A Fast Non-Linear Symmetry Approach for Guaranteed Consensus in Network of Multi-Agent Systems

: There has been tremendous work on multi-agent systems (MAS) in recent years. MAS consist of multiple autonomous agents that interact with each order to solve a complex problem. Several applications of MAS can be found in computer networks, smart grids, and the modeling of complex systems. Despite numerous beneﬁts, a signiﬁcant challenge for MAS is achieving a consensus among agents in a shared target task, which is di ﬃ cult without applying certain mathematical equations. Non-linear models o ﬀ er better possibility of resolving consensus for MAS; however, existing non-linear models are considerably complicated and present no guarantees for achieving consensus. This paper proposes a non-linear mathematical model of semi symmetry quadratic operator (SSQO) in order to resolve the issue of consensus in networks of MAS. The model is based on stochastic quadratic operator theory, with added new notations. An important feature for the proposed model is low complexity, fast consensus, and a guaranteed capability to reach a consensus. We present an evaluation of the proposed SSQO model and comparison with other existing models. We demonstrate that an average consensus can be achieved with our model in addition to the emulation e ﬀ ects for the MAS consensus.


Introduction
In the last decade, increased attention was directed towards the coordination and control of MAS, which appeared in various situations, such as wireless sensor networks and team cooperation. New studies have recently emerged that focus on the problems related to consensus in MAS [1]. The issue of consensus is considered to be a central problem in multi-output multi-input systems (MIMOs) that have many applications in technology [2]. In this context, the consensus is defined as the way in which individual group members agree to cooperate and generate consensus among themselves in order to achieve one objective at the same time [3].
MAS are defined as a dynamic network topology consisting of several smart agents that communicate with each other to locally exchange information in order to implement a task. Consensus or agreement is one of the most important behaviors required for every group in our natural lives. In recent decades, artificial intelligence has been introduced in various fields that benefit the world in many complex and welfare services. MAS have a great role in applying these services, including applications of physics, engineering, biology, mathematics, and social sciences [4]. The consensus in MAS is the most important factor that focuses on the performance of the functions that are assigned to it.

Background and Methods
In this section, preliminary notions and notations for SSQO and based theories will be provided. The proposed model of SSQO is based on QSO [62]. New rules for the transition matrix of QSO are to be included in order to produce a less complicated model to achieve the consensus for MAS. In this case, the statuses of individual members (s 1 , s 2 , s 3 , . . . , s m ) will be duplicated as row s i and column s j and produced with the transition matrix A ij,k that is based to the following protocol: . . .
where s refers to agent, m the total number of agents, i and j the indexed number for each agent in row and column forms, and A ij,k is the transition matrix of the communicated agents.
is the general operator for QSO to determine the limit behavior for each member of s (t) i , where i is in row form and j in column form; however, A ij,k signifies the distributed matrices i with the other agents.
Subsequently, the process will be updated using the new outputs as new inputs for the next iteration (t + 1). The iterations are continued until the fixed values are reached (consensus), in the form of s (t+1) = s (t) .
The transition matrix A ij,k in QSO is distributed to m matrices containing random values, where their sum is a matrix consisting of elements a ij,k equal to one, as following: a ij,1 + a ij,2 + · · · + a ij,k = 1 However, the values of the initial statuses are random and the sum of them equals one. Because the evolutionary matrices are split into m matrices and have random elements, the structure of the equations system of QSO will consist of long polynomials equations. Hence, it is necessary to establish a new class from QSO that has low computations, such as DeGroot's linear model, as detailed in [63]. Furthermore, some studies on some classes of QSO such as doubly stochastic quadratic operators DSQO [2], modified doubly stochastic quadratic operators MDSQO [51], extreme doubly stochastic quadratic operators EDSQO [54,64] and cubic quadratic stochastic operators CQSO [18] already have a consensus MAS with some restricted conditions and drawbacks in the consensus. Now, let us present the proposed model of SSQO for the consensus in MAS. The SSQO has the new notations in the transition matrix A ij,k in Equation (1). The transition matrix of stochastic matrix (SM) has entries elements (a ii.k = a ji,k ), which are either 0 or 1 2 , where the value 1 2 means that the individual member interacts with itself and 0 means idle state. Furthermore, the sum of each distributed matrices (A ij,1 , A ij,2 , . . . , A ij,k ) should have the same condition focusing on the matrix in QSO, which stipulates that the sum of its elements equals to m, as follows where i and j signify the number of rows and columns, while k is the number of matrices.

