# Opinion Dynamics and Unifying Principles: A Global Unifying Frame

## Abstract

**:**

## 1. Opinion Dynamics, Models and Reality

## 2. The General Probabilistic Frame

- ${a}_{1}=+++$ with probability ${p}_{t,1}$,
- ${a}_{2}=++-$ with probability ${p}_{t,2}$,
- ${a}_{3}=+-+$ with probability ${p}_{t,3}$,
- ${a}_{4}=-++$ with probability ${p}_{t,4}$,
- ${a}_{5}=--+$ with probability ${p}_{t,5}$,
- ${a}_{6}=-+-$ with probability ${p}_{t,6}$,
- ${a}_{7}=+--$ with probability ${p}_{t,7}$,
- ${a}_{8}=---$ with probability ${p}_{t,8}$.

- ${p}_{t,1}={p}_{t}^{3}$
- ${p}_{t,2}={p}_{t,3}={p}_{t,4}={p}_{t}^{2}(1-{p}_{t})$
- ${p}_{t,5}={p}_{t,6}={p}_{t,7}={p}_{t}{(1-{p}_{t})}^{2}$
- ${p}_{t,8}={(1-{p}_{t})}^{3}$.

## 3. Iterating the Dynamics: The Global Unifying Frame

- $D>0$: Equation (7) has are three real roots ${p}_{c,1}<{p}_{c,2}<{p}_{c,3}$, which are distinct. Dealing with proportions, these roots are acceptable only when satisfying the condition $0\le {p}_{c,i}\le 1$ for $i=1,2,3$. By symmetry that happens either for three of them with one separator and two attractors or for only one of them, which is then an attractor.For the first scenario, when ${p}_{t}<{p}_{c,2}$, the update iteration drives the opinion + towards ${p}_{c,1}$ leading to the victory of opinion −. At opposite, for ${p}_{t}>{p}_{c,2}$, the update iteration drives the opinion + towards ${p}_{c,3}$ leading to its victory. One case is shown in the upper part of Figure 1) with ${p}_{c,2}<1/2$. The opposite ${p}_{c,2}>1/2$ may also occur. Each attractor can be a pure phase or a mixed phase with, respectively, ${p}_{c,1}=0$ or ${p}_{c,1}>0$ and ${p}_{c,3}=1$ or ${p}_{c,1}<1$.In case ${p}_{c,1}=0$ and ${p}_{c,3}=1$, the fixed point ${p}_{c,2}$ can also become an attractor with both pure ones being unstable in its direction as shown in the lower part of Figure 1. In that case, due to topological constraint, the transition from ${p}_{c,2}$ being a separator to ${p}_{c,2}$ being an attractor occur via a conservative regime ${p}_{t+1}={p}_{t}$ where each point is a fixed point, thus recovering the voter model as shown in Figure 2. It is worth noting that in case ${p}_{c,1}>0$ and ${p}_{c,3}<1$ such a transformation of a separator into an attractor is still possible, but now there is no transition via the voter model. An illustration of both cases is given in Section 4.
- $D=0$: Equation (7) exhibits three real roots with at least a double one, which corresponds to a transition regime from threshold dynamics to threshold-less dynamics. Two subclasses can occur:
**(i)**- One single attractor and a double fixed point, which is an attractor on one side and separator on the other side, the side where the attractor is located. The upper part of Figure 3 exhibits the case ${p}_{c,1}={p}_{c,2}$ with ${p}_{c,3}$ being the attractor. The symmetric case is also possible with ${p}_{c,1}$ being the attractor and ${p}_{c,2}={p}_{c,3}$.
**(ii)**- Or a triple attractor ${p}_{c,1}={p}_{c,2}={p}_{c,3}$ making the dynamics threshold-less. Whatever the initial conditions are, the repeated updates drive the collective opinion towards the single attractor. Two cases are possible. The first one has ${p}_{c,1}={p}_{c,2}={p}_{c,3}=1/2$, which means the dynamics leads to a coexistence phase with a perfect fifty/fifty equality. The second one is not balanced with ${p}_{c,1}={p}_{c,2}={p}_{c,3}\ne 1/2$, which means the dynamics leads to a a stable majority/minority coexistence phase with a deterministic victory for one specific opinion. If ${p}_{c,1}={p}_{c,2}={p}_{c,3}<1/2$, opinion + is certain to lose, provided some number of updates are completed. Otherwise, when ${p}_{c,1}={p}_{c,2}={p}_{c,3}>1/2$ opinion + wins the competition. One case with ${p}_{c,1}={p}_{c,2}={p}_{c,3}>1/2$ is shown in the lower part of Figure 3. The symmetric situation with ${p}_{c,1}={p}_{c,2}={p}_{c,3}<1/2$ is also possible, making opinion + lose the competition.

