Kertesz Threshold

The Kertesz Threshold model was introduced in 2015 by Ruan et al. [1] and it is an extension of the Watts threshold model [2].

The authors extend the classical model introducing a density r of blocked nodes – nodes which are immune to social influence – and a probability of spontaneous adoption p to capture external influence.

Thus, the model distinguishes three kinds of node: Blocked (B), Susceptible (S) and Adoptiong (A). The latter class breaks into two categories: vulnerable and stable nodes. A node can adopt either under its neighbors’ influence, or spontaneously, due to endogenous effects.


During the simulation a node can experience the following statuses:

Name Code
Susceptible 0
Infected 1
Blocked -1


Name Type Value Type Default Mandatory Description
adopter_rate Model float in [0, 1] 0 False Exogenous adoption rate
percentage_blocked Model float in [0, 1] 0.1 False Blocked nodes
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 initial blocked nodes can be defined via:

  • percentage_blocked: Model Parameter, float in [0, 1]
  • Blocked: Status Parameter, set of nodes

In both cases, the two options are mutually exclusive and the latter takes precedence over the former.


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


class ndlib.models.epidemics.KerteszThresholdModel.KerteszThresholdModel(graph)
Node/Model Parameters to be specified via ModelConfig
  • threshold – The node threshold. As default a value of 0.1 is assumed for all nodes.
  • adopter_rate – The probability of spontaneous adoptions. Defaults value 0.
  • percentage_infected – The percentage of blocked nodes. Default value 0.1.

Model Constructor

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

Set the initial model configuration

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

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



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

Returns:a dictionary containing for each parameter class the values specified during model configuration

Specify the statuses allowed by the model and their numeric code

Returns:a dictionary (status->code)

Execute Simulation


Execute a single model iteration

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

Execute a bunch of model iterations

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

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


In the code below is shown an example of instantiation and execution of a Kertesz Threshold model simulation on a random graph: we set the initial infected as well blocked node sets equals to the 10% of the overall population, assign a threshold of 0.25 to all the nodes and impose an probability of spontaneous adoptions of 40%.

import networkx as nx
import ndlib.models.ModelConfig as mc
import ndlib.models.epidemics.KerteszThresholdModel as ks

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

# Model selection
model = ks.KerteszThresholdModel(g)

# Model Configuration
config = mc.Configuration()
config.add_model_parameter('adopter_rate', 0.4)
config.add_model_parameter('percentage_blocked', 0.1)
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)


# Simulation execution
iterations = model.iteration_bunch(200)
  1. Ruan, G. In ̃iguez, M. Karsai, and J. Kertész, “Kinetics of social contagion,” Phys. Rev. Lett., vol. 115, p. 218702, Nov 2015.
    1. Watts, “A simple model of global cascades on random networks,” Proceedings of the National Academy of Sciences, vol. 99, no. 9, pp. 5766–5771, 2002.