Cooperative opinion dynamics on multiple interdependent topics: Modeling and analysis

Cooperati v e opinion dynamics on multiple interdependent topics: Modeling and analysis Hyo-Sung Ahn 1 , Quoc V an T ran 1 , Minh Hoang T rinh 1 , K e vin L. Moore 2 , Mengbin Y e 3 , and Ji Liu 4 Abstract —T o model the interdependent couplings of multiple topics, we de velop a set of rules for opinion updates of a group of agents. The rules are used to design or assign v alues to the elements of interdependent weighting matrices. The coop- erative and anti-cooperative couplings are modeled in both the in verse-proportional and proportional feedbacks. The behaviors of cooperati ve opinion dynamics are analyzed using a null space property of state-dependent matrix-weighted Laplacian matrices and a L yapunov candidate. V arious consensus properties of state- dependent matrix-weighted Laplacian matrices are pr edicted ac- cording to the intra-agent network topology and interdependency topical coupling topologies. Index T erms —Cooperati ve opinion dynamics, Consensus, Matrix-weighted, Multiple interdependent topics I . I N T R O D U C T I O N The problem of opinion dynamics has attracted a lot of attention recently due to its applications to decision-making processes and evolut ion of public opinions [1]. The opinion dynamics arises between persons who interact with each other to influence others’ opinions or to update his or her opinion [2]. The opinion dynamics has been also studied in control territory or in signal processing recently . For examples, control via leadership with state and time-dependent interactions [3], game theoretical analysis of the Hegselmann-Krause Model [4], Hegselmann-Krause dynamics for the continuous-agent model [5], and the impact of random actions [6] hav e been in vestigated. The opinion dynamics under consensus setups has been also studied [7], [8]. In opinion dynamics under scalar-based consensus laws, the antagonistic interactions in some edges are key considerations [9]–[11]. The antagonis- tic interactions may represent repulsiv e or anti-cooperativ e characteristics between neighboring agents. In traditional con- sensus, all the interactions between agents are attractiv e one; so the dynamics of the traditional consensus has a contrac- tion property , which e ventually ensures a synchronization of agents. Howe ver , if there is an antagonistic interaction, a consensus may not be achieved and the Laplacian matrix may hav e negati ve eigen values [12]. Thus, in the existing opinion dynamics, the antagonistic interactions are modeled such that 1 School of Mechanical Engineering, Gwangju Institute of Science and T echnology (GIST), Gwangju, K orea. E-mails: hyosung@gist.ac.kr; tranvanquoc@gist.ac.kr; trinhhoangminh@gist.ac.kr 2 Department of Electrical Engineering, Colorado School of Mines, Golden, CO, USA. E-mails: kmoore@mines.edu 3 Systems and Control, Faculty of Science and Engineering, Univ ersity of Groningen, Groningen 9747 A G, The Netherlands m.ye@rug.nl 4 Department of Electrical and Computer Engineering, 211 Light Engineering, Stony Brook Uni versity , Stony Brook, NY , USA ji.liu@stonybrook.edu the Laplacian matrix would not have any negati ve eigen values. Specifically , in [9], signs of adjacent weights are used to model antagonistic interactions resulting in Laplacian matrix with absolute diagonal elements, and in [10], the author has extended the model of [9] to the one that allows arbitrary time- dependent interactions. In [11], they have further considered time-varying signed graphs under the setup of the antagonistic interactions. On the other hand, opinion dynamics with state constraints was also examined when the agents are preferred to attach to the initial opinion, i.e., with stubborn agents [13]. Recently , in [14], they hav e examined a joint impact of the dynamical properties of individual agents and the interaction topology among them on polarizability , consensusability , and neutralizability , with a further extension to heterogeneous systems with non-identical dynamics. Unlike the scalar-consensus based updates, there also ha ve been some works on opinion dynamics with matrix weighed interactions. Recently , opinion dynamics with multidimen- sional or multiple interdependent topics have been reported in [15], [16]. In [15], multidimensional opinion dynamics based on Friedkin and Johnsen (FJ) model and DeGroot models were analyzed in the discrete-time domain. The continuous-time version of [15] with stubborn agents was presented and ana- lyzed in [16]. The DeGroot-Friedkin model was also analyzed to conclude that it has an exponential conv ergent equilibrium point [17]. Also in [17], they considered the dynamic network topology to ev aluate the propagation property of the social power . Since the topics are interdependent and coupled with each other, these works may be classified as matrix-weighted consensus problems [18]. Opinion dynamics under leader agents with matrix weighted couplings was studied in [19]. In this paper , we would like to present a ne w model for opinion dynamics on multiple interdependent topics under a state-dependent matrix weighted consensus setup. W e first provide a model for characterizing the coupling ef fects of multiple interdependent topics. W e consider both the propor - tional and in verse proportional feedback effects on diagonal and of f-diagonal terms. The cooperative dynamics and non- cooperativ e dynamics are modeled using the signs of diffusi ve couplings of each topic. Then, we provide some analysis on the con vergence or consensus of the topics. T wo results will be presented according to the property of weighting matrices. The first