Theoretical Result
Theorem 1. Suppose that a team of individual agents in MAS receive information and should perform analysis. The node members will communicate with each other locally via SSQO to exchange the information for analysis and produce a decision. The team will reach to an average agreement to send the decision if at least one node has a positive initial status.
Proof. Consider a group of agents S 0 1 , S 0 2 , . . . , S 0 m that have at least one positive initial state where 0 ≤ s 0 i ≤ 1 , s 0 k ∈ S m−1 , while the others could be nonnegative. These states are MAX(s 0 i ) or MIN(s 0 i ).
Allow for the idea to participate amongst agents themselves via rules of the transition matrix A ij,k of SSQO.
The A ij,k represents sharing ideas in a stochastic matrix for the status of agents s j . Consequently, each agent in the group shares and updates its status idea via protocol SSQO (refer to Equations (1) and (2)), as follows Here, the process for each agent s i * A ij,k is multiplied again by the current statuses of agents, but, this time, in column form s (t) j to produce a new status for the agent. In other words, to produce a new status for the agent, the transition matrix A ij,k is multiplied by the current statuses of all agents in both row and column s (t) i,j forms. We want to show that, by protocol SSQO, each agent of MAS has communications of two agents with a coefficient equal to one or of s with coefficient equal to 1 2 . It means that each group has the probable communications of where s i is 0 ≤ s 0 i ≤ 1 and stochastic, as well as the coefficients for the transition matrix A ij,k in SSQO, then we get that where MIN(s where that means MIN(s  ). Hence, we get that In other words, at time t, the evolutionary stochastic matrix transfers the statuses of i ) increases piecemeal in the Equation (9) and, at the same time, MAX(s  i ) reach the same value equally without a loss of generality, which means that consensus is achieved and reached to the average 1/m.
The implication now is that the s (t+1) i has a (common) factor 1 m of s in any t, and k = [1, m].
Meanwhile, because s which means s (t+1) i is bounded and fixed to 1 m . This also means also that MAX(s Consequently, s (t+1) k converges.

Remark 1.
We should indicate that our model of protocol SSQO has been investigated for the required efficiency to avoid the drawbacks in the models of DeGroot's linear model [63] and non-linear models of CQSO [18], DSQO [2], and EDSQO [54], where the SSQO protocol • is a non-linear model as well as DeGroot's model, which means that it has fast convergence; • is distinguished by a less complicated computation as well as the non-linear stochastic operators of CQSO, DSQO, and EDSQO; and, • avoids the problem of periodic and non-changing initial values as in the case of EDSQO.
In the next section (Section 4: Discussion and Numerical Solution), we provide some practical examples in order to support the theoretic results for Theorem 1 and, for Remark 1, we provide comparison results in Section 5.
Here, the rules of SM are that the elements a ij,k are nonnegative and where i and j are row and column, respectively, and k is distributed evaluation matrix for each agent.
Although the rules of DSM are that elements a ij,k are nonnegative and Now, it is possible to present examples for all cases.

Three Agents
In the case of SM: In the case of DSM: Using Equation (1), we obtain Figure 1 shows the consensus for extreme values of the initial statuses for three agents . From the simulation results presented in Figures 1 and 2, we can see that SSQO achieves the consensus for three agents with initial values, with one agent having a full value while the others have zero value, as shown in Figure 1, and with positive random initial values, as shown in Figure 2. Moreover, the consensus is reached faster using DSM is than using SM. The second figure of

Initial Statuses
The consensus of 3 agents via SSQO-SM S1 S2 S3 Figure 2. The consensus of three agents by SSQO using SM and DSM with initial statuses of (0.162, 0.222, 0.616) (random).

Four Agents
In the case of SM: Symmetry 2020, 12, 1692 10 of 23 In the case of DSM: Using Equation (1), we obtain Figure 3 shows the consensus for extreme values of the initial statuses for 4 agents Symmetry 2020, 12, x FOR PEER REVIEW 10 of 22

Initial Statuses
The consensus of 4 agents via SSQO-SM S1 S2 S3 S4 The simulation result, as shown in in Figures 3 and 4, highlights that SSQO achieves the consensus for four agents with initial values. One agent has full value, while the others have zero value, as shown in Figure 3, while, in another case, the initial values are positive random values, as shown in Figure 4. Moreover, the consensus is reached faster using DSM than using SM. We can see this in Figures 3 and 4.