- $D<0$: Equation (7) has one real root and two imaginary roots. It is thus a single attractor dynamic. The attractor can be located at any value between 0 and 1 depending on the details of the update rule.

## 4. Applying the GUF to Existing Opinion Dynamics Models

#### 4.1. Application to Galam Models

#### 4.1.1. The Local Majority Model (LMM) [12,13,14]

- ${b}_{1}={b}_{2}={b}_{3}={b}_{4}=+++$,
- ${b}_{5}={b}_{6}={b}_{7}={b}_{8}=---$,

- ${k}_{1}={k}_{2}={k}_{3}={k}_{4}=3$,
- ${k}_{5}={k}_{6}={k}_{7}={k}_{8}=0$,

#### 4.1.2. The Contrarian Majority Model (CMM) [15]

- Majority rule yielding,
- −
- ${b}_{1}^{1}={b}_{2}^{1}={b}_{3}^{1}={b}_{4}^{1}=+++$,
- −
- ${b}_{5}^{1}={b}_{6}^{1}={b}_{7}^{1}={b}_{8}^{1}=---$,

and- −
- ${k}_{1}^{1}={k}_{2}^{1}={k}_{3}^{1}={k}_{4}^{1}=3$,
- −
- ${k}_{5}^{1}={k}_{6}^{1}={k}_{7}^{1}={k}_{8}^{1}=0$,

with probability $(1-\alpha )$. - Contrarian shifts yielding,
- −
- ${b}_{1}^{2}={b}_{2}^{2}={b}_{3}^{2}={b}_{4}^{2}=---$,
- −
- ${b}_{5}^{2}={b}_{6}^{2}={b}_{7}^{2}={b}_{8}^{2}=+++$,

and- −
- ${k}_{1}^{2}={k}_{2}^{2}={k}_{3}^{2}={k}_{4}^{2}=0$,
- −
- ${k}_{5}^{2}={k}_{6}^{2}={k}_{7}^{2}={k}_{8}^{2}=3$.

with probability $\alpha $.Then, the two cases must be averages giving, - Average
- −
- ${\overline{k}}_{1}={\overline{k}}_{2}={\overline{k}}_{3}={\overline{k}}_{4}=3(1-\alpha )$,
- −
- ${\overline{k}}_{5}={\overline{k}}_{6}={\overline{k}}_{7}={\overline{k}}_{8}=3\alpha $,

- $\alpha <1/6\to D>0\to $ three real roots ${p}_{c,1}<{p}_{c,2}<{p}_{c,3}$.
- $1/6<\alpha <1/2\to D<0\to $ a threshold-less dynamics with one single attractor.
- $\alpha >1/2\to D>0\to $ again three real roots with $\alpha >1/2$, implying an oscillating regime.

#### 4.1.3. The Extended Majority Model (EMM) [16]

- Probability $(1-\beta )$
- −
- ${b}_{2}^{1}={b}_{3}^{1}={b}_{4}^{1}=+++$,
- −
- ${b}_{5}^{1}={b}_{6}^{1}={b}_{7}^{1}=---$,

yielding- −
- ${k}_{2}^{1}={k}_{3}^{1}={k}_{4}^{1}=3$,
- −
- ${k}_{5}^{1}={k}_{6}^{1}={k}_{7}^{1}=0$.

