General Threshold

The General Threshold model was introduced in 20003 by Kempe [1].

In this model, during an epidemics, a is allowed to change its status from Susceptible to Infected.

The model is instantiated on a graph having a non-empty set of infected nodes.

The model is defined as follows:

At time t nodes become Infected if the sum of the weight of the infected neighbors is greater than the threshold

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
weight	Edge	float in [0, 1]	0.1	False	Edge weight

The initial infection status can be defined via:

• fraction_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.GeneralThresholdModel.GeneralThresholdModel(graph, seed=None)

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.
• weight – The edge weight. If not specified otherwise a value of 0.1 is assumed for all edges.

GeneralThresholdModel.__init__(graph)

Model Constructor

Parameters:graph – A networkx graph object

GeneralThresholdModel.set_initial_status(self, configuration)

Set the initial model configuration

Parameters:configuration – a `ndlib.models.ModelConfig.Configuration` object

GeneralThresholdModel.reset(self)

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

Describe

GeneralThresholdModel.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

GeneralThresholdModel.get_status_map(self)

Specify the statuses allowed by the model and their numeric code

Returns:a dictionary (status->code)

Execute Simulation

GeneralThresholdModel.iteration(self)

Execute a single model iteration

Returns:Iteration_id, Incremental node status (dictionary node->status)

GeneralThresholdModel.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.
• progress_bar – whether to display a progress bar, default False

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 as ep

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

# Model selection
model = epd.GeneralThresholdModel(g)

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

# Setting node and edges parameters
threshold = 0.25
weight = 0.2
if isinstance(g, nx.Graph):
    nodes = g.nodes
    edges = g.edges
else:
    nodes = g.vs['name']
    edges = [(g.vs[e.tuple[0]]['name'], g.vs[e.tuple[1]]['name']) for e in g.es]


for i in nodes:
    config.add_node_configuration("threshold", i, threshold)
for e in edges:
    config.add_edge_configuration("weight", e, weight)


model.set_initial_status(config)

# Simulation execution
iterations = model.iteration_bunch(200)

[1]

János Török and János Kertész “Cascading collapse of online social networks” Scientific reports, vol. 7 no. 1, 2017 David Kempe , Jon Kleinberg, and Éva Tardos. “Maximizing the spread of influence through a social network.” Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining. ACM, 2003.