result is developed when the coupling matrices are positiv e semidefinite. When the coupling matrices are positive semidefinite, exact conditions for complete opinion consensus and cluster consensus are provided. Then, as the second result, when the coupling matrices are indefinite, we provide a suf ficient condition for a complete opinion consensus. Consequently , the main contributions of this paper can be summarized as follows. First, a model for opinion dynamics is established. The connectivities are characterized by interac- tion topology between agents and coupling topology among the topics. Thus, the overall system has two-layer netw ork topologies. Second, analysis for complete opinion consensus and partial opinion consensus is presented for both the cases when the coupling matrices are positive semidefinite and indefinite. As far as the authors are concerned, this is the first paper that presents a detailed model for in verse-proportional and proportional feedback opinion dynamics along with the con vergence analysis. This paper is organized as follows. Section II provides a detailed process for building models for opinion dynamics. Section III presents the analysis for con vergence of cooperative opinion dynamics. Section IV is dedicated to simulation results and Section V concludes this paper with some discussions. I I . M O D E L I N G There are d dif ferent topics that may be of interests to the members of a society . Let the set of topics be denoted as T = { 1 , . . . , d } and let the opinion v ector associated with the member i be written as x i = ( x i, 1 , x i, 2 , . . . , x i,d ) T . W e can write the i -th agent’ s opinion about the p -th topic as x i,p . Each member (or can be called agent) has its initial opinion on the topics as x i,k ( t 0 ) = ( x i, 1 ( t 0 ) , x i, 2 ( t 0 ) , . . . , x i,d ( t 0 )) T . The opinion dynamics of agent i can be modeled as      ˙ x i, 1 ˙ x i, 2 . . . ˙ x i,d      = n X j ∈N i       a i,j 1 , 1 . . . a i,j 1 ,d a i,j 2 , 1 . . . a i,j 2 ,d . . . . . . . . . a i,j d, 1 . . . a i,j d,d            x j, 1 − x i, 1 x j, 2 − x i, 2 . . . x j,d − x i,d      , ˙ x i = n X j ∈N i A i,j ( x j − x i ) (1) where A i,j ∈ R d × d is the matrix weighting for the edge ( i, j ) ∈ E and i ∈ V , and |E | = m and |V | = n . The practical meaning of (1) is that each member of a society may hav e its own opinion about the topics, and the opinions are inter- coupled with the opinions of the neighboring agents. Thus, the matrix A i,j characterizes the logical reasoning of agent i with opinions from agent j . The neighborhoods of agents are determined by the interaction graph G = ( V , E ) . If a topic in member i has at least one connection to another topic or the same topic of another agent j , then two agents i and j are called connected . The terminology connection or connected is used for defining the connection in the level of agents. When there are connections between the topics, it is called coupled or couplings between topics. Thus, the terminology coupled or couplings is used in the lev el of topics. Therefore, based on the terminological definitions, if there is at least one coupling between the topics of agents i and j , then two agents i and j can be considered as connected. Howe ver , ev en though two agents are connected, it does not mean that a topic in an agent is connected to another topic of the other agent. The formal definitions are gi ven as follows. Agent i Agent j T opic p T opic q T opic r T opic p T opic q T opic r Fig. 1. Connected vs. Coupled: T opics p and q , and q and r are coupled in the coupling graph G i,j ; so the agents i and j are connected. But, although the agents i and j are connected, for example, the topics p and r are not coupled. Definition 1. T wo agents i and j ar e considered connected if A i,j is not identically zer o, i.e., A i,j 6 = 0 . The topology for overall network connectivities is repr esented by the interaction graph G = ( V , E ) wher e the edge set E characterizes the connectivities between agents. If there is a spanning tree in the network G , it is called connected. F or a topic p ∈ T , the graph is called p -coupled if the elements of the set { a i,j p,p , ∀ ( i, j ) ∈ E } ar e connected for the topic p . The topology for the topic p is defined by the graph G p = ( V p , E p ) , where p ∈ T , and V p = { x 1 ,p , x 2 ,p , . . . , x n,p } and E p = { ( i, j ) : a i,j p,p 6 = 0 } . If it is p -coupled for all topics p ∈ T , it is called all-topic coupled. Definition 2. F or the edge ( i, j ) , let the topology for the couplings among topics be denoted as G i,j = ( V i,j , E i,j ) , which is called coupling graph for the edge ( i, j ) , where V i,j includes all the topics contained in the agents i and j , and E i,j includes all the couplings. If G i 1 ,j 1 = G i 2 ,j 2 for all edges ( i 1 , j 1 ) 6 = ( i 2 , j 2 ) , then all the coupling topologies of the society ar e equivalent. If all the coupling topologies between agents are equivalent, it is called homogeneous-coupling net- work. Otherwise, it is called heter og eneous-coupling network. Based on the above definitions, we can see that every G p is disconnected even though G is connected. If the union of all G p is connected, then G is also connected. Also, since each agent has the same set of topics, V i,j = T for all ( i, j ) ∈ E . Assumption 1. The coupling between neighboring agents is symmetric, i.e., if ther e exists a coupling ( p, q ) in E i,j , ther e also exists a coupling ( q , p ) in E i,j . Example 1. F ig. 1 shows the concepts of “connected” and “coupled” in neighboring agents. The coupling graph G i,j can be determined as G i,j = ( V i,j , E i,j ) wher e V i,j = { p, q , r } and E i,j = { ( p, q ) , ( q, r ) , ( q , p ) , ( r , q ) } . Each agent updates its coef ficients in the matrix A i,j in the direction of cooperation or in the direction of antagonism. For a cooperativ e update, the rules for opinion update are formulated as: • The diagonal terms: If a i,j k,k is positive and as it increases, the tendency of agreement between x j,k and x i,k in- creases. Otherwise, if a i,j k,k is negati ve and as it increases to bigger negati ve value, the tendency of anti-agreement between x j,k and x i,k becomes significant. • The off-diagonal terms: Let us consider the ef fect of a i,j 2 , 1 . W e can consider the following four cases: 1) Case 1 : ( x j, 2 − x i, 2 ) ≥ 0 and ( x j, 1 − x i, 1 ) ≥ 0 2) Case 2 : ( x j, 2 − x i, 2 ) ≥ 0 and ( x j, 1 − x i, 1 ) < 0 3) Case 3 : ( x j, 2 − x i, 2 ) < 0 and ( x j, 1 − x i, 1 ) ≥ 0 4) Case 4 : ( x j, 2 − x i, 2 ) < 0 and ( x j, 1 − x i, 1 ) < 0 When ( x j, 2 − x i, 2 ) ≥ 0 , agent i needs to increase the value of x i, 2 to reach a consensus to x j, 2 . Otherwise, if ( x j, 2 − x i, 2 ) < 0 , agent i needs to decrease the value of x i, 2 to reach a consensus to x j, 2 . So, for the cases 1 and 2 , to enhance the agreement tendency , it needs to increase the value of x i, 2 , by way of multiplying a i,j 2 , 1 and ( x j, 1 − x i, 1 ) . Thus, when ( x j, 1 − x i, 1 ) ≥ 0 , we can select a i,j 2 , 1 > 0 ; but when ( x j, 1 − x i, 1 ) < 0 , we can select a i,j 2 , 1 < 0 . On the other hand, in the case of ( x j, 2 − x i, 2 ) < 0 , we can select a i,j 2 , 1 < 0 when ( x j, 1 − x i, 1 ) ≥ 0 , or we can select a i,j 2 , 1 > 0 when ( x j, 1 − x i, 1 ) < 0 . For the anti- consensus update, a i,j 2 , 1 should be selected with opposite signs. The ef fects of diagonal terms can be modeled as follows: Definition 3. Dir ect coupling effects in diagonal terms: • Pr oportional feedbacks: A close opinion between two agents acts as for increasing the consensus tendency between them. • In verse proportional feedbacks: A quite differ ent opinion between two agents acts as for increasing the consensus tendency between them. The of f-diagonal terms need to be designed carefully taking account of the coupling effects in different topics. Definition 4. Cr oss coupling effects in off-diagonal terms: • Pr oportional feedbacks: A close opinion in one topic acts as for incr easing the consensus tendency of other topics. • In verse proportional feedbacks: A quite differ ent opinion in one topic acts as for increasing the consensus tendency of other topics. Definition 5. (Completely and partial opinion consensus, and clusters) If a consensus is achieved for all topics, i.e., x j,p = x i,p for all p ∈ T , it is called a complete opinion consensus. In this case, ther e exists only one cluster . Otherwise, if a part of topics is agreed, it is called partial opinion consensus. When only a partial opinion consensus is achie ved, ther e could exist clusters C k , k = 1 , . . . , q such that C i ∩ C j = ∅ for differ ent i and j , and P q k =1 C k = { x 1 , x 2 , . . . , x n } , and in each cluster , x i = x j when x i and x j ar e elements of the same cluster , i.e., x i , x j ∈ C k . Definition 6. (Complete clustered consensus) If the opinions of agents are completely divided without ensuring any par- tial opinion consensus between them, it is called completely cluster ed consensus. In the case of a partial opinion consensus as per Definition 5 , the clusters are not completely divided clusters, i.e., in two different clusters, some topics may reach a consensus. Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 C 1 C 2 C 3 T opic 1 T opic 2 T opic 3 Fig. 2. Partial opinion consensus and clusters. The topic 3 reaches a consensus, while topics 1 and 2 do not reach a consensus. Example 2. In F ig. 2, ther e are five agents, with three topics. The agents reac h a consensus on the topic p = 3 . But, on other topics p = 1 , 2 , the y do not r each a consensus. F or the topic 1 , ther e are two clusters (i.e., agents 1 , 2 , 3 in one cluster , and agents 4 , 5 in another cluster), and for the topic 2 , ther e ar e also two clusters (i.e., agents 1 in one cluster , and agents 2 , 3 , 4 , 5 in another cluster). So, overall, the network has a consensus in a part of topics, b ut the y do not reac h a consensus on the other topics. So, no complete opinion consensus is achieved, and no complete cluster ed consensus is achie ved. Consequently , there ar e thr ee clusters as C 1 = { x 1 } , C 2 = { x 2 , x 3 } , and C 3 = { x 4 , x 5 } . The consensus and coupling effects giv en in the Definition 3 and Definition 4 can be mathematically modeled as follows. 1) In verse-pr oportional feedbacks: When the v alues of opinions of two agents are quite different, the coupling effects are more significant, which may be against from a natural phenomenon (ex, gravitational force). That is, when two opinions are close, there could be more attraction force. In in verse-proportional feedbacks, there will be more coupling effects when the v alues of opinions are quite different. • Direct coupling in diagonal terms: a i,j p,p = k i,j p,p (2) where k i,j p,p = k j,i p,p > 0 . • Cross coupling in off-diagonal terms: a i,j p,q = k i,j p,q × sign ( x j,p − x i,p ) × sign ( x j,q − x i,q ) (3) where k i,j p,q = k j,i p,q = k i,j q ,p > 0 , and sign ( x j,p − x i,p ) = 1 when x j,p − x i,p ≥ 0 and sign ( x j,p − x i,p ) = − 1 when x j,p − x i,p < 0 . 