# Independent Cascades with Community Embeddedness¶

The Independent Cascades with Community Embeddedness 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. The probability p(u,v) of the IC model is replaced by the edge embeddedness.

The embeddedness of an edge $$(u,v)$$ with $$u,v \in C$$ is defined as: $$e_{u,v} = \frac{\phi_{u,v}}{|\Gamma(u) \cup \Gamma(v)|}$$ where $$\phi_{u,v}$$ is the number of common neighbors of u and v within $$C$$, and $$\Gamma(u)$$ ( $$\Gamma(v)$$) is the set of neighbors of the node u (v) in the analyzed graph G.

The ICE 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, it succeeds with a probability $$e_{u,v}$$; otherwise with probability $$\min\{e_{z,v}|(z, v)\in E\}$$.
• 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.

## Statuses¶

During the simulation a node can experience the following statuses:

Name Code
Susceptible 0
Infected 1
Removed 2

## Parameters¶

The model is parameter free

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.ICEModel.ICEModel(graph)

Parameter free model: probability of diffusion tied to community embeddedness of individual nodes

ICEModel.__init__(graph)

Model Constructor

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

Set the initial model configuration

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

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

### Describe¶

ICEModel.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
ICEModel.get_status_map(self)

Specify the statuses allowed by the model and their numeric code

Returns: a dictionary (status->code)

### Execute Simulation¶

ICEModel.iteration(self)

Execute a single model iteration

Returns: Iteration_id, Incremental node status (dictionary node->status)
ICEModel.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 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 an ICE model simulation on a random graph: we set the initial set of infected nodes as 1% of the overall population.

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.ICEModel(g)

# Model Configuration
config = mc.Configuration()