Edge Stochastic

Edge Stochastic compartments are used to evaluate stochastic events attached to network edges.

Consider the transition rule Susceptible->Infected that, to be triggered, requires a direct link among an infected node and a susceptible one. Moreover, it can happens subject to probability beta, a parameter tied to the specific edge connecting the two nodes. Such rule can be described by a simple compartment that models Edge Stochastic behaviors. Let’s call il ES.

The rule will take as input the initial node status (Susceptible), the final one (Infected) and the ES compartment. ES will thus require a probability (beta) of edge activation and a triggering status. In advanced scenarios, where the probability threshold vary from edge to edge, it is possible to specify it using the model configuration object.

During each rule evaluation, given a node n and one of its neighbors m

  • if the actual status of n equals the rule initial one and the one of m equals the triggering one
    • a random value b in [0,1] will be generated
    • if b <= beta then ES is considered satisfied and the status of n changes from initial to final.

Parameters

Name Value Type Default Mandatory Description
threshold float in [0, 1] 1/N True Event probability
triggering_status string None False Trigger

Where N is the number of nodes in the graph.

Example

In the code below is shown the formulation of a Cascade model using EdgeStochastic compartments.

The compartment, c1, is used to implement the transition rule Susceptible->Infected. It requires a probability threshold - here set equals to 0.02 - and restrain the rule evaluation to all those nodes that have at least an Infected neighbors.

import networkx as nx
import ndlib.models.ModelConfig as mc
import ndlib.models.CompositeModel as gc
import ndlib.models.compartments.EdgeStochastic as es

# Network generation
g = nx.erdos_renyi_graph(1000, 0.1)

# Composite Model instantiation
model = gc.CompositeModel(g)

# Model statuses
model.add_status("Susceptible")
model.add_status("Infected")
model.add_status("Removed")

# Compartment definition
c1 = ns.EdgeStochastic(0.02, triggering_status="Infected")

# Rule definition
model.add_rule("Susceptible", "Infected", c1)

# Model initial status configuration
config = mc.Configuration()
config.add_model_parameter('percentage_infected', 0.1)

# Simulation execution
model.set_initial_status(config)
iterations = model.iteration_bunch(100)

In case of an heterogeneous edge threshold distribution the same model can be expressed as follows

import networkx as nx
import ndlib.models.ModelConfig as mc
import ndlib.models.CompositeModel as gc
import ndlib.models.compartments.EdgeStochastic as es

# Network generation
g = nx.erdos_renyi_graph(1000, 0.1)

# Composite Model instantiation
model = gc.CompositeModel(g)

# Model statuses
model.add_status("Susceptible")
model.add_status("Infected")
model.add_status("Removed")

# Compartment definition
c1 = es.EdgeStochastic(triggering_status="Infected")

# Rule definition
model.add_rule("Susceptible", "Infected", c1)


# Model initial status configuration
config = mc.Configuration()

# Threshold specs
for e in g.edges():
        config.add_edge_configuration("threshold", e, np.random.random_sample())

config = mc.Configuration()
config.add_model_parameter('percentage_infected', 0.1)

# Simulation execution
model.set_initial_status(config)
iterations = model.iteration_bunch(100)