Generalized Dynamics in Social Networks With Antagonistic Interactions
Shidong Zhai Wei Xing Zheng
math.OC














Generalized Dynamics in Social Networks With Antagonistic Interactions
    Shidong Zhai  and  Wei Xing Zheng



S. Zhai is with the Chongqing Key Laboratory of Complex Systems and Bionic Control, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China, and also with the School of Computing, Engineering and Mathematics, Western Sydney University, Sydney, NSW 2751, Australia (e-mail: zhaisd@cqupt.edu.cn).

W. X. Zheng is with the School of Computing, Engineering and Mathematics, Western Sydney University, Sydney, NSW 2751, Australia (e-mail: w.zheng@westernsydney.edu.au).

    
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In this paper, we investigate a general nonlinear model of opinion dynamics in which both state-dependent susceptibility to persuasion and antagonistic interactions are considered. According to the existing literature and socio-psychological theories, we examine three specializations of state-dependent susceptibility, that is, stubborn positives scenario, stubborn neutrals scenario, and stubborn extremists scenario. Interactions among agents form a signed graph, in which positive and negative edges represent friendly and antagonistic interactions, respectively. Based on Perron-Frobenius property of eventually positive matrices and LaSalle invariance principle, we conduct a comprehensive theoretical analysis of the generalized nonlinear opinion dynamics. We obtain some sufficient conditions such that the states of all agents converge into the subspace spanned by the right positive eigenvector of an eventually positive matrix. When there exists at least one entry of the right positive eigenvector which is not equal to one, the derived results can be used to describe different levels of an opinion. Finally, we present two examples to demonstrate the effectiveness of the theoretical findings.



Opinion dynamics, state-dependent susceptibility, signed graphs, eventually positive matrices, invariance principle.







§ INTRODUCTION



The study of opinion dynamics has attracted much attention in recent years. Based on some social psychology theories such as social comparison theory <cit.>, cognitive dissonance theory <cit.> and balance theory <cit.>, various mathematical models have been presented to model the evolution of opinion forming <cit.>. Among these models, the basis is DeGroot model <cit.>, which is a discrete-time and weighted averaging model. The corresponding continuous-time version of DeGroot model was studied in <cit.>. The system matrix of DeGroot model is the row-stochastic adjacency matrix of an interaction graph, and cannot change with the states. Moreover, DeGroot model cannot model social behavior of stubborn agents, which is often hard to be affected by other agents. In <cit.> DeGroot model, in which some agents' states are unchanged, was investigated.

To overcome the limitation of DeGroot model, Friedkin-Johnsen model <cit.> permits the agents to have different susceptibilities to persuasion. Although Friedkin-Johnsen model can model the behavior of stubborn agents, the behavior of agents cannot change with time and do not depend on current opinion or attitude. Various extensions of Friedkin-Johnsen model have been considered in <cit.>.
A typical opinion model with state-dependent interactions is the bounded confidence model <cit.>.
This model assumes that two agents can interact with each other when they are close enough. Some special bounded confidence models have been studied in <cit.>.

Recently, Amelkin et al. <cit.> proposed a continuous-time nonlinear model of polar opinion dynamics in which the agents' susceptibilities to persuasion depend on the agents' current opinion or attitude at hand, and interactions among agents are cooperative. According to different theories of social psychology, they investigated three specialized scenarios, that is, stubborn extremists, stubborn positives and stubborn neutrals. The scenario with stubborn extremists, based on social-psychological references <cit.>, assumes that the extreme opinions are resistant to change and neutral opinions will be attracted by extreme opinions, while the scenario with stubborn positives, based on <cit.>, is a special version of model with stubborn extremists. The scenario with stubborn neutrals, based on social comparison theory <cit.> and social norms <cit.>, supposes that the extreme opinions are more likely to change than the neutral opinions. Using the non-smooth analysis tool, they made a thorough theoretical analysis of the novel opinion dynamics model in <cit.>.

In this paper, we will study a generalized nonlinear opinion dynamics which includes both state-dependent susceptibility to persuasion and antagonistic interactions. Cooperative and antagonistic interactions usually coexist in social networks <cit.>. The signed graphs are often used to describe the cooperative and antagonistic interactions in social networks, with positive and negative edges denoting respectively the cooperative and antagonistic interactions. In social networks, the signed graphs are generally structurally balanced <cit.> in the sense that all nodes of the graph can be divided into two subsets, where the edges in each subset are positive and the edges connecting the two subsets are negative. Based on the balance theory <cit.>, the bipartite consensus for structurally balanced signed graphs has been investigated in <cit.>.
Generally speaking, when the signed graph is structurally balanced, the bipartite consensus problem can be transformed to a classical consensus or synchronization problem. However, when the signed graph is not structurally balanced, one should not expect the network to achieve bipartite consensus. Instead, the unanimity of opinion over a signed graph was studied in <cit.> when the adjacency matrix satisfies the eventually positive property <cit.>.

Our introduced model can be seen as a generalization of the opinion forming models in <cit.>. Comparing it with the model in <cit.>, our model can deal with the antagonistic interactions and accurately describe the degree of agent's opinion. For example, ten people voted for the Green Party in an election, but they did not support it at the same level. Hence, it is reasonable to use different positive or negative values to differentiate the opinion of these electors. Comparing it with the model in <cit.>, our model introduces state-dependent susceptibility to persuasion. This modification makes the dynamics of our model become more complex than the one in <cit.>. There also exist some unobservable behaviors in the linear model in <cit.> (see Example <ref> in Section <ref> for details).

This paper considers three specializations of state-dependent susceptibility which are drawn from <cit.>. We assume that all eigenvalues of the coupled system matrix have non-positive real parts, and the dominant eigenvalue is real and the corresponding eigenvector is a positive vector. We consider the directed and undirected signed graphs, respectively. Two cases about the adjacency matrix of a signed graph are examined: 1) the adjacency matrix is eventually positive; 2) the adjacency matrix is not eventually positive, but there exists a matrix such that the adjacency matrix has Perron-Frobenius property by adding it. When the dominant eigenvalue of the coupled system matrix is zero, utilizing Perron-Frobenius property of matrix with some negative entries and LaSalle invariance principle, we obtain sufficient conditions such that the states of the studied model will converge into the subspace spanned by the right positive eigenvector of the adjacency matrix. On the other hand, when the dominant eigenvalue of the coupled system matrix is non-positive, we derive sufficient conditions such that the general equilibrium points of the underlying model are asymptotically stable.

The remainder of the paper is organized as follows. Section <ref> presents notations and the problem statement. Section <ref> undertakes a comprehensive theoretical analysis of three specializations of state-dependent susceptibility, that is, stubborn positives scenario, stubborn neutrals scenario, and stubborn extremists scenario in generalized nonlinear opinion dynamics. Section <ref> gives two numerical examples to verify the obtained theoretical results. Finally, Section <ref> summarizes our conclusions and describes future work. Besides,  Appendix <ref> provides some definitions and facts about signed graphs and Appendix <ref> states some results of eventually positive matrices and Perron-Frobenius property, followed by some lemmas about positive semidefinite matrices given in Appendix <ref>.






§ PRELIMINARIES






 §.§ Notations


Let 1  (0) be a vector of appropriate dimension with all elements equal to 1 (0). Let ℝ^n_+:={x| x≥0}, ℝ^n_-:={x| x≤0}, and ℝ^n_-,+:={x| x≤0 or x≥0}, where x=[x_1 x_2 ⋯ x_n]^ T.  is the signum function. For a vector x∈ℝ^n, let (x):={x_1,x_2,⋯,x_n}, (x)^2:={x_1^2,x_2^2,⋯,x_n^2}, and ||x||_∞:=max_i|x_i| which is the infinity norm of x. For vectors x, y∈ℝ^n, x<y if x_i<y_i, i=1,2,⋯,n. For a real square matrix A=[a_ij]∈ℝ^n× n, A≥0 if a_ij≥0, and A>0 if a_ij>0, i,j=1,2,⋯,n. If A≥0  (>0), then A is called a nonnegative (positive) matrix. For a real symmetric square matrix B, B≽0  (≼0) means that B is positive semidefinite (negative semidefinite), and B≻0  (≺0) means that B is positive definite (negative definite). I represents an identity matrix of appropriate dimension. Let sp(A)={λ_1(A),⋯,λ_n(A)} denote the spectrum of matrix A, and max{λ_1(A),⋯,λ_n(A)} denote the dominant eigenvalue of matrix A, where λ_i(A), i=1,⋯,n are the eigenvalues of A. ρ(A) stands for the spectral radius of matrix A, which is the smallest real positive number such that ρ(A)≥ |λ_i(A)|, ∀ i=1,⋯,n. Denote the transpose of matrix A by A^ T, and let H(A)=1/2(A+A^ T). A matrix A is irreducible if and only if it cannot be transformed into a block upper-triangular form by simultaneous row/column permutations.





