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.
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()
config.add_model_parameter('fraction_infected', 0.1)
model.set_initial_status(config)
# Simulation execution
iterations = model.iteration_bunch(200)
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