Five Agents
In the case of SM: In the case of DSM: Using Equation (1), we obtain  Figure 5 shows the consensus for extreme values of the initial statuses for five agents Figure 6 shows the consensus for random values of the initial statuses for five agents Figures 5 and 6 present the consensus for five agents, where one agent has full value while the others have zero value, as shown in Figure 5, while, in another case, the initial values are positive and random, as shown in Figure 6. Additionally, the consensus is reached faster using DSM than using SM. We can see this in the second figure of Figures 5 and 6.     Figures 5 and 6 present the consensus for five agents, where one agent has full value while the others have zero value, as shown in Figure 5, while, in another case, the initial values are positive and random, as shown in Figure 6. Additionally, the consensus is reached faster using DSM than using SM. We can see this in the second figure of Figures 5 and 6.
The consensus achieved in the initial statuses utilizing SSQO rules with SM and DSM is outlined in Figures 1-6. It can be observed that the consensus is reached faster by DSM than by SM. We can see in Figures 1-6 that the number of iterations in DSM required to reach a consensus is fewer than by SM, with 2-5 iterations as compared to 35-52 iterations, respectively.
Here, we show some examples of three, four, and five agents (although the simulation can also be applied for consensus of finite agents). Figure 7 shows the consensus of 10 agents and Figure 8 for 1000 agents.
The consensus achieved in the initial statuses utilizing SSQO rules with SM and DSM is outlined in Figures 1-6. It can be observed that the consensus is reached faster by DSM than by SM. We can see in Figures 1-6 that the number of iterations in DSM required to reach a consensus is fewer than by SM, with 2-5 iterations as compared to 35-52 iterations, respectively.
Here, we show some examples of three, four, and five agents (although the simulation can also be applied for consensus of finite agents). Figure 7 shows the consensus of 10 agents and Figure 8 for 1000 agents.     To conclude, in this section, we present some examples for three, four, and five agents using SSQO with the methods of calculation in order to prove that the SSQO reach a consensus. Figures 1-6 emphasize the result for Theorem 1, where Figures 1 and 2 for three agents, Figures 3 and 4 for four agents, and Figures 5 and 6 for five agents. Finally, the simulation results presented in Figures 7 and 8 show the results for 10 and 100 agents, respectively.

Comparison of the Consensus SSQO Model with Other Consensus Models
It is worthwhile to evaluate the efficiency of the proposed SSQO model in comparison to other models of the same structure such as DeGroot [63], CQSO [18], DSQO [2], and EDSQO [54]. The abstracted Table 1 illustrates the advantages and disadvantages of each model. It also shows the evolutionary operator process and the transition matrix rules for each model; whereas Table 2 indicates the special cases of drawbacks. Figures 9 and 10 clearly identify the weaknesses of the existing models: Figures 9 and 10 show that no consensus is reached by three agents for one and 10 times with different initial statuses. No consensus is reached due to the transition matrix being periodic (DeGroot and CSQO), and selfish communication in the transition matrix (DSQO and EDSQO). As mentioned in [63], the DeGroot model reaches a consensus if and only if the transition matrix is not periodic. As mentioned in [18], the CSQO model is based on the DeGroot model and, thus, also cannot reach a consensus in the periodic case of in the transition matrix. As mentioned in [2], the models DSQO and EDSQO cannot reach a consensus in the case of the selfish communication in the transition matrix. Moreover, in terms of the limit behavior of the agent statuses, we can observe that DSQO displays smoother and long-period oscillations, while DeGroot, CSQO, and EDSQO show a period one cycle with iterations, due to the fact that DSQO has no vertices (extreme) values in the transition matrix, while DeGroot, CSQO, and EDSQO have such values in the transition matrix. After 50 iterations in DSQO, the agent statuses may diverge; the elements of each agent obtain a share factor from the product of the statuses in the transition matrix; in other words, the communication among all agents only with one agent.
Thus, in this paper, we propose a new non-linear consensus model that converges to a consensus and involves less computation. In fact, Figures 9 and 10 show the weakness for the existing models of DeGroot, CSQO, DSQO, and EDSQO in one and 10 times for different random initial statuses for three agents (S1, S2, S3), respectively.
In general, we have noted that the transition matrix for SSQO could be any value for nondiagonal elements that have an interaction, because the sum of a ji.k and a ij,k under the conditions of SSQO is stochastic and equals one in any way for those agents who interact with others. Because the a ji.k and a ij,k are stochastic, it makes the transition matrix fixable. It means that the interactions among agents are not under restricted conditions.
The contribution of this work is the development of a new consensus model that combines four distinct advantages. The first advantage is that the model is non-linear, the second is that the consensus is achieved fast, the third is that it requires less computation, and the fourth is that the consensus is achieved in all cases following the condition rules of the transition matrix.

EDSQO [54]
• a ii,k = (0|1), a ij,k = 0 1 2 1 , a ji.k = a ij,k , Convergence in all cases • Some restricted conditions s This is a transition matrix under the rules of the DeGroot linear consensus model, which is the drawback that cannot reach to consensus.

CSQO [18]
This is a transition matrix under the rules of the CSQO non-linear consensus model, which is the drawback that cannot reach to consensus.

DSQO [2]
A i j,1 = This is a transition matrix under the rules of the proposed SSQO non-linear consensus model, which avoids the drawback that cannot reach to consensus.   Thus, in this paper, we propose a new non-linear consensus model that converges to a consensus and involves less computation. In fact, Figures 9 and 10 show the weakness for the existing models of DeGroot, CSQO, DSQO, and EDSQO in one and 10 times for different random initial statuses for three agents ( 1, 2, 3), respectively.

Conclusions
In this paper, we have established a new less complicated non-linear convergence protocol of SSQO to resolve the consensus problem in MAS. This protocol generalizes both the linear model of DeGroot's and non-linear model of QSO. We have considered several new concepts for the SSQO