 §.§ Problem Statements


Consider a general model of opinion dynamics as follows:

    ẋ_i=a_i(x_i)(-σ_ix_i+∑_j=1^Nb_ijx_j), i=1,⋯,N,

where x_i(t)∈[-1,1] is the agent state, a_i(x_i)∈[0,1] is the susceptibility function which is state-dependent and Lipschitz in [-1,1], and σ_i>0 represents forgetting factors for the ith agent. B=[b_ij]∈ℝ^N× N is the adjacency matrix of a strongly connected signed digraph 𝒢(B), where b_ii=0, b_ij>0 indicates that agents i and j are friendly, and b_ij<0 indicates that agents i and j are antagonistic.

Let x=[x_1,⋯,x_N]^ T and Σ={σ_1,⋯,σ_N}. Then the generalized opinion dynamics (<ref>) can be written in matrix form as

    ẋ=A(x)Ex,

where A(x)={a_1(x_1),⋯,a_N(x_N)}, and E=B-Σ.

If there do not exist the susceptibility functions, then the generalized opinion dynamics (<ref>) becomes ẋ=Ex, which was investigated in <cit.>. However, when there exist the susceptibility functions a_i(x_i)∈[-1,1], the dynamic behavior of the generalized opinion dynamics (<ref>) will become more complex when compared to ẋ=Ex. If Σ=I and the adjacency matrix B  (b_ij∈{0,1}) is row-stochastic, then the generalized opinion dynamics (<ref>) becomes the model (1) considered in <cit.>. In <cit.>, the dynamic behavior of its model (1) was studied for some special susceptibility functions, that is, stubborn positives scenario A(x)=0.5(I-(x)), stubborn neutrals scenario A(x)=(x)^2, and stubborn extremists scenario A(x)=(I-(x)^2).

Our model (<ref>) can be viewed as a generalization of the models considered in <cit.>. In the next section, we will study the dynamic behavior of the generalized opinion dynamics (<ref>) under three special scenarios, that is, stubborn extremists, stubborn positives, and stubborn neutrals. In this paper, we assume that matrix E is eventually positive and all eigenvalues have non-positive real parts. Precisely, we adopt the following two assumptions.



The adjacency matrix B is eventually positive, and σ_i=σ_j≥ρ(B), ∀ i,j=1,⋯,N.




The adjacency matrix B is not eventually positive, and E can be represented as E=C-dI, where C=B+D is eventually positive, D=dI-Σ, and d≥ρ(C).








§ ANALYSIS OF GENERALIZED OPINION DYNAMICS



This section will study the dynamic behavior of the generalized opinion dynamics (<ref>) under three special scenarios (stubborn extremists, stubborn positives, and stubborn neutrals) respectively. Before moving forward, we first consider equilibrium points of the generalized opinion dynamics (<ref>).

All equilibrium points of the generalized opinion dynamics (<ref>) satisfy A(x)Ex=0. Suppose that ℐ_a,0={i| a_i(x_i)=0}, ℐ_a,+={i| a_i(x_i)>0}. First, if ℐ_a,0=∅, then all a_i(x_i)>0, i=1,⋯,N, and all equilibrium points satisfy Ex=0. Second, if ℐ_a,0≠∅ and ℐ_a,+≠∅, then there exists a permutation matrix P such that

    PA(x)P^ T=[[ A_1(x)      0;      0 A_2(x) ]],

where A_1(x)=0, A_2(x)>0. Let

    y=Px=[[ y_1; y_2 ]],

where y_1=|ℐ_a,0|. Then

    ẏ   =Pẋ=PA(x)Ex
       =PA(x)P^ TPEP^ Ty
       =PA(x)P^ TE̅y
       =[[        0        0;        0 A_2(y_2) ]][[ E̅_11 E̅_12; E̅_21 E̅_22 ]][[ y_1; y_2 ]],

where A_2(y_2)>0. Hence, all equilibrium points of (<ref>) satisfy y^∗_1∈{s| a_i(s)=0,∀ i}^|ℐ_a,0|, E̅_22y^∗_2=-E̅_21y^∗_1. In this case, all equilibrium points are x^∗=P^ Ty^∗. Third, if ℐ_a,+=∅, then all equilibrium points of the generalized opinion dynamics (<ref>) are x^∗∈{s| a_i(s)=0,∀ i}^N.





 §.§ Stubborn Positives Scenario


This subsection focuses on the dynamic behavior of the generalized opinion dynamics (<ref>) under the stubborn positives scenario, that is, A(x)=0.5(I-(x)).



Suppose that Assumption <ref> (Assumption <ref>) holds, 𝒢(B) is the signed digraph, and there exists a positive diagonal matrix Γ={γ_1,⋯,γ_N} such that H(Γ E)≼0 and H(Γ E)= E.


  * If x(0)<1, then lim_t→∞x(t)=α v_r, where v_r is the right positive eigenvector of B (C), α∈[-1/||v_r||_∞,1/||v_r||_∞).


  * If ℐ_a,0≠∅, ℐ_a,+≠∅, there exists a permutation matrix P such that the network (<ref>) has the form of (<ref>), E̅_22 is nonsingular, and -1≤-E̅_22^-1E̅_211<1, then lim_t→∞x(t)=x^∗ for all x(0)∈{x| x_i=1, i∈ℐ_a,0, x_i∈[-1,1), ∀ i∉ℐ_a,0}, where x^∗=P^ Ty^∗, and y^∗ is the equilibrium point of (<ref>).


  * If σ_i=σ_j=ρ(B) (d=ρ(C)), ∀ i,j=1,⋯,N, v_r≠1, then every solution starting in {x| x_i=1, ∀ i∈𝒥, x_i∈[-1,1), ∀ i∉𝒥} approaches β v_r, where 𝒥={j| v_rj=||v_r||_∞}, β=1/||v_r||_∞.



First, we consider the case that Assumption <ref> holds. The partition of the state space is illustrated in Fig. <ref>.

1) Because B is an eventually positive matrix, by Lemma <ref>, we have that matrix B is irreducible, ρ(B) is a simple positive eigenvalue of B and the corresponding right eigenvector v_r is positive. When Assumption <ref> holds, σ_i=σ_j≥ρ(B), ∀ i,j=1,⋯,N. If σ_i=ρ(B), ∀ i=1,⋯,N, then all eigenvalues of E=B-Σ have non-positive real parts. Moreover, 0 is a simple eigenvalue of E with a right positive eigenvector v_r. If σ_i=σ_j>ρ(B), ∀ i,j=1,⋯,N, then all eigenvalues of E have negative real parts, and its dominant eigenvalue has the right positive eigenvector v_r.



Let 𝒮(ε)=[-1,1-ε]^N, where 0<ε<2. Suppose that

    V(x)=∑_i=1^Nγ_i[-2x_i-2ln(1-x_i)].

Then V(x) is positive definite in 𝒮(ε). For all x∈𝒮(ε), the time derivative of V(x) along (<ref>) is given by

    V̇   =∑_i=1^N2γ_ix_iẋ_i/1-x_i
       =∑_i=1^Nx_iγ_ie_ix
       =x^ TH(Γ E)x,

where e_i is the ith row of matrix E. Since H(Γ E)≼0, we have

    V̇   =∑_i=1^Nx_iγ_ie_ix
       =x^ TH(Γ E)x
       ≤0

for all x∈𝒮(ε). Hence, 𝒮(ε) is positively invariant with respect to (<ref>). By Lemma <ref>, V̇=0 if and only if H(Γ E)x=0. Because H(Γ E)= E, H(Γ E) and E have the same null space. Thus, V̇=0 if and only if Ex=0. If σ_i=ρ(B), ∀ i=1,⋯,N, then V̇=0 if and only if x∈𝒫(ε)={x |  x=α v_r, α∈[-1/||v_r||_∞,1-ε/||v_r||_∞]}. If σ_i=σ_j>ρ(B), ∀ i,j=1,⋯,N, then V̇=0 if and only if x=0. By LaSalle invariance principle, the solution of (<ref>) with initial condition x(0)∈𝒮(ε) converges into 𝒫(ε). If x(0) is sufficiently close to 1, then ε→ 0. Therefore, for all x(0)<1, lim_t→∞x(t)=α v_r, α∈[-1/||v_r||_∞,1/||v_r||_∞).

2) Let 𝒮_ℐ_a,0(ε)={x| x_i=1, ∀ i∈ℐ_a,0,  x_i∈[-1,1-ε], ∀ i∉ℐ_a,0}, where 0<ε<2. When x(0)∈𝒮_ℐ_a,0(ε), the solution of (<ref>) satisfies x_i(t)≡ 1, ∀ i∈ℐ_a,0. P is a permutation matrix such that

    y=Px=[[ y_1; y_2 ]],

where y_1=1,  y_1=|ℐ_a,0|. Then the network (<ref>) becomes

    ẏ   =[[            0            0;            0 0.5(I-(y_2)) ]][[ E̅_11 E̅_12; E̅_21 E̅_22 ]][[ y_1; y_2 ]].

