# Nonlinear Consensus Protocol Modified from Doubly Stochastic Quadratic Operators in Networks of Dynamic Agents

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## Abstract

**:**

## 1. Introduction

- To find the optimal decision of the group.
- To investigate simple mathematical nonlinear consensus protocol.

## 2. Background

## 3. Materials and Methods

**(a) The Problem Domain**

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**(b) The Solution Domain**

**Definition**

**4.**

- (i)
- The sum of all m matrices is a matrix with all entries as ones as follows:$$\sum}_{i=1}^{m}{a}_{ij,k}\ne {\displaystyle \sum}_{j=1}^{m}{a}_{ij,k}\ne 1and{\displaystyle \sum}_{k=1}^{m}{a}_{ij,k}=1,{a}_{ij,k}0,\forall i,j,k=\overline{1,m},$$
- (ii)
- The sum of each row or column in each matrix is stochastic and the sum of all m matrices is a matrix with all entries as ones as follows:$$\sum}_{i=1}^{m}{a}_{ij,k}or{\displaystyle \sum}_{j=1}^{m}{a}_{ij,k}=1and{\displaystyle \sum}_{k=1}^{m}{a}_{ij,k}=1,{a}_{ij,k}0,\forall i,j,k=\overline{1,m},$$
- (iii)
- Triple stochastic matrix, where the sum of each row and column in each matrix is stochastic and the sum of all m matrices is a matrix with all entries as ones as follows:$$\sum}_{i=1}^{m}{a}_{ij,k}={\displaystyle \sum}_{j=1}^{m}{a}_{ij,k}=1and{\displaystyle \sum}_{k=1}^{m}{a}_{ij,k}=1,{a}_{ij,k}0,\forall i,j,k=\overline{1,m},$$

## 4. Results

**Theorem**

**1.**

**Proof.**

_{ij,k}with other agents and the sum of these interactions equals to the number of agents $\sum}_{ij=1}^{m}{a}_{ij,k}=m$ and $t$ is the sequence of the states.

**Corollary**

**1.**

**Proof.**

## 5. Numerical Solution

## 6. Comparison Study between the Nonlinear Model of MDSQO and DeGroot’s Linear Model

## 7. The Comparison of the Convergence of MDSQO with DSQO

## 8. Discussion and Critical Reflection of the Research

## 9. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 11.**The consensus of MPDSQO for 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400 and 500 agents, respectively.

**Figure 12.**Comparison between the convergence of the DeGroot Linear Model and the Nonlinear Model of MDSQO with SM and DSM.

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**MDPI and ACS Style**

Abdulghafor, R.; Almotairi, S.; Almohamedh, H.; Turaev, S.; Almutairi, B.
Nonlinear Consensus Protocol Modified from Doubly Stochastic Quadratic Operators in Networks of Dynamic Agents. *Symmetry* **2019**, *11*, 1519.
https://doi.org/10.3390/sym11121519

**AMA Style**

Abdulghafor R, Almotairi S, Almohamedh H, Turaev S, Almutairi B.
Nonlinear Consensus Protocol Modified from Doubly Stochastic Quadratic Operators in Networks of Dynamic Agents. *Symmetry*. 2019; 11(12):1519.
https://doi.org/10.3390/sym11121519

**Chicago/Turabian Style**

Abdulghafor, Rawad, Sultan Almotairi, Hamad Almohamedh, Sherzod Turaev, and Badr Almutairi.
2019. "Nonlinear Consensus Protocol Modified from Doubly Stochastic Quadratic Operators in Networks of Dynamic Agents" *Symmetry* 11, no. 12: 1519.
https://doi.org/10.3390/sym11121519