_degree:
The mean degree `k` of every network.
p:
The share of nodes the attack leaves functional.
random_state:
The seed or generator of the networks and the attack.
Returns:
system:
The system, with the functional nodes `nodes.active` over the graph of the first network and the layer
`dependent` of the second.
Examples:
>>> from skcomplex.core.engine.runtime import Simulation
>>> from skcomplex.systems.mathematics.networks import interdependent_network_cascade
>>> system = interdependent_network_cascade(200, p=0.3)
>>> int(Simulation(system).prepare().run(until=20).trajectory_.values('nodes.active')[-1].sum())
0
"""
rng = np.random.default_rng(random_state)
probability = mean_degree / (n_nodes - 1)
first = nx.gnp_random_graph(n_nodes, probability, seed=int(rng.integers(2**31)))
second = LayerSpec('dependent', nx.gnp_random_graph(n_nodes, probability, seed=int(rng.integers(2**31))))
space = NetworkSpace(first, layers=[second])
functional = (rng.uniform(size=n_nodes) < p).astype(float)
nodes = LatticeElement('nodes', element=DependentNodeElement, space=space, values={'active': functional})
def _get_cascade(s: SimpleNamespace) -> object:
"""Return the activity of a node after the failures of the network of the step."""
both = s.target.active * s.target.other.active
first_giant = components(both, reduce='giant')
second_giant = components(both, over=second, reduce='giant')
return s.target.active * where(mod(s.t, 2) < 1, first_giant, second_giant)
cascade = MapInteraction('cascade', target='nodes', state='active', function=_get_cascade)
return System(elements=[nodes], interactions=[cascade], time=DiscreteTime())
Components
Elements:
DependentNodeElement, fromElement
Interactions:
MapInteraction
class DependentNodeElement(Element):
"""Node of two interdependent networks, functional while it belongs to the giant component of both.
The node stands for a pair of nodes, one of each network, that depend on each other, so it fails when either of
them leaves the giant component of the functional nodes of its network.
Args:
label:
The label of the node group.
visual:
How the element is drawn, or `None` for `default_visual_`.
active:
Whether the node is functional, 1 or 0.
Examples:
>>> from skcomplex.systems.mathematics.networks import DependentNodeElement
>>> DependentNodeElement('nodes', active=0.0).active
0.0
"""
def __init__(
self,
label: str,
visual: VisualSpec | None = ACTIVITY_VISUAL,
*,
active: Annotated[float, StateSpec(domain=FiniteEnumeratedSet([0, 1]))] = 1.0,
) -> None:
super().__init__(label, visual=visual)
self.active = active
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
nodes.active[i]: active_nodes(t + 1) == active_nodes(t)*where(mod(t, 2) < 1, _neighbours_1, _neighbours_0)