Hence,

    ẏ_2=0.5(I-(y_2))(E̅_22y_2+E̅_211),

where y_2∈[-1,1-ε]^|ℐ_a,+|.

Since E̅_22 is nonsingular, we have that (E̅_22)≠ 0 and there exists a unique y_2^∗ such that E̅_22y_2^∗=-E̅_211. Let s=y_2-y_2^∗. Then we obtain

    ṡ=0.5(I-(s+y_2^∗))E̅_22s.

where s∈[-1-y_2^∗,1-ε-y_2^∗]^|ℐ_a,+|. Assume that

    V(s)
      =-∑_i=1^N̅γ̅_i+|ℐ_a,0|[2s_i+2(1-y_2i^∗)ln(1-y_2i^∗-s_i)],

where N̅=N-|ℐ_a,0|, Γ̅={γ̅_1,⋯,γ̅_N}=PΓ P^ T. Let Γ̅_1={γ̅_1,⋯,γ̅_|ℐ_a,0|} and Γ̅_2={γ̅_|ℐ_a,0|+1,⋯,γ̅_N}. It is easy to see from (<ref>) that V(s) is continuously differentiable in  [-1-y_2^∗,1-ε-y_2^∗]^|ℐ_a,+|. The time derivative of V(s) along (<ref>) is calculated as follows:

    V̇   =∑_i=1^N-|ℐ_a,0|2γ̅_i+|ℐ_a,0|s_iṡ_i/1-y_2i^∗-s_i
       =∑_i=1^N-|ℐ_a,0|s_iγ̅_i+|ℐ_a,0|e̅_22is
       =s^ TH(Γ̅_2E̅_22)s,

where e̅_22i is the ith row of matrix E̅_22. Since H(Γ E)≼0 and H(Γ̅_2E̅_22) is a principal submatrix of H(Γ E), we have H(Γ̅_2E̅_22)≼0.
Hence,

    V̇   =∑_i=1^N-|ℐ_a,0|s_iγ̅_i+|ℐ_a,0|e̅_22is
       =s^ TH(Γ̅_2E̅_22)s
       ≤ 0

for all s∈ [-1-y_2^∗,1-ε-y_2^∗]^|ℐ_a,+|.

Thus, [-1-y_2^∗,1-ε-y_2^∗]^|ℐ_a,+| is positively invariant with respect to (<ref>). By LaSalle invariance principle, the solution of (<ref>) with initial condition s(0)∈ [-1-y_2^∗,1-ε-y_2^∗]^|ℐ_a,+| converges to y_2^∗. When ε→ 0^+, we have 1-ε→1^-. Hence, if x(0)∈{x| x_i=1, i∈ℐ_a,0,  x_i∈[-1,1), ∀ i∉ℐ_a,0}, then lim_t→∞x(t)=x^∗.

3) Let 𝒮_0(ε)={x| x_i=1, ∀ i∈𝒥, x_i∈[-1,1-ε], ∀ i∉𝒥}, where 0<ε<2, 𝒥={j| v_rj=||v_r||_∞}. There exists a permutation matrix P such that the network (<ref>) takes the following form

    ẏ   =[[            0            0;            0 0.5(I-(y_2)) ]][[ E̅_11 E̅_12; E̅_21 E̅_22 ]][[ y_1; y_2 ]],

where

    y=Px=[[ y_1; y_2 ]]

with y_1=1,  y_1=|𝒥|. Then we can get

    ẏ_2=0.5(I-(y_2))(E̅_22y_2+E̅_211).

Because σ_i=σ_j=ρ(B) (d=ρ(C)), ∀ i,j=1,⋯,N, v_r≠1 and 𝒥={j| v_rj=||v_r||_∞}, there exists a unique solution y_2^∗ such that E̅_22y_2^∗+E̅_211=0. From a geometric point of view, the point

    x^∗=P^ T[[     1; y_2^∗ ]]

is the intersection point of the line α v_r with [-1,1]^N. Similar to the arguments presented in 2), we can obtain that every solution starting in 𝒮_0(ε) approaches x^∗= β v_r, where β=1/||v_r||_∞. When ε→ 0^+, we can get that every solution starting in {x| x_i=1, ∀ i∈𝒥,  x_i∈[-1,1), ∀ i∉𝒥} tends to β v_r.

Second, we consider the case that Assumption <ref> holds. Because C=B+D is eventually positive, by  Lemma <ref>, matrix C is irreducible, ρ(C) is a simple positive eigenvalue of C and the corresponding right eigenvector v_r is positive. Hence, E=C-dI has the eigenvalue ρ(C)-d with an eigenvector v_r. If d=ρ(C), then all eigenvalues of E have non-positive real parts, and 0 is a simple eigenvalue of E with a right positive eigenvector v_r. If d>ρ(C), then all eigenvalues of E have negative real parts, and its dominant eigenvalue has the right positive eigenvector v_r.

On this basis, one can use the similar methods presented above to acquire the results in 1), 2) and 3) in the case of Assumption <ref>, and details are omitted here for brevity. This completes the proof of Theorem <ref>.




Suppose that Assumption <ref> (Assumption <ref>) holds and the signed graph 𝒢(B) is undirected.


  * If x(0)<1, then lim_t→∞x(t)=α v_r, where v_r is the right positive eigenvector of B (C), α∈[-1/||v_r||_∞,1/||v_r||_∞).


  * If ℐ_a,0≠∅, ℐ_a,+≠∅, there exists a permutation matrix P such that the network (<ref>) has the form of (<ref>), E̅_22 is nonsingular, and -1≤-E̅_22^-1E̅_211<1, then lim_t→∞x(t)=x^∗ for all x(0)∈{x| x_i=1, i∈ℐ_a,0,  x_i∈[-1,1), ∀ i∉ℐ_a,0}, where x^∗=P^ Ty^∗, and y^∗ is the equilibrium points of (<ref>).


  * If σ_i=σ_j=ρ(B) (d=ρ(C)), ∀ i,j=1,⋯,N, v_r≠1, then every solution starting in {x| x_i=1, ∀ i∈𝒥, x_i∈[-1,1), ∀ i∉𝒥} approaches β v_r, where 𝒥={j| v_rj=||v_r||_∞}, β=1/||v_r||_∞.



Because the signed graph 𝒢(B) is undirected, it can be known that matrix E is symmetric and diagonalizable. By Remark <ref> given in Subsection <ref>, we have H(E)≼0 and H(E)= E. Hence, it follows from Theorem <ref> that 1), 2) and 3) of this corollary hold. This completes the proof of Corollary <ref>.


The stubborn positives scenario describes the case that the agents only at one end of the opinion spectrum are stubborn. For this scenario, an example about two smartphone brands was presented in <cit.>. The opinion values 1 and -1 mean an aggressive marketing of the brand and a neutral marketing of the brand, respectively. Opinion -1 may be changed by other opinions, but opinion 1 is stubborn and cannot be influenced by other opinions. The above Theorem <ref> and Corollary <ref> show that the states of the generalized opinion dynamics (<ref>) are convergent under different initial conditions.





 §.§ Stubborn Neutrals Scenario


This subsection examines the dynamic behavior of the generalized opinion dynamics (<ref>) under the stubborn neutrals scenario, that is, A(x)=(x)^2.



Consider the generalized opinion dynamics (<ref>) under the stubborn neutrals scenario A(x)=(x)^2. Then x(t) has nonnegative derivatives.


Let x_t(z) be the solution of the generalized opinion dynamics (<ref>) with initial condition z∈ℝ^N. Define M(t)=Dx_t(z), F(x)=(x)^2Ex, where Dx_t(z) is the Jacobian matrix. Then M(t) satisfies the variational equation

    Ṁ=DF(x)M, M(0)=I,

where DF(x) is the Jacobian matrix. It is easy to get

    DF(x)=(x)[[    η _1 e_12x_1       ⋯ e_1Nx_1; e_21x_2     η_2       ⋯ e_2Nx_2;       ⋯       ⋯       ⋯       ⋯; e_N1x_N e_N2x_N       ⋯     η_N;         ]],

where η_i=e_iix_i+2∑_j=1^Ne_ijx_j, i=1,2,⋯,N. We can obtain that M(t)=0 when x=0. Hence, M(t)≥0, that is, x(t) has nonnegative derivatives.




Suppose that Assumption <ref> (Assumption <ref>) holds, 𝒢(B) is the signed digraph, and there exists a positive diagonal matrix Γ={γ_1,⋯,γ_N} such that H(Γ E)≼0 and H(Γ E)= E.
If σ_i=σ_j=ρ(B) (d=ρ(C)), ∀ i,j=1,⋯,N, then


  * lim_t→∞x(t)=α v_r when x(0)>0, where v_r is the right positive eigenvector of B (C), α∈(0,1/||v_r||_∞];


  * lim_t→∞x(t)=α v_r when x(0)<0, where α∈[-1/||v_r||_∞,0);


  * lim_t→∞x(t)=0 otherwise.

