Stability of a Fractional Opinion Formation Model with and without Leadership Using the New Generalized Hattaf Fractional Derivative

In this paper, we propose and analyze the dynamical behaviors of two opinion formation models, one with leadership and the other without leadership. The two proposed models are formulated by fractional differential equations (FDEs) with the frame of the new generalized Hattaf fractional (GHF) derivative. The stability in the sense of Mittag – Lef ﬂ er is rigorously established for both models. The convergence of agents ’ opinions to the consensus opinion is fully investigated. Numerical simulations are given to illustrate the analytical results.


Introduction
Opinion formation is a complex process that involves the development or change of beliefs, opinions, or viewpoints on a specific subject.It is influenced by various factors such as education, personal experience, culture, media, social interactions, and external influences.In the field of education, formal education acquired in schools and universities can have a significant impact on people's opinions on various issues.The media also plays a major role in opinion formation, as the information and messages disseminated by the media can influence an individual's perception of a particular issue.Social interactions, including those with family, friends, work colleagues, and the community, can also influence opinion formation.Additionally, the way opinions are formed can be influenced by the culture in which an individual resides.Furthermore, external influences such as opinion leaders, celebrities, interest groups, political groups, educational campaigns, and advertising can also play a role in shaping opinions [1][2][3][4][5][6][7].
Opinion formation models have been established in the discipline of mathematical modeling to describe the dynamics of how opinions arise and change over time within a population, taking into account factors, such as personal experiences, social interactions, and external sources of information.Such models are highly relevant in various fields such as politics [8], consumerism [9], and social media [10][11][12], as they provide organizations with valuable insights into public opinion and enable them to adapt their strategies accordingly.For instance, Degroot [9] presented the initial agent-based opinion formation model to discuss the importance of consensus in group decision-making processes, he suggested different approaches to achieving it and discussed methods of voting, compromise, negotiation, and the use of mathematical and statistical models.The author also pointed out that the appropriate method depends on the specific context of the group decision-making situation.Chen et al. [13] studied a fractional-order memristor neural network model that describes the effects of memristor memory and nonlinearity.They proved that the system is Mittag-Leffler stable.Furthermore, they proposed synchronization conditions for neural memory networks.Almeida et al. [14] used the theory of fractional differential equations (FDEs) to describe the opinion formation model and the proposed leader-follower control method.They showed that the proposed method can minimize the gap between the opinions of group members and the leader while limiting the influence of the leader on the opinions of other members.In [15], the authors proposed a leader-follower control approach to a fractional opinion formation model, in which the leader is responsible for directing the opinion formation of group members.The goal of the proposed approach was to minimize the discrepancy between the opinions of group members and the leader while limiting the influence of the leader on the opinions of other members.Somjaiwang and Ngiamsunthorn [16] investigated the exponential stability of a leader-follower opinion formation model based on a nonlinear system of FDEs.
It is important to note that the FDEs used in [13][14][15][16] have been formulated by the classical Caputo [17] fractional derivative.To avoid the singularity of such derivative, Caputo and Fabrizio [18] replaced the singular kernel with an exponential kernel.The version of Caputo-Fabrizio (FC) derivative was extended by Atangana and Baleanu [19] to Mittag-Leffler kernel.In 2020, Hattaf [20] developed a new generalized Hattaf fractional (GHF) derivative that includes the CF derivative [18] and Atangana-Baleanu (AB) derivative [19].This study aims to extend the exponential stability defined in [21] and used in [16], as well as the Mittag-Leffler stability introduced in [13] to FDEs with GHF derivative.
The second objective of this work is to propose two mathematical models incorporating the new GHF derivative for opinion formation with and without leadership.In order to accomplish this, the fundamental ideas and the development of the two fractional models are covered in Section 2. The Mittag-Leffler stability of the fractional model in the absence of leadership is investigated in Section 3. The stability and consensus of the second fractional model with leadership are established in Section 4. Three examples are provided in Section 5 to illustrate the analytical results.Finally, we end our current paper with a conclusion and some future works.

