Threshold

The Threshold model was introduced in 1978 by Granovetter [1].

In this model during an epidemics, a node has two distinct and mutually exclusive behavioral alternatives, e.g., the decision to do or not do something, to participate or not participate in a riot.

Node’s individual decision depends on the percentage of its neighbors have made the same choice, thus imposing a threshold.

The model works as follows: - each node has its own threshold; - during a generic iteration every node is observed: iff the percentage of its infected neighbors is grater than its threshold it becomes infected as well.

Statuses

During the simulation a node can experience the following statuses:

Name Code
Susceptible 0
Infected 1

Parameters

Name Type Value Type Default Mandatory Description
threshold Node float in [0, 1] 0.1 False Individual threshold

The initial infection status can be defined via:

  • percentage_infected: Model Parameter, float in [0, 1]
  • Infected: Status Parameter, set of nodes

The two options are mutually exclusive and the latter takes precedence over the former.

Methods

The following class methods are made available to configure, describe and execute the simulation:

Configure

class ndlib.models.epidemics.ThresholdModel.ThresholdModel(graph)
Node Parameters to be specified via ModelConfig
Parameters:threshold – The node threshold. If not specified otherwise a value of 0.1 is assumed for all nodes.
ThresholdModel.__init__(graph)

Model Constructor

Parameters:graph – A networkx graph object
ThresholdModel.set_initial_status(self, configuration)

Set the initial model configuration

Parameters:configuration – a `ndlib.models.ModelConfig.Configuration` object
ThresholdModel.reset(self)

Reset the simulation setting the actual status to the initial configuration.

Describe

ThresholdModel.get_info(self)

Describes the current model parameters (nodes, edges, status)

Returns:a dictionary containing for each parameter class the values specified during model configuration
ThresholdModel.get_status_map(self)

Specify the statuses allowed by the model and their numeric code

Returns:a dictionary (status->code)

Execute Simulation

ThresholdModel.iteration(self)

Execute a single model iteration

Returns:Iteration_id, Incremental node status (dictionary node->status)
ThresholdModel.iteration_bunch(self, bunch_size)

Execute a bunch of model iterations

Parameters:
  • bunch_size – the number of iterations to execute
  • node_status – if the incremental node status has to be returned.
Returns:

a list containing for each iteration a dictionary {“iteration”: iteration_id, “status”: dictionary_node_to_status}

Example

In the code below is shown an example of instantiation and execution of a Threshold model simulation on a random graph: we set the initial set of infected nodes as 1% of the overall population, and assign a threshold of 0.25 to all the nodes.

import networkx as nx
import ndlib.models.ModelConfig as mc
import ndlib.models.epidemics.ThresholdModel as th

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

# Model selection
model = th.ThresholdModel(g)

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

# Setting node parameters
threshold = 0.25
for i in g.nodes():
    config.add_node_configuration("threshold", i, threshold)

model.set_initial_status(config)

# Simulation execution
iterations = model.iteration_bunch(200)
[1]
  1. Granovetter, “Threshold models of collective behavior,” The American Journal of Sociology, vol. 83, no. 6, pp. 1420–1443, 1978.