# Generalised Threshold¶

The Generalised Threshold model was introduced in 2017 by Török and Kertesz [1].

In this model, during an epidemics, a node 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:

1. At time t nodes become Infected with rate mu t/tau
2. Nodes for which the ratio of the active friends dropped below the threshold are moved to the Infected queue
3. Nodes in the Infected queue become infected with rate tau. If this happens check all its friends for 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
tau Model int   True Adoption threshold rate
mu Model int   True Exogenous timescale

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.GeneralisedThresholdModel.GeneralisedThresholdModel(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.
GeneralisedThresholdModel.__init__(graph)

Model Constructor :param graph: A networkx graph object

GeneralisedThresholdModel.set_initial_status(self, configuration)

Set the initial model configuration

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

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

### Describe¶

GeneralisedThresholdModel.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
GeneralisedThresholdModel.get_status_map(self)

Specify the statuses allowed by the model and their numeric code

Returns: a dictionary (status->code)

### Execute Simulation¶

GeneralisedThresholdModel.iteration(self)

Execute a single model iteration :return: Iteration_id, Incremental node status (dictionary node->status)

GeneralisedThresholdModel.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. 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.GeneralisedThresholdModel as gth

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

# Model selection
model = gth.GeneralisedThresholdModel(g)

# Model Configuration
config = mc.Configuration()