2) Pr oportional feedbacks: In proportional feedbacks, there will be less coupling effects when the v alues of opinions are quite dif ferent. • Direct coupling in diagonal terms: a i,j p,p = k i,j p,p c 2 k x j,p − x i,p k 2 + c 1 k x j,p − x i,p k + c 0 (4) where k i,j p,p = k j,i p,p > 0 . • Cross coupling in off-diagonal terms: a i,j p,q = k i,j p,q × sign ( x j,p − x i,p ) × sign ( x j,q − x i,q ) ( c 1 k x j,p − x i,p k + c 0 )( c 1 k x j,q − x i,q k + c 0 ) (5) where k i,j p,q = k i,j q ,p > 0 and k i,j p,q = k j,i p,q > 0 , and c 1 and c 0 are positi ve constants. Then, we can have ( A i,j ) T = A i,j and A i,j = A j,i . Thus, with the model (5), the Laplacian matrix is of symmetric. Note that in the above coupling models, if ( p, q ) ∈ E i,j , then a i,j p,q 6 = 0 , otherwise, a i,j p,q = 0 . So, the matrix A i,j = [ a i,j p,q ] is the weighting matrix for the topics between two agents i and j . But, the matrix A i,j is state- and sign-dependent, while the matrix K i,j = [ k i,j p,q ] is a matrix which defines the topological characteristics between topics of the neighboring agents. The matrix K i,j is called coupling matrix, and it is a constant matrix. It is remarkable that the matrix A i,j is not an adjacency matrix, and neither is the matrix K i,j . But, they are similar to an adjacenc y matrix. For e xample, if there is no direct coupling between the same topics, then K i,j is the adjacency matrix for characterizing the couplings between the topics of two neighboring agents. On the other hand, if all the topics are coupled (i.e., a topic of agent i is coupled to all the topics of neighboring agent j ), then K i,j − I d is the adjacency matrix ignoring the self-loops. The direct coupling in the p -th topic implies that there is a self-loop in the p -th topic node. Example 3. Let us consider the following coupling matrices. K i,j 1 =       2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 3 1 1 1 1 1 1       , K i,j 2 =       0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0       The matrix K i,j 1 means that any topic in agent i is coupled to all the topics in j , while the matrix K i,j 2 means that any topic in agent i is coupled to all the topics in j , but x i,p is not coupled to x j,p (i.e., no dir ect coupling). The anti-consensus can be simply modeled by adding the minus sign to the elements of the coupling matrix, i.e., k i,j p,q . Thus, there are four types of couplings: proportional coupling, proportional anti-coupling, in verse-proportional coupling, and in verse-proportional anti-coupling. The dynamics with anti- consensus terms is called non-cooperative opinion dynamics , while the dynamics without anti-consensus terms is called cooperative opinion dynamics . Note that in existing traditional consensus works, the in verse proportional diagonal terms, i.e., (2), are only used for the consensus couplings. The dynamics (1) can be concisely rewritten as: ˙ x = − L ( x 1 , . . . , x n ) x (6) where the Laplacian is computed as L =      P j ∈N i A 1 ,j − A 1 , 2 . . . − A 1 ,n − A 2 , 1 P j ∈N i A 2 ,j . . . − A 2 ,n . . . . . . . . . . . . − A n, 1 − A n, 2 . . . P j ∈N i A n,n      (7) Note that in the dynamics (6), the Laplacian L is dependent upon the sign of x j,p − x i,p ; thus, it is discontinuous when the sign changes abruptly . T o be a continuous function, the sign function may be modified as a sigmoid function as: sig n ( x j,p − x i,p ) , 2 1 + e − k e ( x j,p − x i,p ) − 1 (8) where k e is a sufficiently large positive constant. W e remark that the sign function can be also changed to a signum function sig ( x j,p − x i,p ) | x j,p − x i,p | α , where 0 < α < 1 . In the case of the in verse-proportional feedback laws, we can see that a i,j p,q = a i,j q ,p , and a i,j p,q = a j,i p,q . Then, the Laplacian matrix L is of symmetric. Consequently , for the inv erse- proportional consensus couplings, we can rewrite (1) as:      ˙ x i, 1 ˙ x i, 2 . . . ˙ x i,d      = n X j ∈N i       sgn i,j 1 , 1 k i,j 1 , 1 . . . sgn i,j 1 ,d k i,j 1 ,d sgn i,j 2 , 1 k i,j 2 , 1 . . . sgn i,j 2 ,d k i,j 2 ,d . . . . . . . . . sgn i,j d, 1 k i,j d, 1 . . . sgn i,j d,d k i,j d,d       ×      x j, 1 − x i, 1 x j, 2 − x i, 2 . . . x j,d − x i,d      (9) where sgn i,j p,q , sign ( x j,p − x i,p ) × sign ( x j,q − x i,q ) and sgn i,j p,q = sgn i,j q ,p . If there are some in verse-proportional anti- consensus couplings between some topics, then some elements in (9) will hav e negati ve signs. For example, if the 1 -st topic and 2 -nd topic are anti-consensus coupled, then the terms sgn i,j 1 , 2 k i,j 1 , 2 and sgn i,j 2 , 1 k i,j 2 , 1 need to be modified as − sgn i,j 1 , 2 k i,j 1 , 2 and − sgn i,j 2 , 1 k i,j 2 , 1 . But, in this case, the Laplacian matrix L may hav e ne gati ve eigen v alues; thus, the stability or con ver gence may not be ensured any more. Thus, in this paper , we focus on only the cooperativ e opinion dynamics. For a matrix A , we use N ( A ) and R ( A ) to denote the nullspace and the range of A , respecti vely . I I I . A N A L Y S I S It will be sho wn in this section that the positiv e definite- ness of the Laplacian matrix in (7) is closely related with the positive definiteness of the coupling matrix K i,j . When the coupling matrices are positi ve semidefinite, we pro vide exact conditions for complete opinion consensus and cluster consensus. Howe ver , when L ( x ) is indefinite, since L ( x ) is time-varying, the system (6) can still be stable and a consensus might be reached. Let us first focus on the case of positiv e semidefinite Laplacian, and then, we consider general cases that include indefinite Laplacian matrices. A. Case of P ositive Semidefinite Laplacian It is not straightforward to verify whether the Laplacian L ( x ) in (7) is positi ve semidefinite or not since it is a block matrix. In L ( x ) , the element matrices could be posi- tiv e definite, positive semidefinite, negati ve definite, negativ e semidefinite, or indefinite. Thus, an