If σ_i=σ_j>ρ(B) (d>ρ(C)), ∀ i,j=1,⋯,N, then lim_t→∞x(t)=0, ∀ x(0)∈ [-1,1]^N.


First, we consider the case that Assumption <ref> holds. The corresponding state space is partitioned as depicted in Fig. <ref>.



When σ_i=σ_j=ρ(B), ∀ i,j=1,⋯,N, and B is eventually positive matrix, by Lemma <ref>, we have that matrix B is irreducible, ρ(B) is a simple positive eigenvalue of B and the corresponding right eigenvector v_r is positive. In this case, the equilibrium points are x^∗=α v_r, α∈[-1/||v_r||_∞,1/||v_r||_∞].
Specifically, if ℐ_a,0=∅, then all equilibrium points satisfy Ex=0. Hence, the equilibrium points are x^∗=α v_r. If ℐ_a,+=∅, then the equilibrium points are x^∗=0. If ℐ_a,0≠∅ and ℐ_a,+≠∅, then there exists a permutation matrix P such that

    E̅   =PEP^ T,
       =[[ E̅_11 E̅_12; E̅_21 E̅_22 ]],

and all equilibrium points are x^∗=P^ Ty^∗, y^∗_1=0, E̅_22y^∗_2=0. If E̅_22 is nonsingular, then y^∗_2=0 and x^∗=0. Assume that E̅_22 is singular, and there exists a nonzero vector ξ such that E̅_22ξ=0. Let x=[0^ T ξ^ T]^ T. Since there exists a positive diagonal matrix Γ such that H(Γ E)≼0, and H(Γ E)= E, H(Γ E) and E have the same null space. As P is a permutation matrix, H(PΓ EP^ T) and PEP^ T have the same null space. Suppose that Γ̅={Γ̅_1,Γ̅_2}=PΓ P^ T. Then x^ TH(PΓ EP^ T)x=ξ^ TH(Γ̅_2E̅_22)ξ=0. Thus, PEP^ Tx=0. Because x=[0^ T ξ^ T]^ T, PEP^ Tx=0 if and only if ξ=0. This is a contradiction with the assumption that ξ is a nonzero vector. Therefore, if ℐ_a,0≠∅ and ℐ_a,+≠∅, then the equilibrium points are x^∗=0.

If σ_i=σ_j>ρ(B), ∀ i,j=1,⋯,N, then we can use the same method to obtain that the equilibrium points are x^∗=0.

1) If x(0)>0, then there exists ε∈(0,1) such that x(0)∈[ε,1]^N. Let

    V(x)=2∑_i=1^Nγ_iln(1+x_i).

So V(x) is continuously differentiable in 𝒮_+=[ε,1]^N. By Lemma <ref>, x(t) has nonnegative derivatives, that is, ẋ_i≥0, ∀ i=1,⋯,N. For all x∈𝒮_+, the time derivative of V(x) along (<ref>) is given by

    V̇   =2∑_i=1^Nγ_iẋ_i/1+x_i
       ≤ 2∑_i=1^Nγ_iẋ_i/2x_i
       =x^ TH(Γ E)x
       ≤0

for all x∈𝒮_+. Hence, 𝒮_+ is positively invariant with respect to (<ref>). V̇=0 if and only if x∈𝒫_+={x  |  x=α v_r, α∈[ε/||v_r||_∞, 1/||v_r||_∞]}. By LaSalle invariance principle, the solution of (<ref>) with initial condition x(0)∈𝒮_+ converges into 𝒫_+. Hence, if x(0)>0, then lim_t→∞x(t)=α v_r, α∈(0,1/||v_r||_∞].

2) If x(0)<0, then there exists ε∈(0,1) such that x(0)∈[-1,-ε]^N. Let

    V(x)=2∑_i=1^Nγ_iln(ε-x_i).

So V(x) is continuously differentiable in 𝒮_-=[-1,-ε]^N. Because x(t) has nonnegative derivatives according to Lemma <ref>, the time derivative of V(x) along (<ref>) is found to be

    V̇   =2∑_i=1^Nγ_iẋ_i/x_i-ε
       ≤2∑_i=1^Nγ_iẋ_i/2x_i
       =x^ TH(Γ E)x
       ≤0

for all x∈𝒮_-. Hence, 𝒮_- is positively invariant with respect to (<ref>). V̇=0 if and only if x∈𝒫_-={x  |  x=α v_r, α∈[-1/||v_r||_∞,-ε/||v_r||_∞]}. From LaSalle invariance principle, it follows that the solution of (<ref>) with initial condition x(0)<0 converges into 𝒫_-.

3) Let 𝒮_±={x  |  x∈[-1,1]^N, ∏_i=1^Nx_i=0}⋃{x  |  x∈[-1,1]^N, ∃ i,j  s.t. (x_ix_j)=-1}. By Lemma <ref>, x(t) has nonnegative derivatives, that is, ẋ_i≥0, ∀ i=1,⋯,N. From the generalized opinion dynamics (<ref>) with A(x)=(x)^2, it can be seen that (x_i(t))=(x_i(0)). For arbitrary x(0)∈𝒮_± with x(0)≠0, without loss of generality, assume that x_i(0)>0, i=1,2,⋯,l_1, x_i(0)=0, i=l_1+1,⋯,l_2, x_i(0)<0, i=l_2+1,⋯,N, where 1<l_1<l_2<N are positive constants. Define

    V(x)=2∑_i=1^l_2γ_iln(x_i+1)+2∑_i=l_2+1^Nγ_iln(ε-x_i),

where x(t) is the solution of the generalized opinion dynamics (<ref>) with initial condition x(0), and ε is a sufficiently small positive constant. Let 𝒮(ε)={x| x∈[-1,1]^N, x_i∈[-1,-ε], i=l_2+1,⋯,N}. Then V(x) is continuously differentiable in 𝒮_±⋂𝒮(ε). The time derivative of V(x) along (<ref>) is given by

    V̇   =2∑_i=1^l_1γ_iẋ_i/1+x_i+2∑_i=l_2+1^Nγ_iẋ_i/x_i-ε
       ≤ 2∑_i=1^l_1γ_iẋ_i/2x_i+2∑_i=1^Nγ_iẋ_i/2x_i
       =x^ TH(Γ E)x

for all x∈𝒮_±⋂𝒮(ε). Since x^ TH(Γ E)x=0, x∈𝒮_± if and only if x=0, V̇<0 for all x∈𝒮_±⋂𝒮(ε). Hence, V(x) is strictly monotonically decreasing on 𝒮_±⋂𝒮(ε), and V(x) has a minimum value at x̅ with x̅_i=0, i=1,⋯,l_2, x̅_i=-ε, i=l_2+1,⋯,N. Moreover, V̇(x̅)<0. Thus, the solution of (<ref>) with initial condition x(0) converges to 0. For arbitrary x(0)∈𝒮_± , lim_t→∞x(t)=0.

If σ_i=σ_j>ρ(B), ∀ i,j=1,⋯,N, then H(E)≺0, and Ex=0 if and only if x=0. Hence, using the method shown above, it is easy to obtain that lim_t→∞x(t)=0, ∀ x(0)∈ [-1,1]^N.

Second, we turn to the case that Assumption <ref> holds.
Since E=C-dI, C=B+D is eventually positive, and d≥ρ(C), with the above-presented similar method we arrive at that the equilibrium points are x^∗=α v_r, α∈[-1/||v_r||_∞,1/||v_r||_∞].
Precisely, if ℐ_a,0=∅, then the equilibrium points are x^∗=α v_r. If ℐ_a,+=∅, then the equilibrium points are x^∗=0. If ℐ_a,0≠∅ and ℐ_a,+≠∅, then all equilibrium points are x^∗=0. Therefore, the results of this theorem under Assumption <ref> can be derived by means of the method given above, but the actual derivation is skipped here. The proof of Theorem <ref> is thus complete.


When the signed graph 𝒢(B) is undirected, we have that matrix E is symmetric and diagonalizable. Then we can obtain the following corollary from Theorem <ref>. Its proof is similar to that of Corollary <ref>, so it is omitted here.



Suppose that Assumption <ref> (Assumption <ref>) holds and the signed graph 𝒢(B) is undirected.
If σ_i=σ_j=ρ(B) (d=ρ(C)), ∀ i,j=1,⋯,N, then


  * lim_t→∞x(t)=α v_r when x(0)>0, where v_r is the right positive eigenvector of B (C), α∈(0,1/||v_r||_∞];


  * lim_t→∞x(t)=α v_r when x(0)<0, where α∈[-1/||v_r||_∞,0);


  * lim_t→∞x(t)=0 otherwise.