and with - Probability $\beta $
- −
- ${b}_{2}^{2}={b}_{3}^{2}={b}_{4}^{2}=---$,
- −
- ${b}_{5}^{2}={b}_{6}^{2}={b}_{7}^{2}=+++$,

yielding- −
- ${k}_{2}^{2}={k}_{3}^{2}={k}_{4}^{2}=0$,
- −
- ${k}_{5}^{2}={k}_{6}^{2}={k}_{7}^{2}=3$,

Both cases lead to the - Averages
- −
- ${\overline{k}}_{2}^{1}={\overline{k}}_{3}^{1}={\overline{k}}_{4}^{1}=3(1-\beta )$,
- −
- ${\overline{k}}_{5}^{1}={\overline{k}}_{6}^{1}={\overline{k}}_{7}^{1}=3\beta $.

#### 4.2. Application to Sznajd Models

#### 4.2.1. The Original Outflow Model (OOM) [17]

- ${S}_{2,t}={S}_{3,t}\to {S}_{1,t+1}={S}_{4,t+1}={S}_{2,t+1}={S}_{3,t+1}={S}_{2,t}={S}_{3,t}$.
- ${S}_{2,t}=-{S}_{3,t}\to {S}_{1,t+1}=-{S}_{2,t+1}=-{S}_{2,t}$ and ${S}_{4,t+1}=-{S}_{3,t+1}=-{S}_{3,t}$,

- making $r=4$ and ${r}_{u}=2$.

- ${a}_{1}=++(+)\to {b}_{1}=++(+)$, with ${k}_{1}=1$,
- ${a}_{2}=++(-)\to {b}_{2}=++(+)$, with ${k}_{2}=1$,
- ${a}_{3}=+-(+)\to {b}_{3}=+-(+)$, with ${k}_{3}=1$,
- ${a}_{4}=-+(+)\to {b}_{4}=-+(-)$, with ${k}_{4}=0$,
- ${a}_{5}=--(+)\to {b}_{5}=--(-)$, with ${k}_{5}=0$,
- ${a}_{6}=-+(-)\to {b}_{6}=-+(-)$, with ${k}_{6}=0$,
- ${a}_{7}=+-(-)\to {b}_{7}=+-(+)$, with ${k}_{7}=1$,
- ${a}_{8}=--(-)\to {b}_{8}=--(-)$, with ${k}_{8}=0$,

#### 4.2.2. The Modified Outflow Model (MOM) [18]

- ${a}_{1}=++(+)\to {b}_{1}=++(+)$, with ${k}_{1}=1$,
- ${a}_{2}=++(-)\to {b}_{2}=++(+)$, with ${k}_{2}=1$,
- ${a}_{3}=+-(+)\to {b}_{3}=+-(+)$, with ${k}_{3}=1$,
- ${a}_{4}=-+(+)\to {b}_{4}=-+(+)$, with ${k}_{4}=1$,
- ${a}_{5}=--(+)\to {b}_{5}=--(-)$, with ${k}_{5}=0$,
- ${a}_{6}=-+(-)\to {b}_{6}=-+(-)$, with ${k}_{6}=0$,
- ${a}_{7}=+-(-)\to {b}_{7}=+-(-)$, with ${k}_{7}=0$,
- ${a}_{8}=--(-)\to {b}_{8}=--(-)$, with ${k}_{8}=0$,

#### 4.2.3. The Modified Inflow Model (MIM) [18,21]

- ${a}_{1}=+(+)+\to {b}_{1}=+(+)+$, with ${k}_{1}=1$,
- ${a}_{2}=+(+)-\to {b}_{2}=+(+)-$, with ${k}_{2}=1$,
- ${a}_{3}=+(-)+\to {b}_{3}=+(+)+$, with ${k}_{3}=1$,
- ${a}_{4}=-(+)+\to {b}_{4}=-(+)+$, with ${k}_{4}=1$,
- ${a}_{5}=-(-)+\to {b}_{5}=-(-)+$, with ${k}_{5}=0$,
- ${a}_{6}=-(+)-\to {b}_{6}=-(-)-$, with ${k}_{6}=0$,
- ${a}_{7}=+(-)-\to {b}_{7}=+(-)-$, with ${k}_{7}=0$,
- ${a}_{8}=-(-)-\to {b}_{8}=-(-)-$, with ${k}_{8}=0$,