analysis for the dynamics (6) would be more difficult than the traditional scalar-based consensus. For the analysis, let us define the incidence matrix H = [ h ij ] ∈ R m × n for the interaction graph G = ( V , E ) as: h ki =      − 1 if verte x i is the tail of the k -th edge 1 if verte x i is the head of the k -th edge 0 otherwise (10) where the direction of the edge k is arbitrary . Let us also define the incidence matrix in d -dimensional space as ¯ H = H ⊗ I d and write the weighting matrix for the k -th edge as A k -th ∈ R d × d . Let us also write the coupling matrix K i,j corresponding to A k -th as K k -th . As aforementioned, if there is no direct coupling between the same topics of two neighboring agents, the coupling matrix K i,j can be considered as a constant adjacency matrix for the coupling graph G i,j . The block diagonal matrix composed of A k -th , k = 1 , . . . , m is denoted as blkdg ( A k -th ) and the block diagonal matrix composed of K k -th , k = 1 , . . . , m is denoted as blkdg ( K k -th ) . Lemma 1. F or the in verse pr oportional coupling, the Lapla- cian L ( x ) is positive semidefinite if and only if blkdg ( K k -th ) is positive semidefinite. Pr oof. Due to the same reason as Lemma 1 of [20], we can write L = ¯ H T blkdg ( A k -th ) ¯ H . It is shown that the weighting matrix A k -th can be written as A k -th = diag ( S i,j ) K k -th diag ( S i,j ) (11) where the edge ( i, j ) is the k -th edge and diag ( S i,j ) is giv en as diag ( S i,j ) = diag ( sign ( x j,k − x i,k )) =    sign ( x j, 1 − x i, 1 ) · · · 0 . . . . . . . . . 0 · · · sign ( x j,d − x i,d )    (12) Hence, the Laplacian matrix can be written as L = ¯ H T blkdg ( diag ( S i,j )) blkdg ( K k -th ) blkdg ( diag ( S i,j )) ¯ H (13) Therefore, it is obvious that the Laplacian matrix L is positiv e semidefinite if and only if the matrix blkdg ( K k -th ) can be decomposed as blkdg ( K k -th ) = ¯ K T ¯ K with a certain matrix ¯ K . It means that the Laplacian matrix L is positive semidefinite if and only if blkdg ( K k -th ) is positi ve semidefinite. Theorem 1. The Laplacian L ( x ) is positive semidefinite if and only if the coupling matrices K i,j ar e positive semidefinite. Pr oof. The proof is immediate from Lemma 1 . If is well-kno wn that the adjacenc y matrix of a complete graph with d nodes has eigen v alues d − 1 with multiplicity 1 and − 1 with multiplicity d − 1 . Then, with this fact, we can obtain the follo wing result. Theorem 2. Let us suppose that ther e is no dir ect coupling between the same topics of two neighboring ag ents; but a topic is coupled to all other topics. Then, under the condition that all diag ( S i,j ) are not equal to zer o (i.e., there exists at least one topic p such that x j,p 6 = x i,p ), the Laplacian L ( x ) has ne gative eigen values. Pr oof. It is clear that the matrix K k -th can be considered as an adjacency matrix characterizing the topic couplings of the k -th edge. Let us denote this matrix as K k -th − . Then, the matrix K k -th − has an eigen v alue d − 1 with multiplicity 1 and the eigen value − 1 with multiplicity d − 1 . Thus, the matrix L = ¯ H T blkdg ( diag ( S i,j )) blkdg ( K k -th − ) blkdg ( diag ( S i,j )) ¯ H has eigen v alues located in the open left half plane because blkdg ( K k -th − ) has eigen value − 1 with multiplicity of m ( d − 1) where m = |E | . The opposite circumstance occurs when all the topics are coupled, including the same topics, which is summarized in the next result. Corollary 1. Suppose that all the topics ar e coupled for all edges. Then, the Laplacian L ( x ) is positive semidefinite. Pr oof. In this case, the matrix K k -th can be considered as a rank 1 matrix defined as K k -th , K k -th + = K k -th − + I d , because the matrix K k -th + is a matrix with all elements being equal to 1 . Thus, the eigenv alues of K k -th + are d with multiplicity 1 and all others being equal to zero. Therefore, the matrix L positiv e semidefinite. Now , let us suppose that the matrix blkdg ( K k -th ) is pos- itiv e semidefinite; then it can be written as blkdg ( K k -th ) = U T U for some matrix U . Then, by denoting U = U blkdg ( diag ( S i,j )) ¯ H , we can write L ( x ) = U T U . It is clear that nullspace ( ¯ H ) ⊆ nullspace ( L ) = nullspace ( U ) , because blkdg ( K k -th ) is positive semidefinite. Noticing that the null space of incidence matrix is N ( ¯ H ) = R ( 1 n ⊗ I d ) , R , we can see that the set R is always a subspace of N ( L ) . T o find the null space of L , the following lemma will be employed. Lemma 2. [21] When a matrix A is positive semidefinite, for any vector x , it holds that Ax = 0 if and only if x T Ax = 0 . W e remark that if a matrix A is indefinite, the abov e lemma (i.e., Lemma 2 ) does not hold. For example, let us consider the follo wing matrix: A =     1 0 0 0 0 1 0 0 0 0 − 4 0 0 0 0 − 4     (14) which is nonsingular and has 1 , 1 , − 4 , − 4 as its eigenv alues. Then, the vector x = (1 , 1 , 1 / 2 , 1 / 2) T makes that x T Ax = 0 , while Ax 6 = 0 . It may be also important to note that, since the elements of the coupling matrix K i,j are all non-negati ve, and sign ( x j,p − x i,p )( x j,p − x i,p ) ≥ 0 , Lemma 2 may be further generalized. Howe ver , a further generalization is not obvious. From Lemma 2 , we can see that a vector x in the null space of A is equiv alent to a vector x that makes x T Ax = 0 , when the matrix A is positi ve semidefinite. With this fact, since a i,j p,q = k i,j p,q sgn i,j p,q = k i,j p,q sign ( x j,p − x i,p ) × sign ( x j,q − x i,q ) = k i,j p,q h 2 1+ e − k e ( x j,p − x i,p ) − 1 i h 2 1+ e − k e ( x j,q − x i,q ) − 1 i , we can write x T L x as follo ws: x T L x = X ( i,j ) ∈E ( x j − x i ) T A ij ( x j − x i ) = X ( i,j ) ∈E d X p =1 d X q =1 a i,j p,q ( x j,p − x i,p )( x j,q − x i,q ) = X ( i,j ) ∈E d X p =1 d X q =1 k i,j p,q ×  2 1 + e − k e ( x j,p − x i,p ) − 1  ( x j,p − x i,p ) ×  2 1 + e − k e ( x j,q − x i,q ) − 1  ( x j,q − x i,q ) = X ( i,j ) ∈E σ ( x j − x i ) T K i,j σ ( x j − x i ) (15) where σ ( x j − x i ) is defined as σ ( x j − x i ) , ( σ ( x j, 1 − x i, 1 ) , . . . , σ ( x j,d − x i,d )) T (16) with σ ( x j,p − x i,p ) , sig n ( x j,p − x i,p )( x j,p − x i,p ) =    2 1+ e − k e ( x j,p − x i,p ) − 1    | x j,p − x i,p | . Thus, to hav e P ( i,j ) ∈E σ ( x j − x i ) T K i,j σ ( x j − x i ) = 0 , based on Lemma 2 , it is required to satisfy σ ( x j − x i ) ∈ N ( K i,j ) for all ( i, j ) ∈ E . Therefore, summarizing this discussion, we can obtain the follo wing lemma. Lemma 3. Suppose that the blkdg ( K k -th ) is positive semidef- inite, which is equivalent to the positive semi-definiteness of L . Then, the null space of L is given as: N ( L ) = span {R , { x = ( x T 1 , x T 2 , · · · , x T n ) T ∈ R dn | σ ( x j − x i ) ∈ N ( K i,j ) , ∀ ( i, j ) ∈ E }} (17) Remark 1. Lemma 3 implies that if blkdg ( K k -th ) is positive definite, then N ( L ) = R . Thus, a complete opinion consensus is achie ved. Remark 2. In (15) , if K i,j is nonsingular , then it has only the trivial null space. Thus, it appears that a complete opinion consensus might be achieved. However , as discussed with the A matrix in (14) , the set of vectors x making x T L x = 0 is not equivalent to the set of vectors x making L x = 0 . In Lemma 3 , there are possibly two subspaces for the null space. The subspace R is the standard consensus space; b ut the subspace spanned by x satisfying σ ( x j − x i ) ∈ N ( K i,j ) needs to be elaborated since the elements of the coupling matrix K i,j are zero or positive constants, and the elements of the vector σ ( x j − x i ) are also positiv e e xcept the zero. The following example provides some intuitions for the coupling matrix. Example 4. Let us consider that ther e ar e five topics, and the coupling matrix between agents i and j is given as: K i,j =       0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0       (18) which means that only the topics 1 and 2 ar e coupled. But, the above matrix is not positive semidefinite. Thus, the basic condition of Lemma 3 is not satisfied. Let us consider another coupling matrix as K i,j =       1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1       (19) which is positive semidefinite. It follows that K i,j σ ( x j − x i ) =       1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1             σ ( x j, 1 − x i, 1 ) σ ( x j, 2 − x i, 2 ) σ ( x j, 3 − x i, 3 ) σ ( x j, 4 − x i, 4 ) σ ( x j, 5 − x i, 5 )       =       σ ( x j, 1 − x i, 1 ) + σ ( x j, 2 − x i, 2 ) σ ( x j, 1 − x i, 1 ) + σ ( x j, 2 − x i, 2 ) 0 σ ( x j, 4 − x i, 4 ) + σ ( x j, 5 − x i, 5 ) σ ( x j, 4 − x i, 4 ) + σ ( x j, 5 − x i, 5 )       Consequently , to satisfy K i,j σ ( x j − x i ) = 0 , we need to have x j, 1 = x i, 1 , x j, 2 = x i, 2 , x j, 4 = x i, 4 , and x j, 5 = x i, 5 ; but, x j, 3 and x i, 3 can be chosen arbitrarily . From the above example, we can observe that the coupling matrices K i,j , ( i, j ) ∈ E would provide all possible consensus solutions in the topics among agents. Let us add another coupling between topics 1 and 3 into K i,j in (19) as: K i,j =       1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 1 1       (20) Then, against from an intuition, the matrix K i,j is no more positiv e semidefinite (actually it is indefinite since it has negati ve, zero, and positive eigenv alues). As there are more couplings between topics, the Laplacian matrix L has lost the positiv e semidefinite property . Thus, as far as the coupling matrix K i,j is positiv e semidefinite, the null space of K i,j enforces x j and x i to be synchronized. That is, in the multi- plication K i,j σ ( x j − x i ) , if the term σ ( x j,p − x i,p ) appears, then the y will be synchronized; otherwise, if it does not appear , the synchronization x j,p → x i,p is not enforced. Now , we can summarize the discussions as follows. Lemma 4. Let us assume that the Laplacian L ( x ) is positive semidefinite, and for two neighboring ag ents i and j , the topics p and q ar e coupled, i.e., K i,j p,q 6 = 0 . Then, the opinion values x i,p and x j,p will r each a consensus and the opinion values x i,q and x j,q will also reac h a consensus. If the same topic p is coupled between neighboring agents, i.e., K i,j p,p 6 = 0 , then x i,p and x j,p will r each a consensus. Agent 1 Agent 2 Agent 3 Agent 4 T opic 1 T opic 2 T opic 3 Fig. 3. Interaction topology of a network and coupling between neighboring topics. Giv en a coupling matrix K i,j , let us define a consensus matrix, C i,j = [ c i,j p,q ] , between agents i and j as c i,j p,p = c i,j q ,q = ( 1 if K i,j p,q = 1 0 if K i,j p,q = 0 (21) Now , we define the topic consensus graph G p,con = ( T p,con , E p,con ) for the topic p as follows: T p,con , { x 1 ,p , x 2 ,p , . . . , x n,p } (22) E p,con , { ( i, j ) | ∃ ( i, j ) if c i,j p,p = 1; @ ( i, j ) otherwise if c i,j p,p = 0 } (23) So, x j,p is a neighbor of x i,p if and only if ( i, j ) ∈ E p,con . That is, x j,p ∈ N x i,p if and only if c i,j p,p = 1 . W ith the above definition, we can make the following main theorem. Theorem 3. Let us assume that the Laplacian L ( x ) is positive semidefinite. Then, for the topic p , the ag ents in T p,con will r each a consensus if and only if the topic consensus gr aph G p,con is connected. Pr oof. If the consensus graph G p,con is connected, it can be considered that there is at least one path between x i,p and x j,p for any