Weighted Hegselmann-Krause

The Weighted Hegselmann-Krause was introduced by Milli et al. in 2021 [1].

This model is a variation of the well-known Hegselmann-Krause (HK). During each interaction a random agenti is selected and the set \(\Gamma_{\epsilon}\) of its neighbors whose opinions differ at most \(\epsilon\) (\(d_{i,j}=|x_i(t)-x_j(t)|\leq \epsilon\)) is identified. Moreover, to account for the heterogeneity of interaction frequency among agent pairs, WHK leverages edge weights, thus capturing the effect of different social bonds’ strength/trust as it happens in reality. To such extent, each edge \((i,j) \in E\), carries a value \(w_{i,j}\in [0,1]\). The update rule then becomes:

\[\begin{split}x_i(t+1)= \left\{ \begin{array}{ll} x_i(t) + \frac{\sum_{j \in \Gamma_{\epsilon}} x_j(t)w_{ij}}{\#\Gamma_{\epsilon}} (1-x_i(t)) \quad \quad \text{\quad if x_i(t) \geq 0}\\ x_i(t) + \frac{\sum_{j \in \Gamma_{\epsilon}} x_j(t)w_{ij}}{\#\Gamma_{\epsilon}} (1+x_i(t)) \quad \text{if x_i(t) < 0 } \end{array} \right.\end{split}\]

The idea behind the WHK formulation is that the opinion of agent \(i\) at time \(t+1\), will be given by the combined effect of his previous belief and the average opinion weighed by its, selected, \(\epsilon\)-neighbor, where \(w_{i,j}\) accounts for \(i\)’s perceived influence/trust of \(j\).

Statuses

Node statuses are continuous values in [-1,1].

Parameters

Name	Type	Value Type	Default	Mandatory	Description
epsilon	Model	float in [0, 1]	—	True	Bounded confidence threshold
perc_stubborness	Model	float in [0, 1]	0	False	Percentage of stubborn agent
similarity	Model	int in {0, 1}	0	False	The method use the feature of the nodes ot not
option_for_stubbornness	Model	int in {-1,0, 1}	0	False	Define distribution of stubborns
weight	Edge	float in [0, 1]	0.1	False	Edge weight
stubborn	Node	int in {0, 1}	0	False	The agent is stubborn or not
vector	Node	Vector of float in [0, 1]	[]	False	Vector represents the character of the node

Example

In the code below is shown an example of instantiation and execution of an WHK model simulation on a random graph: we an epsilon value of 0.32 and a weight equal 0.2 to all the edges.

import networkx as nx
import ndlib.models.ModelConfig as mc
import ndlib.models.opinions as opn

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

# Model selection
model = opn.WHKModel(g)

# Model Configuration
config = mc.Configuration()
config.add_model_parameter("epsilon", 0.32)

# Setting the edge parameters
weight = 0.2
if isinstance(g, nx.Graph):
    edges = g.edges
else:
    edges = [(g.vs[e.tuple[0]]['name'], g.vs[e.tuple[1]]['name']) for e in g.es]

for e in edges:
    config.add_edge_configuration("weight", e, weight)


model.set_initial_status(config)

# Simulation execution
iterations = model.iteration_bunch(20)

[1]	Milli and G. Rossetti. “Opinion Dynamic Modeling of News Perception”.