If σ_i=σ_j>ρ(B) (d>ρ(C)), ∀ i,j=1,⋯,N, then lim_t→∞x(t)=0, ∀ x(0)∈ [-1,1]^N.


The stubborn neutrals scenario represents the situation that the neutral opinions cannot be altered, while the extreme opinions may be easily affected by other opinions. Theorem <ref> and Corollary <ref> establish the convergence of the generalized opinion dynamics (<ref>) under different initial conditions for this scenario.





 §.§ Stubborn Extremists Scenario


This subsection concentrates on the dynamic behavior of the generalized opinion dynamics (<ref>) under the stubborn extremists scenario, that is, A(x)=(I-(x)^2).



Suppose that Assumption <ref> (Assumption <ref>) holds, 𝒢(B) is the signed digraph, and there exists a positive diagonal matrix Γ={γ_1,⋯,γ_N} such that H(Γ E)≼0 and H(Γ E)= E.


  * If x(0)∈ (-1,1)^N, then lim_t→∞x(t)=α v_r, where v_r is the right positive eigenvector of B (C), and α∈(-1/||v_r||_∞,1/||v_r||_∞).


  * If ℐ_a,0≠∅, ℐ_a,+≠∅, there exists a permutation matrix P such that the network (<ref>) has the form of (<ref>), E̅_22 is nonsingular, and -1<-E̅_22^-1E̅_21y_1^∗<1, then lim_t→∞x(t)=x^∗ for all x(0)∈{x| |x_i|=1, i∈ℐ_a,0,  x_i∈(-1,1), ∀ i∉ℐ_a,0}, where x^∗=P^ Ty^∗, and y^∗ is the equilibrium point of (<ref>).


  * If σ_i=σ_j=ρ(B) (d=ρ(C)), ∀ i,j=1,⋯,N, v_r≠1, then every solution starting in {x| x_i=1, ∀ i∈𝒥, x_i∈(-1,1), ∀ i∉𝒥} approaches 1/||v_r||_∞v_r, where 𝒥={j| v_rj=||v_r||_∞}.


  * If σ_i=σ_j=ρ(B) (d=ρ(C)), ∀ i,j=1,⋯,N, v_r≠1, then every solution starting in {x| x_i=-1, ∀ i∈𝒥,  x_i∈(-1,1), ∀ i∉𝒥} approaches -1/||v_r||_∞v_r.



First, we consider the case under Assumption <ref>, and the partition of the state space is shown in Fig. <ref>.

If Assumption <ref> holds, then 0 is a simple eigenvalue of E with a right positive eigenvector v_r when σ_i=ρ(B), ∀ i=1,⋯,N. When σ_i=σ_j>ρ(B), ∀ i,j=1,⋯,N, all eigenvalues of E have negative real parts, and its dominant eigenvalue has the right positive eigenvector v_r.



1) Let 𝒮(ε)={x| -1+ε≤ x_i≤1-ε}, where 0<ε<1. Introduce

    V(x)=-1/2∑_i=1^Nγ_iln(1-x_i^2).

For x∈𝒮(ε), V(x) is positive definite, and the time derivative of V(x) along (<ref>) is found to be

    V̇   =∑_i=1^Nγ_ix_iẋ_i/1-x_i^2
       =x^ TH(Γ E)x.

Since H(Γ E)≼0, we have V̇≤ 0. Hence, 𝒮(ε) is positively invariant with respect to (<ref>). By Lemma <ref>, V̇=0 if and only if H(Γ E)x=0. Because H(Γ E)= E, V̇=0 if and only if Ex=0. If σ_i=ρ(B), ∀ i=1,⋯,N, then V̇=0 if and only if x∈𝒫(ε)={x  |  x=α v_r, α∈[--1+ε/||v_r||_∞,1-ε/||v_r||_∞]}. If σ_i=σ_j>ρ(B), ∀ i,j=1,⋯,N, then V̇=0 if and only if x=0. By LaSalle invariance principle, the solution of (<ref>) with initial condition x(0)∈𝒮(ε) converges into 𝒫(ε). If x(0) is sufficiently close to 1 or -1, then ε→ 0. Hence, for all x(0)∈(-1,1)^N, lim_t→∞x(t)=α v_r, where α∈(-1/||v_r||_∞,1/||v_r||_∞).

2) Let 𝒮_ℐ_a,0(ε)={x| |x_i|=1, ∀ i∈ℐ_a,0,  x_i∈[ε-1,1-ε], ∀ i∉ℐ_a,0}, where 0<ε<1. When x(0)∈𝒮_ℐ_a,0(ε), the solution of (<ref>) satisfies |x_i(t)|≡ 1, ∀ i∈ℐ_a,0. P is a permutation matrix such that

    y=Px=[[ y_1; y_2 ]],

where |y_1i|=1,  y_1=|ℐ_a,0|. Then the network (<ref>) becomes

    ẏ   =[[           0           0;           0 (I-(y_2)^2) ]][[ E̅_11 E̅_12; E̅_21 E̅_22 ]][[ y_1; y_2 ]].

Hence,

    ẏ_2=(I-(y_2)^2)(E̅_22y_2+E̅_21y_1^∗),

where y_2∈[ε-1,1-ε]^|ℐ_a,+|.

Since E̅_22 is nonsingular, there exists a unique y_2^∗ such that E̅_22y_2^∗=-E̅_21y_1^∗. Let s=y_2-y_2^∗. Then we have

    ṡ=(I-(s+y_2^∗)^2)E̅_22s,

where s∈[ε-1-y_2^∗,1-ε-y_2^∗]^|ℐ_a,+|. Assume that

    V(s)=-1/2∑_i=1^N̅γ̅_i+|ℐ_a,0|[y_2i^∗lnΦ_1(s_i)/Φ_2(s_i)-lnΦ_2(s_i)],

where N̅=N-|ℐ_a,0|,
Φ_1(s_i)=1+(s_i+y_2i^∗), Φ_2(s_i)=1-(s_i+y_2i^∗), and Γ̅={γ̅_1,⋯,γ̅_N}=PΓ P^ T. Let Γ̅_1={γ̅_1,⋯,γ̅_|ℐ_a,0|} and Γ̅_2={γ̅_|ℐ_a,0|+1,⋯,γ̅_N}.
It is easy to see from (<ref>) that V(s) is continuously differentiable in [ε-1-y_2^∗,1-ε-y_2^∗]^|ℐ_a,+|. The time derivative of V(s) along (<ref>) is calculated as follows

    V̇   =∑_i=1^N-|ℐ_a,0|γ̅_i+|ℐ_a,0|s_iṡ_i/1-(s_i+y_2i^∗)^2
       =∑_i=1^N-|ℐ_a,0|s_iγ̅_i+|ℐ_a,0|e̅_22is
       =s^ TH(Γ̅_2E̅_22)s,

where e̅_22i is the ith row of matrix E̅_22.
Since H(Γ E)≼0 and H(Γ̅_2E̅_22) is a principal submatrix of H(Γ E), we have H(Γ̅_2E̅_22)≼0. Hence,

    V̇   =∑_i=1^N-|ℐ_a,0|s_iγ̅_i+|ℐ_a,0|e̅_22is
       =s^ TH(Γ̅_2E̅_22)s
       ≤ 0

for all s∈ [ε-1-y_2^∗,1-ε-y_2^∗]^|ℐ_a,+|.

Thus, [ε-1-y_2^∗,1-ε-y_2^∗]^|ℐ_a,+| is positively invariant with respect to (<ref>). By LaSalle invariance principle, the solution of (<ref>) with initial condition s(0)∈ [ε-1-y_2^∗,1-ε-y_2^∗]^|ℐ_a,+| converges to y_2^∗. When ε→ 0^+, it leads to 1-ε→1^- and ε-1→-1^+. Hence, if x(0)∈{x| |x_i|=1, i∈ℐ_a,0,  x_i∈(-1,1), ∀ i∉ℐ_a,0}, then lim_t→∞x(t)=x^∗.

3) Let 𝒮_1(ε)={x| x_j=1,∀ j∈𝒥, x_i∈ [ε-1,1-ε], ∀ i∉𝒥}, where 0<ε<1. For x∈𝒮_1(ε), there exists a permutation matrix P such that the network (<ref>) has the following form

    ẏ   =[[           0           0;           0 (I-(y_2)^2) ]][[ E̅_11 E̅_12; E̅_21 E̅_22 ]][[ y_1; y_2 ]],

where

    y=Px=[[ y_1; y_2 ]]

with y_1=1,  y_1=|𝒥|. Then we can get

    ẏ_2=(I-(y_2)^2)(E̅_22y_2+E̅_211).