pair of i and j . Thus, the set of p -topic agents, i.e., T p,con , { x 1 ,p , x 2 ,p , . . . , x n,p } , is connected, and the p -topic will reach a consensus by Lemma 4 . The only if is also direct from Lemma 4 . That is, if the topic opinions in T p,con are not connected, it means that there are clusters, which are not connected. Thus, the consensus of the agents in T p,con is not possible. Example 5. Let us consider F ig. 3 that illustrates the in- teraction topology of a network. In this case, the elements of coupling matrix are given as K 1 , 2 1 , 1 = K 3 , 4 1 , 1 = K 2 , 3 1 , 2 = K 2 , 3 2 , 1 = K 1 , 2 2 , 2 = K 1 , 2 3 , 3 = K 3 , 4 2 , 3 = K 3 , 4 3 , 2 = 1 . Thus, we have c 1 , 2 1 , 1 = c 3 , 4 1 , 1 = c 2 , 3 1 , 1 = c 2 , 3 2 , 2 = c 1 , 2 2 , 2 = c 1 , 2 3 , 3 = c 3 , 4 2 , 2 = c 3 , 4 3 , 3 = 1 . Then, fr om this consensus matrix, we can obtain the consensus graphs G p,con = ( T p,con , E p,con ) for p = 1 , 2 , 3 with the following edge sets: E 1 ,con , { (1 , 2) , (3 , 4) , (2 , 3) } E 2 ,con , { (2 , 3) , (1 , 2) , (3 , 4) } E 3 ,con , { (1 , 2) , (3 , 4) } Ther efore, the consensus graphs G 1 ,con and G 2 ,con ar e con- nected, while the consensus graph G 3 ,con is not connected. Now , by virtue of Theor em 3 , we can conclude that if the Laplacian L ( x ) is positiv e semidefinite and all the topics are connected in the sense of Theorem 3 (i.e., from the topic consensus graph G p,con = ( T p,con , E p,con ) , then a complete consensus will be ensured. Otherwise, given G p,con , although the Laplacian L ( x ) is positi ve semidefinite, if the topic p is not connected, then a partial opinion consensus will be achieved. The number of partial opinion clusters will be dependent on the number of clusters on the topic p . For example, in Fig. 2, the topic p = 1 has two clusters, the topic p = 2 has two clusters, and the topic p = 3 has one cluster . Let us define disconnection as follo ws. Definition 7. F or a topic p , we call there is no disconnection if and only if the opinion values x 1 ,p , x 2 ,p , . . . , x n,p ar e con- nected. If the opinion values are divided into c p components (ther e is no connection between components), then ther e ar e c p − 1 disconnections. Then, for Fig. 2, we can say that the topic p = 1 has one disconnection (i.e., between agents 3 and 4 ), the topic p = 1 also has one disconnection (i.e., between agents 1 and 2 ), and the topic p = 3 does not have a disconnection. W ith the above definition, although it looks tri vial, we can obtain the follo wing observation. Observation 1. Let ther e be d topics, and each topic has c i , i = 1 , . . . , d, clusters. Then ther e ar e T c = P d k =1 ( c i − 1) + 1 partial opinion clusters at maximum. Pr oof. Suppose that for the topic d = 1 , we have c 1 clusters. It means that there are c 1 − 1 disconnections in the set T 1 ,con , { x 1 , 1 , x 2 , 1 , . . . , x n, 1 } . Similarly , for the topic d = 2 with c 2 clusters, there are c 2 − 1 disconnections in the set T 2 ,con . Thus, by combining the topics d = 1 and d = 2 , there could be at maximum ( c 1 − 1) + ( c 2 − 1) disconnections in T 1 ,con and T 2 ,con . Thus, if we consider all the topics, there are at maximum ( c 1 − 1) +( c 2 − 1) disconnections in P d k =1 ( c i − 1) , which implies that there could be T c = P d k =1 ( c i − 1) + 1 partial opinion clusters at maximum. The results thus far are dev eloped for the in verse- proportional feedbacks. For the proportional feedbacks, we use (5). The weighting matrix can be decomposed as (11), with the diagonal matrix diag ( S i,j ) gi ven as: diag ( S i,j ) = diag  sign ( x j,k − x i,k ) c 1 k x j,k − x i,k k + c 0  (24) Also, x T L x can be expressed as follows: x T L x = X ( i,j ) ∈E η ( x j − x i ) T K i,j η ( x j − x i ) , (25) where η ( x j − x i ) ,  σ ( x j, 1 − x i, 1 ) c 1 k x j, 1 − x i, 1 k + c 0 , σ ( x j, 2 − x i, 2 ) c 1 k x j, 2 − x i, 2 k + c 0 , . . . , σ ( x j,d − x i,d ) c 1 k x j,d − x i,d k + c 0  T . Thus, the null space of the Laplacian L in (25) is same to the null space of L in (15). Consequently , all the results in the inv erse-proportional feedback couplings are exactly applied to the cases of the proportional feedback couplings. B. General Cases The results in the previous section are quite clear and provide precise conditions for the characterization of opinion dynamics. Howe ver , the results are de veloped when the ma- trix blkdg ( K k -th ) is positiv e semidefinite. As shown in (20), when the matrix blkdg ( K k -th ) is not positiv e semidefinite, although it is against from intuition, there can be no the- oretical guarantee for opinion consensus. For general case, we would like to directly analyze the stability of the inv erse- proportional feedbacks modeled by (2) and (3). Let us take the L yapunov candidate V = 1 2 k x k 2 , which is radially unbounded and continuously dif ferentiable, for the in verse-proportional feedbacks. The deri v ativ e of V is computed as: ˙ V = − x T L x = − X ( i,j ) ∈E σ ( x j − x i ) T K i,j σ ( x j − x i ) = − X ( i,j ) ∈E d X p =1 k i,j p,p ( σ ( x j,p − x i,p )) 2 | {z } , φ − X ( i,j ) ∈E d X p =1 d X q =1 , q 6 = p k i,j p,q σ ( x j,p − x i,p ) σ ( x j,q − x i,q ) | {z } , ψ (26) ≤ 0 From the abov e inequality , it is clear that ˙ V = 0 if and only if x j,p = x i,p for all topics, i.e., ∀ p ∈ T , the opinion consensus is achie ved, which is summarized as follows: Theorem 4. Let us suppose that the underlying interaction graph G is all-topic coupled, i.e, G p is p -coupled for all p ∈ { 1 , . . . , d } . Then, a complete opinion consensus is achieved. Pr oof. T o make ˙ V = 0 , it is required to have φ = 0 and ψ = 0 . Since G is all-topic coupled, φ = 0 implies