DeGroot Stubborn Opinion Limits

The DeGroot model describes how opinions spread through a network as individuals average their neighbors' beliefs. Adding stubbornness creates an anchored limit.

Introduction

In a network, individuals often update their opinions by taking a weighted average of their neighbors' opinions. We study a two-node network.

Model Description

We use a weight matrix W to describe the listening weights between two individuals. With stubbornness λ, the updating rule becomes x_(k+1) = (1-λ)Wx_k + λ x_0.

Experiment

You can interact with the network by adjusting listening weights, stubbornness, and rounds in the interactive DeGroot model.

Set First listening weight to 0.25 and Second listening weight to 1.
Set Stubbornness to 0 and Rounds to 2.
Observe that the First current opinion is 0.1875 and the Second current opinion is 0.25.

Worked Example

Let a = 0.25, b = 1, λ = 0, k = 2, with initial opinions x_0 = (0, 1).

Round 0: x_0 = (0, 1).
Round 1: x_1 = Wx_0 = (0.25, 0).
Round 2: x_2 = Wx_1 = (0.1875, 0.25).

Limitations

This is a toy linear model. Real people do not obey strict linear averaging and networks are far more complex.

References

Primary literature on consensus and opinion dynamics:

DeGroot Stubborn Opinion Limits