Because v_r≠1 and 𝒥={j| v_rj=||v_r||_∞}, there exists a unique solution y_2^∗ such that E̅_22y_2^∗+E̅_211=0. From a geometric point of view, the point

    x^∗=P^ T[[     1; y_2^∗ ]]

is the intersection point of the line α v_r with (-1,1]^N. Similar to the arguments presented in 2), we can obtain that every solution starting in 𝒮_1(ε) approaches x^∗=1/||v_r||_∞v_r. When ε→ 0^+, we can have that every solution starting in {x| x_i=1, ∀ i∈𝒥, x_i∈(-1,1), ∀ i∉𝒥} approaches 1/||v_r||_∞v_r.

4) Let 𝒮_2(ε)={x| x_i=-1,∀ i∈𝒥, x_i∈ [ε-1,1-ε], ∀ i∉𝒥}, where 0<ε<1. For x(0)∈𝒮_2(ε), x_i≡-1, ∀ i∈𝒥. There exists a permutation matrix P such that the network (<ref>) takes the following form

    ẏ   =[[           0           0;           0 (I-(y_2)^2) ]][[ E̅_11 E̅_12; E̅_21 E̅_22 ]][[ y_1; y_2 ]],

where

    y=Px=[[ y_1; y_2 ]]

with y_1=-1,  y_1=|𝒥|. Hence,

    ẏ_2=(I-(y_2)^2)(E̅_22y_2-E̅_211).

Because σ_i=σ_j=ρ(B), ∀ i,j=1,⋯,N, v_r≠1, there exists a unique solution y_2^∗ such that E̅_22y_2^∗-E̅_211=0. From a geometric viewpoint, the point

    x^∗=P^ T[[    -1; y_2^∗ ]]

is the intersection point of the line α v_r with [-1,1)^N. Similar to 3), we can derive that every solution starting in {x| x_i=-1, ∀ i∈𝒥, x_i∈(-1,1), ∀ i∉𝒥} approaches -1/||v_r||_∞v_r.

Second, we deal with the case that Assumption <ref> holds. Similar to Theorem <ref>, we can know that E=C-dI has the eigenvalue ρ(C)-d with eigenvector v_r. If d=ρ(C), then all eigenvalues of E have non-positive real parts, and 0 is a simple eigenvalue of E with a right positive eigenvector v_r. If d>ρ(C), then all eigenvalues of E have negative real parts, and its dominant eigenvalue has a right positive eigenvector v_r.

Finally, we can adopt the similar methods presented above to get 1), 2), 3) and 4) under Assumption <ref>, and we omit details here. This completes the proof of Theorem <ref>.


When the signed graph 𝒢(B) is undirected, the next corollary is an immediate result from Theorem <ref>. The proof is skipped here since it is similar to that of Corollary <ref>.



Suppose that Assumption <ref> (Assumption <ref>) holds and the signed graph 𝒢(B) is undirected.


  * If x(0)∈ (-1,1)^N, then lim_t→∞x(t)=α v_r, where v_r is the right positive eigenvector of B (C), and α∈(-1/||v_r||_∞,1/||v_r||_∞).


  * If ℐ_a,0≠∅, ℐ_a,+≠∅, there exists a permutation matrix P such that the network (<ref>) has the form of (<ref>), E̅_22 is nonsingular, and -1<-E̅_22^-1E̅_211<1, then lim_t→∞x(t)=x^∗ for all x(0)∈{x| |x_i|=1, i∈ℐ_a,0, x_i∈(-1,1), ∀ i∉ℐ_a,0}, where x^∗=P^ Ty^∗, and y^∗ is the equilibrium point of (<ref>).


  * If σ_i=σ_j=ρ(B) (d=ρ(C)), ∀ i,j=1,⋯,N, v_r≠1, then every solution starting in {x| x_i=1, ∀ i∈𝒥, x_i∈(-1,1), ∀ i∉𝒥} approaches 1/||v_r||_∞v_r, where 𝒥={j| v_rj=||v_r||_∞}.


  * If σ_i=σ_j=ρ(B) (d=ρ(C)), ∀ i,j=1,⋯,N, v_r≠1, then every solution starting in {x| x_i=-1, ∀ i∈𝒥, x_i∈(-1,1), ∀ i∉𝒥} approaches -1/||v_r||_∞v_r.



The stubborn extremists scenario illustrates the circumstance that the extreme opinions cannot be persuaded by other opinions, while the other opinions are readily changed. This scenario is adequate when there exists two competitive extreme opinions, such as two opposite political parties. In this scenario it is demonstrated in Theorem <ref> and Corollary <ref> that the generalized opinion dynamics (<ref>) has the assured convergence under different initial conditions.






 §.§ Further Discussion


In this subsection, we will discuss several important aspects of the results derived in the preceding subsections.



For different state-dependent susceptibility, the dynamical behaviors of the generalized opinion model (<ref>) are very different. It can be seen from the proofs of Theorem <ref>–Theorem <ref> that the number of equilibrium points is different for different state-dependent susceptibility. Specifically, the number of equilibrium points is smallest in the stubborn neutrals scenario, while the number of equilibrium points is biggest in the stubborn extremists scenario. As a result, distinctive Lyapunov functions (compare (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>)) are constructed to effectively handle these different scenarios with a view to proving the convergence of the generalized opinion dynamics (<ref>).




If for each x_0∈ℝ^n, there exists T>0 such that x∈ℝ^n_+ or x∈ℝ^n_- ∀ t≥ T, then the system is said to achieve an unanimous opinion. Our Theorem <ref>–Theorem <ref> reveal that the network (<ref>) achieves a unanimous opinion. Moreover, although all agents achieve a unanimous opinion, the degree of opinion for each agent is different when the right positive eigenvector v_r≠1.




In Theorem <ref>–Theorem <ref>, one may use the condition “E is real orthogonally diagonalizable" to replace the condition “H(Γ E)≼0 and H(Γ E)= E." Because E is real orthogonally diagonalizable, there exists a real orthogonal matrix U such that UEU^ T is a diagonal matrix. By Lemma <ref> and Lemma <ref>, we have that H(E)≼0 and H(E)= E.




If the signed graph 𝒢(B) is undirected, then matrix E is a real symmetric matrix and orthogonally diagonalizable. If the signed graph 𝒢(B) is directed and B is real orthogonally diagonalizable, then  matrix E is orthogonally diagonalizable.




If matrix E is not symmetric, then it is not easy to verify whether the matrix is real orthogonally diagonalizable. However, by using the Matlab LMI (linear matrix inequality) toolbox, it is straightforward to find a positive diagonal matrix Γ such that H(Γ E)≼0 and H(E)= E. See Example <ref> in next section for details.




For a special class of matrices, there exists a positive diagonal matrix Γ such that H(Γ E)≼0. If σ_i=σ_j>ρ(B), ∀ i,j=1,⋯,N, and B≥ 0, then E=B-Σ is a nonsingular M-matrix. It is known from the property of M-matrices <cit.> that there exists a positive diagonal matrix Γ such that H(Γ E)≺0. Moreover, H(E)= E.




Unlike the model in <cit.>, our model requires that all eigenvalues of matrix E have non-positive real parts. If there exists an eigenvalue of E which has the positive real part, then the evolution of one agent may approach infinity as time goes to infinity. This will contradict the assumption in the stubborn neutrals scenario A(x)=(x)^2. Moreover, in practice, the opinion about a specific matter should be measured as a finite number rather than infinity.




The major difference between our work and <cit.> is that the work in this paper can deal with antagonistic interactions in social networks. Moreover, the model in this paper is more general than the one in <cit.>. For example, all eigenvalues of matrix E may have negative real parts in our model. However, when all interactions among agents are cooperative, our results may be more conservative than those in <cit.>, especially in the case of directed graphs.







§ NUMERICAL EXAMPLES



This section will provide two examples to illustrate the effectiveness of the theoretical results. The first example comes from the real-world, that is, Zachary's Karate Club <cit.>, while the second example is taken from <cit.>.



In <cit.>, Zachary studied the relationships in a karate club, and obtained some results about fission. The club consists of 34 members. There are two important persons in the club: one is the club president John A., and the other is Mr. Hi who is a part-time karate instructor. There is an conflict between John A., and Mr. Hi over the price of karate lessons. As the instructor, Mr. Hi wishes to raise prices. However, as the club's chief administrator, John A. wishes to stabilize prices. Assume that the viewpoint of John A. is 1, the viewpoint of Mr. Hi is -1, and the viewpoint of other members is included in [-1,1]. In <cit.>, the three scenarios were studied, that is, stubborn positives scenario A(x)=0.5(I-(x)), stubborn neutrals scenario A(x)=(x)^2, and stubborn extremists scenario A(x)=(I-(x)^2). However, antagonistic interactions among 34 members are also likely, but were not considered in <cit.>. In this example, we assume that there exist three pairs of antagonistic interactions, that is, Members 1 and 2, Members 1 and 32, and Members 33 and 34. We examine three different scenarios, that is, stubborn positives scenario, stubborn neutrals scenario, and stubborn extremists scenario.

