Belief System Dynamics Simulation Framework

A modular platform for generating and simulating belief system dynamics on social influence networks, based on:

Ye et al., IEEE TAC 2020 — "Consensus and Disagreement of Heterogeneous Belief Systems in Influence Networks".

The framework produces:

• Social influence matrices $W$
• Heterogeneous logic matrices $C_i$
• Initial beliefs $X_0$

and supports exporting, visualization, and downstream simulation.

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Model Overview

Each agent $i$ holds an $m$-dimensional belief vector $x_i(t)$.
Belief updates follow the extended DeGroot model:

$$ x_i(t+1) = \sum_{j=1}^n W_{ij} , C_i , x_j(t). $$

• $W$: row-stochastic influence matrix
• $C_i$: logic matrix encoding topic dependencies
• Heterogeneity in $C_i$ may lead to consensus or persistent disagreement

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Repository Structure

1. social_network.py

Generates the influence matrix $W$ using ER, WS, BA, or Random-Regular models.

• Beta-distributed edge weights
• Ensures row-stochasticity
• Provides network summary + Gephi export

2. logic_matrix.py

Creates baseline and heterogeneous logic matrices $C_i$.

• Lower-triangular structure
• Beta-distributed coefficients
• Supports sparsity & heterogeneity
• Exports baseline $C_{\text{base}}$

3. init_belief.py

Generates initial beliefs $X_0$.
Modes: uniform, beta.
Exports all results into a timestamped folder.

4. belief_simulation.ipynb

Interactive notebook for testing:

• Component generation
• Simple belief dynamics
• Visualizations

Simulation Pipeline

1. Generate $W$, $C_i$, and $X_0$
2. Update beliefs over time using

$$ X(t+1) = \sum_{j=1}^n W_{ij} , C_i , x_j(t). $$

3. Visualise belief trajectories

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Reference

[1] M. Ye, J. Liu, L. Wang, B. D. O. Anderson, and M. Cao, “Consensus and Disagreement of Heterogeneous Belief Systems in Influence Networks,” IEEE Transactions on Automatic Control, vol. 65, no. 11, pp. 4679–4694, Nov. 2020.