Basic Concepts and Models Formulation
In this section, we introduce the essential concepts and formulate models both with and without leadership.
Here, H 1 ða; bÞ: ¼ f f 2 L 2 ða; bÞ : f 0 2 L 2 ða; bÞg: be the Sobolev space of order one.Denote C D η; θ; θ a; t; w by D η; θ a; w .It follows from Hattaf's [20] study that the GHF integral associated with D η; θ a; w is given by the following definition: Definition 2 (Hattaf [20]).The GHF integral associated with D η; θ a; w is given by the following equation: where RL I θ a; w is the weighted Riemann-Liouville fractional integral of order θ defined by the following formula: Now, we consider the following FDE: where OðtÞ: 2 R n and g : ½0; þ1Þ × Ω → R n is a continuous locally Lipschitz function.
Lemma 2 (Hattaf [23]).Let λ>0 and OðtÞ: be a function satisfying the following inequality: Next, we propose two fractional opinion formation models.The first one without leadership, which is formulated by the following system: with the given initial conditions O i ð0Þ: ¼ ξ i 2 R, for i ¼ 1; 2; …; N. The state variable O i ðtÞ: represents the opinion of the ith follower at time t.The matrix A ¼ ½a ij N×N denotes the adjacency matrix of the social network connecting the followers, and f j ð⋅Þ: is the opinion function of the jth follower.
The second opinion formation fractional model with leadership is given by the following equation: where I i ðtÞ: represents external inputs, c i is a constant that indicates the leader's influence on the ith follower and O 0 ðtÞ: is the leader's opinion at time t.As in Equation ( 8), we consider System (9) with initial conditions O i ð0Þ:

Stability of the Model without Leadership
where XðtÞ: ¼ OðtÞ: − ŌðtÞ: .So, we have the following equation: where λ 1 >0 and λ 2 >0 are the minimum and the maximum eigenvalues of the matrix P, respectively.It follows from Lemma 1 of [24] that: Let us denote by δ the minimum eigenvalue of the matrix Q.
Since Q is a positive definite matrix, we have δ>0.We therefore have using Equation (11): By applying Lemma 2, we deduce the following equation: By Equation (11), we have the following equation: Therefore, it follows from Definition 3 that System (8) is Mittag-Leffler stable.This completes the proof.

Stability and Consensus of the Model with Leadership
In this section, we discuss the stability and consensus of the second opinion formation fractional model with leadership presented by System (9).
In the following, we consider the following assumptions: (A1) The functions f i , i ¼ 1; 2; …; N, are continuous and satisfy the following Lipschitz condition on R: where L i >0 are the Lipschitz constants.
(A2) The constants c i are positive for i ¼ 1; 2; …; N and satisfy the following equation: (A3) The functions I i are bounded, i.e., there exists a M i >0 such that: for every i ¼ 1; 2; …; N.
If u * i ðtÞ: is positive, then: If u * i ðtÞ: is negative, then: Hence: Consider the following function: Then: where Λ ¼ min 1≤i≤N Λ i .It follows from Lemma 2 that: By using Definition 3, we conclude that System (9) is Mittag-Leffler stable.
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Numerical Simulations
This section presents three examples as in [16] and [14] to demonstrate our analytical results using numerical simulations based on the numerical method presented in [25].
Example 6.Consider the following system: with the initial conditions O We notice that each agent is influenced by the opinions of its neighbors.Also, in a leaderless environment where each agent acts independently, the agents' opinions do not converge to a consensus state.
Example 7. Consider the following system: In this case, the leader O 0 ðtÞ: ¼ ξ 0 ¼ 0, f i ðt; ξ 0 Þ: ¼ 0, lim t→1 I i ðtÞ: ¼ 0, and L i ¼ 1 for i ¼ 1; 2; …; 6.As a result, the fractional opinion formation Model ( 39  In this example, we have f i ðt; ξ 0 Þ: ¼ 0, lim t→1 I i ðtÞ: ¼ 0, and L i ¼ 1 for i ¼ 1; 2; 3. System (40) is Mittag-Leffler stable, as well as the states of O 0 , O 1 , and O 2 converge to a consensus state ð1; 1; 1Þ: , in which all agents share the same opinion.Figures 7-9 demonstrate these results.In addition, we observe that the speed of consensus depends on the values of the fractional orders η and θ.More precisely, the time for arriving at the consensus with large fractional derivative order value is faster than that with small fractional derivative order value.Mathematical Problems in Engineering

Conclusion
In this work, we have presented two fractional opinion formation models involving the new GHF derivative.The first model is without leadership, while the second one is with leadership and time-dependent external inputs.At the beginning, we have proposed a definition for Mittag-Leffler stability in order to investigate the dynamical behaviors of such fractional models.The proposed definition extends that introduced in [13] and generalizes the exponential stability given in [21] and used in [16].Furthermore, the convergence of agents' opinions toward the consensus opinion was carefully examined.At the last, three numerical examples and corresponding numerical simulations have been given to illustrate our main analytical results.On the other hand, it would be very interesting to extend our work to the case of delay systems and to model the dynamics of opinion formation using the novel mixed fractional derivative [26] that covers the GHF and Caputo fractional derivatives.These issues will be the topics of future work.

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