Fig. <ref> shows the evolution of each person's viewpoint in the stubborn positives scenario, and illustrates the results of Theorem <ref>. It is clear from Fig. <ref> that the evolution of each person's viewpoint achieves the same viewpoint with different levels of the opinion. Fig. <ref> depicts the evolution of each person's viewpoint in the stubborn neutrals scenario, exemplifying the results of Theorem <ref>. When all persons' initial viewpoints are the same as John A's, that is, x_i(0)>0, the evolution of each person's viewpoint reaches the same viewpoint as John A but at different levels of the opinion. Furthermore, as an illustration of the results of Theorem <ref>, Fig. <ref> displays the evolution of each person's viewpoint in the stubborn extremists scenario. In particular, when there exist i,j such that x_i(0)=1, x_j(0)=-1, the evolutions of these persons' viewpoint are unable to attain the same opinion, as seen from Fig. <ref>(d). Note that this phenomenon cannot be deduced from Theorem <ref>, which is left for future research.










Consider the opinion dynamics

    ẋ=A(x)(-Σ+B)x,

where B is the adjacency matrix of signed digraph 𝒢(B), and

    B=[[       0  1.7877 -0.6743; -0.7678       0  0.7354;  0.5878       0       0 ]].

As pointed out in <cit.>, B is not eventually nonnegative. But one can choose D={0.2688,1.002,1.3272} such that C=B+D is eventually positive. The matrix C has the spectral radius ρ(C)=1.5817 and the corresponding right positive eigenvector v_c=[0.3350 0.5378 0.7737] with ||v_c||_∞=1. If we select  Σ=1.5817I-D, then

    E=[[ -1.3129  1.7877 -0.6743; -0.7678 -0.5797  0.7354;  0.5878       0 -0.2545 ]]≼0,

and 0 is a simple positive eigenvalue of E=-Σ+B with the corresponding right positive eigenvector being v_c. We can find a positive diagonal matrix Γ={4.2681,8.1972,11.5733} such that H(Γ E)≼0 and H(Γ E)= E. Fig. <ref>–Fig. <ref> plot the evolution curves of each agent under the stubborn positives scenario A(x)=0.5(I-(x)), the stubborn neutrals scenario A(x)=(x)^2, and the stubborn extremists scenario A(x)=(I-(x)^2), respectively. So Fig. <ref>–Fig. <ref> illustrate the results of Theorem <ref>–Theorem <ref>, respectively.

It is interesting to note that system (<ref>) exhibits some new behaviors when compared with the following linear model given in <cit.>:

    ẋ=(-Σ+B)x.

For example, as pointed out in <cit.>, the states of (<ref>) hold to the orthant pair ℝ^3_-,+. However, when A(x)=(x)^2, the states of (<ref>) hold to the orthant ℝ^3_+ for initial condition x_i(0)>0 and to the orthant ℝ^3_- for initial condition x_i(0)<0 (see Fig. <ref>). Moreover, when A(x)=(I-(x)^2), every solution starting in {x| x_i=1, ∀ i∈𝒥, x_i∈(-1,1), ∀ i∉𝒥} approaches v_c, where 𝒥={3} (see Fig. <ref>(b)). And every solution starting in {x| x_i=-1, ∀ i∈𝒥, x_i∈(-1,1), ∀ i∉𝒥} approaches -v_c (see Fig. <ref>(c)).













§ CONCLUSION



A generalized nonlinear opinion dynamics with state-dependent susceptibility to persuasion and antagonistic interactions has been introduced in this paper. The introduced model has been comprehensively analyzed, particularly with a thorough investigation into three specializations of state-dependent susceptibility drawn from the existing literature. The main theoretical analysis tools adopted have been Perron-Frobenius property of eventually positive matrix and LaSalle invariance principle. It has been shown that all agents of the opinion network converge into the subspace spanned by the positive eigenvector of an eventually positive matrix. Generally speaking, all entries of the positive eigenvector are not the same unless the eventually positive matrix is a Laplacian matrix. Therefore, the obtained results can represent different levels of an opinion, which is a common phenomenon in social networks. Finally, in this paper, it has been assumed that all eigenvalues of the system matrix are non-positive. An interesting topic for future work will be to study the case that there does not exist any restriction on eigenvalues of a network matrix.






§ SIGNED GRAPHS



Let 𝒢=(𝒱,ℰ,B) denote a weighted directed signed graph (signed digraph), where 𝒱={1,2,⋯,N} is the set of all nodes, ℰ∈𝒱×𝒱 is the set of edges, and B is an adjacency matrix which assigns real numbers to the edges. Let 𝒢(B) denote a graph with adjacency matrix B. An edge (i,j)∈ℰ is directed from node i to node j. The entry b_ij>0  (or <0) is the weight corresponding to the edge (i,j). Define ℰ^+={(i,j)| b_ij>0} and ℰ^-={(i,j)| b_ij<0}. Then ℰ=ℰ^+∪ℰ^-. For signed digraph 𝒢(B), let L=C_r-B denote the Laplacian matrix of 𝒢(B), where C_r={∑_j=1^N |b_1j|,⋯,∑_j=1^N |b_Nj|}. There exists an undirected graph 𝒢(B_u) with adjacency matrix B_u=(B+B^ T)/2, and the corresponding Laplacian is L_u=(L+L^ T)/2=C_r-B_u. Define C_c={∑_i=1^N |b_i1|,⋯,∑_i=1^N |b_iN|}. It is obvious that C_r=C_c when B^ T=B. A signed digraph 𝒢(B) is said to be weight balanced if C_r=C_c.

A directed path (length l-1) is a sequence of directed edges of the form (i_1,i_2),(i_2,i_3),⋯,(i_l−1,i_l) with distinct nodes. A signed digraph has a spanning tree if there is a root node, which has directed paths to all other nodes. A signed digraph is said to be strongly connected if there is a directed path between any two distinct nodes. 𝒢(B) is strongly connected if and only if B is irreducible.





§ EVENTUALLY POSITIVE MATRICES AND PERRON–FROBENIUS PROPERTY





A matrix A∈ℝ^n× n is said to possess Perron-Frobenius property if ρ(A) is a positive eigenvalue of A and the corresponding right eigenvector is nonnegative.




A matrix A∈ℝ^n× n is said to possess the strong Perron-Frobenius property if ρ(A) is a simple positive eigenvalue of A and the corresponding right eigenvector is positive.




A matrix A∈ℝ^n× n is said to be eventually positive (eventually nonnegative) if there exists a positive integer k_0 such that A^k>0  (A^k≥ 0), ∀ k≥ k_0.


The following lemma presents some connection between the eventual positivity and strong Perron-Frobenius property.


For a matrix A∈ℝ^n× n, the following statements are equivalent:


  * Both matrices A and A^ T possess the strong Perron-Frobenius property.

  * A is an eventually positive matrix.

  * A^ T is an eventually positive matrix.





Consider an eventually positive matrix A, and denote a right eigenvector of A by v_r>0. Then any eigenvector v_1 of A with v_1>0 must be a multiple of v_r.






§ SOME FACTS ABOUT SEMIDEFINITE MATRICES





Let A∈ℝ^n× n be negative semidefinite and let x∈ℝ^n. Then x^ TAx=0 if and only if Ax=0.




Suppose that A∈ℝ^n× n and H(A)=1/2(A+A^ T) is negative semidefinite. Then


  * nullspaceA⊂nullspaceH(A); nullspaceA^ T⊂nullspaceH(A).


  * H(A)≤ A.


  * The following statements are equivalent:


  * A and H(A) have the same null space.

  * A^ T and H(A) have the same null space.

  * H(A)= A.






Suppose that A∈ℝ^n× n and H(A)=1/2(A+A^ T) is negative semidefinite. If A is orthogonally diagonalizable, then A and H(A) have the same null space, and A^ T and H(A) have the same null space.


If A is orthogonally diagonalizable, then there exists a real orthogonal matrix B such that BAB^ T is a diagonal matrix. Meanwhile, A^ T is orthogonally diagonalizable, and BA^ TB^ T=[BAB^ T]^ T. Hence, H(A)= A. By Lemma <ref>, A and H(A) have the same null space, and A^ T and H(A) have the same null space.










