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.

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.