## 5. A Discrepancy

- ${p}_{t,3}={p}_{t,4}={p}_{t,6}={p}_{t,7}=0$,
- ${p}_{t,1}=\frac{{p}^{2}}{{p}^{2}+{(1-p)}^{2}}p$,
- ${p}_{t,2}=\frac{{p}^{2}}{{p}^{2}+{(1-p)}^{2}}(1-p)$,
- ${p}_{t,5}=\frac{{(1-p)}^{2}}{{p}^{2}+{(1-p)}^{2}}p$,
- ${p}_{t,8}=\frac{{(1-p)}^{2}}{{p}^{2}+{(1-p)}^{2}}(1-p)$,

- ${p}_{t,2}={p}_{t,4}={p}_{t,5}={p}_{t,7}=0$,
- ${p}_{t,1}=\frac{{p}^{2}}{{p}^{2}+{(1-p)}^{2}}p$,
- ${p}_{t,3}=\frac{{p}^{2}}{{p}^{2}+{(1-p)}^{2}}(1-p)$,
- ${p}_{t,6}=\frac{{(1-p)}^{2}}{{p}^{2}+{(1-p)}^{2}}p$,
- ${p}_{t,8}=\frac{{(1-p)}^{2}}{{p}^{2}+{(1-p)}^{2}}(1-p)$,

## 6. Mean Field Versus the GUF

## 7. Conclusions

- United we stand, divided we fall (MOM);
- If you do not know what to do, just do nothing (MIM);
- Follow the opinion of anybody else (VM);
- Follow the majority (LMM),

## Funding

## Conflicts of Interest

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**Figure 1.**Case $D>0$ with three fixed points. The upper part of the Figure shows a threshold dynamics with two attractors ${p}_{c,1}$ and ${p}_{c,3}$ separated by an unstable fixed point ${p}_{c,2}$. A case with ${p}_{c,2}\le 1/2$, $0\le {p}_{c,1}<{p}_{c,2}$ and ${p}_{c,2}<{p}_{c,3}\le 1$ is exhibited. The lower part of the Figure shows the threshold dynamics with two separators ${p}_{c,1}$ and ${p}_{c,3}$ with an attractor ${p}_{c,2}$ in between. In such a case, ${p}_{c,1}=0$, $0<{p}_{c,2}<1$ and ${p}_{c,3}=1$.

**Figure 2.**The transformation of a threshold dynamics with the two attractors ${p}_{c,1}=0$ and ${p}_{c,3}=1$ and the separator ${p}_{c,2}=1/2$ (

**higher part**) into a threshold-less like dynamics with the two separators ${p}_{c,1}=0$ and ${p}_{c,3}=1$ and the attractor ${p}_{c,2}=1/2$ (

**lower part**). It must pass via a voter model (

**middle part**) where each point is conserved by the dynamics.

**Figure 3.**Upper part of the Figure shows a case $D=0$ with a double fixed points ${p}_{c,1}={p}_{c,2}$, which is an attractor below it and separator above it. The attractor of the dynamics is ${p}_{c,3}$. The Case $D=0$ with a single triple fixed point ${p}_{c,1}={p}_{c,2}={p}_{c,3}$, which is an attractor, is shown in the lower part of the figure. Whatever the initial ${p}_{t}$ is, the dynamics lead towards ${p}_{c,1}={p}_{c,2}={p}_{c,3}$ to reach it, provided the required number of updates has been performed. Otherwise, it stops before. The attractor can be located any place with $0\le {p}_{c,1}={p}_{c,2}={p}_{c,3}\le 1$.

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Galam, S.
Opinion Dynamics and Unifying Principles: A Global Unifying Frame. *Entropy* **2022**, *24*, 1201.
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**AMA Style**

Galam S.
Opinion Dynamics and Unifying Principles: A Global Unifying Frame. *Entropy*. 2022; 24(9):1201.
https://doi.org/10.3390/e24091201

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Galam, Serge.
2022. "Opinion Dynamics and Unifying Principles: A Global Unifying Frame" *Entropy* 24, no. 9: 1201.
https://doi.org/10.3390/e24091201