# Weighted Hegselmann-Krause¶

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

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

## Methods¶

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

### Configure¶

class ndlib.models.opinions.WHKModel.WHKModel(graph)

Model Parameters to be specified via ModelConfig :param epsilon: bounded confidence threshold from the HK model (float in [0,1]) :param perc_stubborness: Percentage of stubborn agent (float in [0,1], default 0) :param option_for_stubbornness: Define distribution of stubborns (in {-1,0,1}, default 0) :param similarity: the method uses the similarity or not ( in {0,1}, default 0) :param weight: the weight of edges (float in [0,1]) :param stubborn: The agent is stubborn or not ( in {0,1}, default 0) :param vector: represents the character of the node (list in [0,1], default [])

WHKModel.__init__(graph)

Model Constructor :param graph: A networkx graph object

WHKModel.set_initial_status(self, configuration)

Override behaviour of methods in class DiffusionModel. Overwrites initial status using random real values.

WHKModel.reset(self)

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

### Describe¶

WHKModel.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
WHKModel.get_status_map(self)

Specify the statuses allowed by the model and their numeric code

Returns: a dictionary (status->code)

### Execute Simulation¶

WHKModel.iteration(self)

Execute a single model iteration

Returns: Iteration_id, Incremental node status (dictionary code -> status)
WHKModel.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 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()

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

for e in edges: