Independent Cascades with Community Permeability

The Independent Cascades with Community Permeability model was introduced by Milli and Rossetti in 2019 [1].

This model is a variation of the well-known Independent Cascade (IC), and it is designed to embed community awareness into the IC model. This model exploits the idea of permeability. A community is “permeable” to a given content if it permits that content to spread from it fast (or vice-versa, if it permits the content to be easily received from nodes outside the community). Conversely, a community has a low degree of permeability if it dampens the diffusion probability across its border.

The ICP model starts with an initial set of active nodes A0; the diffusive process unfolds in discrete steps according to the following randomized rule:

  • When node v becomes active in step t, it is given a single chance to activate each currently inactive neighbor u. If v and u belong to the same community, the method works as a standard IC model (it succeeds with a probability p(v,u)); instead, if the nodes are part of to different communities, the likelihood p(v,u) is dampened of a factor \(\eta\) (the community permeability parameter).
  • If u has multiple newly activated neighbors, their attempts are sequenced in an arbitrary order.
  • If v succeeds, then u will become active in step t + 1; but whether or not v succeeds, it cannot make any further attempts to activate u in subsequent rounds.
  • The process runs until no more activations are possible.


During the simulation a node can experience the following statuses:

Name Code
Susceptible 0
Infected 1
Removed 2


Name Type Value Type Default Mandatory Description
Edge threshold Edge float in [0, 1] 0.1 False Edge threshold
Community permeability Model float in [0, 1] 0.5 True Community Permeability

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.


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



Model Constructor

Parameters:graph – A networkx graph object
ICPModel.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)
ICPModel.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.
  • progress_bar – whether to display a progress bar, default False

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 an ICP model simulation on a random graph: we set the initial set of infected nodes as 1% of the overall population, assign a threshold of 0.1 to all the edges and set the community permeability equal 0.3.

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 = ep.ICPModel(g)

# Model Configuration
config = mc.Configuration()
config.add_model_parameter('fraction_infected', 0.1)
config.add_model_parameter('permeability', 0.3)

# Setting the edge parameters
threshold = 0.1
for e in g.edges():
    config.add_edge_configuration("threshold", e, threshold)


# Simulation execution
iterations = model.iteration_bunch(200)
  1. Milli and G. Rossetti. “Community-Aware Content Diffusion: Embeddednes and Permeability,” in Proceedings of International Conference on Complex Networks and Their Applications, 2019 pp. 362–371.