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festinger1954theory
L. Festinger, “A theory of social comparison processes,” Human
  Relations, vol. 7, no. 2, pp. 117–140, 1954.

festinger1962theory
L. Festinger, A Theory of Cognitive Dissonance. Stanford, CA:
  Stanford University Press, 1962. 

cartwright1956structural
D. Cartwright and F. Harary, “Structural balance: A generalization of heider's
  theory,” Psychological Review, vol. 63, no. 5, pp. 277–293, 1956.

heider1946attitudes
F. Heider, “Attitudes and cognitive organization,” The Journal of
  Psychology, vol. 21, no. 1, pp. 107–112, 1946.

degroot1974reaching
M. H. DeGroot, “Reaching a consensus,” Journal of the American
  Statistical Association, vol. 69, no. 345, pp. 118–121, 1974.

pirani2014spectral
M. Pirani and S. Sundaram, “Spectral properties of the grounded laplacian
  matrix with applications to consensus in the presence of stubborn agents,”
  in Proceedings of 2014 American Control Conference, Portland, Oregon,
  USA, June 2014, pp. 2160–2165.

friedkin1999social
N. E. Friedkin and E. C. Johnsen, “Social influence networks and opinion change,”
  Advances in Group Processes, vol. 16, no. 1, pp. 1–29, 1999.

friedkin2015problem
N. E. Friedkin, “The problem of social control and coordination of complex
  systems in sociology: A look at the community cleavage problem,” IEEE
  Control Systems Magazine, vol. 35, no. 3, pp. 40–51, 2015.

jia2015opinion
P. Jia, A. MirTabatabaei, N. E. Friedkin, and F. Bullo, “Opinion dynamics and
  the evolution of social power in influence networks,” SIAM Review,
  vol. 57, no. 3, pp. 367–397, 2015.

ghaderi2014opinion
J. Ghaderi and R. Srikant, “Opinion dynamics in social networks with stubborn
  agents: Equilibrium and convergence rate,” Automatica, vol. 50,
  no. 12, pp. 3209–3215, 2014.

groeber2014dissonance
P. Groeber, J. Lorenz, and F. Schweitzer, “Dissonance minimization as a
  microfoundation of social influence in models of opinion formation,”
  The Journal of Mathematical Sociology, vol. 38, no. 3, pp. 147–174,
  2014.

xu2015modified
Z. Xu, J. Liu, and T. Başar, “On a modified degroot-friedkin model of
  opinion dynamics,” in Proceedings of 2015 American Control Conference,
  Chicago, Illinois, USA, July 2015, pp. 1047–1052.

Abelson64
R. Abelson, “Mathematical models of the distribution of attitudes under
  controversy,” in Contributions to Mathematical Psychology,
  N. Fredericksen and H. Gullicksen, Eds. Bristol, UK: Holt, Rinehart &
  Winston, 1964.

deffuant2000mixing
G. Deffuant, D. Neau, F. Amblard, and G. Weisbuch, “Mixing beliefs among
  interacting agents,” Advances in Complex Systems, vol. 3, no. 01n04,
  pp. 87–98, 2000.

hegselmann2002opinion
R. Hegselmann and U. Krause, “Opinion dynamics and bounded confidence models,
  analysis, and simulation,” Journal of Artificial Societies and Social
  Simulation, vol. 5, no. 3 (33 pp.), 2002.

lorenz2007continuous
J. Lorenz, “Continuous opinion dynamics under bounded confidence: A survey,”
  International Journal of Modern Physics C, vol. 18, no. 12,
  pp. 1819–1838, 2007.

castellano2009statistical
C. Castellano, S. Fortunato, and V. Loreto, “Statistical physics of social
  dynamics,” Reviews of Modern Physics, vol. 81, no. 2, pp. 591–646,
  2009.

lorenz2005stabilization
J. Lorenz, “A stabilization theorem for dynamics of continuous opinions,”
  Physica A: Statistical Mechanics and Its Applications, vol. 355,
  no. 1, pp. 217–223, 2005.

blondel2009krause
V. D. Blondel, J. M. Hendrickx, and J. N. Tsitsiklis, “On Krause's multi-agent
  consensus model with state-dependent connectivity,” IEEE Transactions
  on Automatic Control, vol. 54, no. 11, pp. 2586–2597, 2009.

mirtabatabaei2012opinion
A. Mirtabatabaei and F. Bullo, “Opinion dynamics in heterogeneous networks:
  Convergence conjectures and theorems,” SIAM Journal on Control and
  Optimization, vol. 50, no. 5, pp. 2763–2785, 2012.

jalili2013effects
M. Jalili, “Effects of leaders and social power on opinion formation in
  complex networks,” Simulation, vol. 89, no. 5, pp. 578–588, 2013.

sobkowicz2015extremism
P. Sobkowicz, “Extremism without extremists: Deffuant model with emotions,”
  Frontiers in Physics, vol. 3, art. 17 (12 pp.), 2015.

chen2016characteristics
S. Chen, D. H. Glass, and M. McCartney, “Characteristics of successful opinion
  leaders in a bounded confidence model,” Physica A: Statistical
  Mechanics and Its Applications, vol. 449, pp. 426–436, 2016.

amelkin2017polar
V. Amelkin, F. Bullo, and A. K. Singh, “Polar opinion dynamics in social
  networks,” IEEE Transactions on Automatic Control, vol. 62, no. 11,
  pp. 5650–5665, 2017.

miller1993attitude
A. G. Miller, J. W. McHoskey, C. M. Bane, and T. G. Dowd, “The attitude
  polarization phenomenon: Role of response measure, attitude extremity, and
  behavioral consequences of reported attitude change,” Journal of
  Personality and Social Psychology, vol. 64, no. 4, pp. 561–574, 1993.

eagly1993psychology
A. H. Eagly and S. Chaiken, The Psychology of Attitudes. Fort Worth,
  TX: Harcourt Brace Jovanovich College Publishers, 1993.

bassili1996meta
J. N. Bassili, “Meta-judgmental versus operative indexes of psychological
  attributes: The case of measures of attitude strength,” Journal of
  Personality and Social Psychology, vol. 71, no. 4, pp. 637–653, 1996.

wasserman1994social
S. Wasserman and K. Faust, Social Network Analysis: Methods and
  Applications. Cambridge, UK: Cambridge University Press, 1994. 

easley2010networks
D. Easley and J. Kleinberg, Networks, Crowds, and Markets: Reasoning
  About A Highly Connected World. Cambridge, UK: Cambridge University Press,
  2010.

altafini2012dynamics
C. Altafini, “Dynamics of opinion forming in structurally balanced social
  networks,” PloS One, vol. 7, no. 6, e38135 (8 pp.), 2012.

altafini2013consensus
C. Altafini, “Consensus problems on networks with antagonistic interactions,”
  IEEE Transactions on Automatic Control, vol. 58, no. 4, pp. 935–946,
  2013.

HuZheng2013CDC
J. Hu and W. X. Zheng, “Bipartite consensus for multi-agent systems on directed
  signed networks,” in Proceedings of the 52nd IEEE Conference on Decision
  and Control, Florence, Italy, December 2013, pp. 3451–3456.






zhang2014bipartite
H. Zhang and J. Chen, “Bipartite consensus of linear multi-agent systems over
  signed digraphs: An output feedback control approach,” in Proceedings of
  the 19th IFAC World Congress, Cape Town, South Africa, August 2014,
  pp. 4681–4686.

hu2014emergent
J. Hu and W. X. Zheng, “Emergent collective behaviors on coopetition networks,”
  Physics Letters A, vol. 378, no. 26, pp. 1787–1796, 2014.

valcher2014consensus
M. E. Valcher and P. Misra, “On the consensus and bipartite consensus in
  high-order multi-agent dynamical systems with antagonistic interactions,”
  Systems & Control Letters, vol. 66, pp. 94–103, 2014.

zhai2016pinning
S. Zhai and Q. Li, “Pinning bipartite synchronization for coupled nonlinear
  systems with antagonistic interactions and switching topologies,”
  Systems & Control Letters, vol. 94, pp. 127–132, 2016.

qin2017TCYB
J. Qin, W. Fu, W. X. Zheng, and H. Gao, “On the bipartite consensus for generic
  linear multiagent systems with input saturation,” IEEE Transactions on
  Cybernetics, vol. 47, no. 8, pp. 1948–1958, 2017.

altafini2015predictable
C. Altafini and G. Lini, “Predictable dynamics of opinion forming for networks
  with antagonistic interactions,” IEEE Transactions on Automatic
  Control, vol. 60, no. 2, pp. 342–357, 2015.

noutsos2006perron
D. Noutsos, “On Perron-Frobenius property of matrices having some negative
  entries,” Linear Algebra and Its Applications, vol. 412, no. 2-3,
  pp. 132–153, 2006.

olesky2009m
D. D. Olesky, M. J. Tsatsomeros, and P. Van den Driessche, “Mv-matrices: A
  generalization of M-matrices based on eventually nonnegative matrices,”
  Electronic Journal of Linear Algebra, vol. 18, pp. 339–351, 2009.

elhashash2008general
A. Elhashash and D. B. Szyld, “On general matrices having the Perron-Frobenius
  property,” Electronic Journal of Linear Algebra, vol. 17,
  pp. 389-413, 2008.

zachary1977information
W. W. Zachary, “An information flow model for conflict and fission in small
  groups,” Journal of Anthropological Research, vol. 33, no. 4,
  pp. 452–473, 1977.

horn2012matrix
R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd Ed. Cambridge, UK:
  Cambridge University Press, 2012.





