Partisan-Confidence-Model

Simulation files for the Partisan Confidence model of opinion evolution written in python programming language. The Project include three folders as below:

1. Corollaries: A jupyter notebook for verifying collaries.
2. One-dimensional: Two jupyter notebooks for simulating the Partisan confidenc-lite (PC-lite) and Partisan confidence (PC) model.

• PC-lite_case_1.ipynb: Three bubbles and some are considered outside the bubbles.
• PC-lite_case_2.ipynb: Two bubbles are considered in the population.
• PC_case_2.ipynb: Two bubbles are considered in the population.

4. Two-dimensional: Two jupyter notebooks for simulating the Partisan confidenc-lite multi-dimensional (MDPC-lite) and Partisan confidenc multi-dimensional (MDPC) model

• MDPC-lite_case_2.ipynb: Two bubbles and two topics are considered.
• MDPC_case_2.ipynb: Two bubbles and two topics are considered.

One-dimensional model

Suppose that $x_i(t)\in [-1,1]$ stands for the opinion of agent $i$ at time $t$ about a specific and fixed subject. Opinion evolution according to the PC-lite and PC model is given below.

PC-lite:

$$x_i(t+1) - x_i(t) = \sum_{j\neq i} w_{ij}(t) |x_j(t)| \left( sgn(x_j(t)) - x_i(t) \right)$$

PC

$$x_i(t+1) - x_i(t) = \sum_{j\neq i} w_{ij}(t) d_i(x_i(t), x_j(t))|x_j(t)| \left( sgn(x_j(t)) - x_i(t) \right)$$

where $d_i: [-1,1]^2 \to [0,1]$ is the discount function accounting for confirmation bias in the model described as below. If two individuals share the same general belief

$$d_i(x_i(t), x_j(t)) = 1 ~~\text{if}~~sgn(x_i(t)) = sgn(x_j(t)),$$

and if two individuals share different general beliefs

$$d_i(x_i(t), x_j(t)) \leq \hat{d}_i(|x_i(t)|) ~~\text{if}~~sgn(x_i(t)) = sgn(x_j(t))$$

where $\hat{d}_i:[-1,1] \to [0,1]$ is a non-increasing discounting function.

multi-dimensional model

Suppose that there are $\mathcal{M} = \{1, ..., m\}$ topics for discussion. Take $x^{(k)}_i(t)\in [-1,1]$ as the opinion of agent $i$ at time $t$ about topic $k \in \mathcal{M}$. Opinion evolution according to the MDPC-lite and MDPC model is given below.

MDPC-lite: $m \geq 2$

$$x^{(k)}_i(t+1) - x^{(k)}_i(t) = \sum_{j\neq i} w_{ij}(t) |x^{(k)}_j(t)| \left( sgn(x^{(k)}_j(t)) - x^{(k)}_i(t) \right)$$

MDPC: $m = 2$

$$x^{(k)}_i(t+1) - x^{(k)}_i(t) = \sum_{j\neq i} w_{ij}(t) d_i(\vec{x}_i(t), \vec{x_j(t)})|x^{(k)}_j(t)| \left( sgn(x^{(k)}_j(t)) - x^{(k)}_i(t) \right)$$

where $d_i: [-1,1]^2 \to [0,1]$ is the discount function accounting for confirmation bias in the model described as below. If two individuals share the same general belief

$$d_i(\vec{x}_i(t), \vec{x}_j(t)) = 1 ~~\text{if}~~sgn(\vec{x}_i(t)) = sgn(\vec{x}_j(t))~~\text{or}~~\vec{x}_i(t) = \mathbf{0}~~\text{or}~~\vec{x}_j(t) = \mathbf{0},$$

and if two individuals share different general beliefs

$$d_i(\vec{x}_i(t), \vec{x}_j(t)) \leq \hat{h}_i(\theta) ~~\text{if}~~sgn(\vec{x}_i(t)) \neq sgn(\vec{x}_j(t))~~\text{and}~~\vec{x}_i(t) \neq \mathbf{0}~~\text{and}~~\vec{x}_j(t) \neq \mathbf{0}.$$

where $\hat{h}_i:[0, \pi] \to [0,1]$ is a non-increasing discounting function and $\theta$ is the angle between $\vec{x}_i$ and $\vec{x}_j$ given as below.

$$\theta = \frac{\vec{x}_i.\vec{x}_j}{\|\vec{x}_i\|_2\times\|\vec{x}_j\|_2}$$