ψ = 0 . But, ψ = 0 does not imply φ = 0 . Thus, it is true that ˙ V = 0 if x j,p = x i,p for all topics and for all ( i, j ) ∈ E . Suppose that there exists an edge such that x j,p 6 = x i,p for a specific topic p . Then, ˙ V 6 = 0 . Thus, ˙ V = 0 only if x j,p = x i,p for all topics and for all ( i, j ) ∈ E . Consequently , the set D = { x : x j,p = x i,p , ∀ i, j ∈ V , ∀ p ∈ T } is the largest in variant set. Finally , by the Barbalat’ s lemma (due to ¨ V exists and is bounded), the proof is completed. Remark 3. It is r emarkable that the above results are true for both the homog eneous-coupling and hetero geneous-coupling networks, as far as the interaction graph G is all-topic coupled. Remark 4. It is noticeable that the condition of Theor em 4 is only a sufficient condition for a complete consensus. Thus, we may be able to achieve a complete opinion consensus even if the network is not all-topic coupled. Let us suppose that two topics ¯ p and ¯ q ar e not p -coupled. F or example, the two topics ar e not directly coupled at the edge ( ¯ i, ¯ j ) . Since the overall network is connected, ther e must be terms suc h as k ¯ i, ¯ j ¯ p, ¯ q σ ( x ¯ j, ¯ p − x ¯ i, ¯ p ) σ ( x ¯ j, ¯ q − x ¯ i, ¯ q ) in ψ . Thus, to make ˙ V = 0 , it is r equired to have either σ ( x ¯ j, ¯ p − x ¯ i, ¯ p ) = 0 or σ ( x ¯ j, ¯ q − x ¯ i, ¯ q ) = 0 . Ther efore, e ven if the two topics ¯ p and ¯ q ar e not p -coupled, the neighboring agents ¯ i and ¯ j may reac h a consensus. W e will illustrate this case by an example in the simulation section. From the equation (26), we can see that if there is no cross couplings, i.e., ψ = 0 , then it is a usual consensus protocol in different layers. On the other hand, if there is no direct coupling, i.e., φ = 0 , then there is no coupling in the same topics among agents. In the case of ψ = 0 with k i,j p,q = 0 whenev er p 6 = q , it is still true that ˙ V = 0 if and only if x j,p = x i,p ; thus, the typical consensus is achieved. Let φ = 0 , with k i,j p,p = 0 for all p . There are some undesired equilibrium cases. For example, gi ven a coupling graph G i,j , let there exist paths from the topic node 1 to all other topic nodes. That is, the graph G i,j is a star graph with root node 1 . Then, ˙ V , with φ = 0 , can be changed as: ˙ V = − X ( i,j ) ∈E σ ( x j, 1 − x i, 1 ) " d X q =2 k i,j 1 ,q σ ( x j,q − x i,q ) # (27) So, if sgmd ( x j, 1 − x i, 1 ) = 0 for all ( i, j ) ∈ E , then we have ˙ V = 0 . Thus, for a star graph, if the root topic has reached a consensus, all other topics may not reach a consensus. Actually , when φ = 0 , a complete consensus is not achieved, due to the following reason: Claim 1. Let us suppose that, ∀ ( i, j ) ∈ E , k i,j p,p = 0 , ∀ p . Then, ˙ V will be almost zer o (for the meaning of “almost”, see the footnote 1 ) with at least one topic having x j,p 6 = x i,p if and only if the coupling graphs G i,j , ∀ ( i, j ) ∈ E , ar e complete graphs. Pr oof. ( If ) When it is a complete graph, without loss of generality , let the first topic, p = 1 , be reached a consen- sus. Then, we need to hav e P d p =2 P d q =2 , q 6 = p k i,j p,q σ ( x j,p − x i,p ) σ ( x j,q − x i,q ) = 0 to make ˙ V = 0 . Similarly , suppose that the second topic has been reached a consensus, i.e., p = 2 . Then, we need to hav e P d p =3 P d q =3 , q 6 = p k i,j p,q σ ( x j,p − x i,p ) σ ( x j,q − x i,q ) = 0 . By induction, when p = d − 1 , we need to have k i,j d,d − 1 σ ( x j,d − x i,d ) σ ( x j,d − 1 − x i,d − 1 ) = 0 . So, to make k i,j d,d − 1 σ ( x j,d − x i,d ) σ ( x j,d − 1 − x i,d − 1 ) = 0 , either σ ( x j,d − x i,d ) or σ ( x j,d − 1 − x i,d − 1 ) needs to be zero. 1 Thus, at least one topic does not need to reach a consensus. ( Only if ) Without loss of generality , let us suppose that there is no couple between the topics p = d − 1 and p = d ; but there are couplings between all other remaining topics. Then, by follo wing the above procedure (“ if ” procedure), when p = d − 2 , we have k i,j d − 1 ,d − 2 σ ( x j,d − 1 − x i,d − 1 ) σ ( x j,d − 2 − x i,d − 2 ) + k i,j d,d − 2 σ ( x j,d − x i,d ) σ ( x j,d − 2 − x i,d − 2 ) = 0 . Thus, if 1 Actually , this does not imply that only one equality holds; the two equalities may hold. Indeed, suppose that, at time t , all other conditions have been satisfied, and i and j are still updating their opinions on topics d and d − 1 using the couplings σ ( x j,d ( t ) − x i,d ( t )) and σ ( x j,d − 1 ( t ) − x i, − 1 d ( t )) . If those two coupling gains are equal, then the consensus speeds of i and j on topics d and d − 1 are equal. Thus, the topics d and d − 1 might achiev e a consensus simultaneously . Although this case might rarely happen, it may occur; that is why we call it “ almost zero with at least one topic having x j,p 6 = x i,p ”. it is assumed that σ ( x j,d − 2 − x i,d − 2 ) = 0 , then the two topics, p = d − 1 and p = d , do not need to reach a consensus. Remark 5. In Claim 1, since k i,j p,p = 0 for all p and for all edges, and the coupling graphs are complete gr aphs, it can be classified as a homogeneous-coupling network. The above claim implies that a complete opinion consensus for all topics is not ensured for general graphs, when φ = 0 . Also under the condition of φ = 0 , when the coupling graphs are not complete graphs, it is likely that more than one topics would not reach consensus. Thus, for a complete opinion consensus, it is required to have φ 6 = 0 . Observation 2. Consider a homogeneous-coupling network. Let φ 6 = 0 ; but k i,j p,p = 0 for some p ∈ T , ∀ ( i, j ) ∈ E . Then, a complete opinion consensus is not ensured. Pr oof. Let us divide the set T as T = T ◦ ∪ T × and T ◦ ∩ T × = ∅ , where k i,j p,p 6 = 0 when p ∈ T ◦ and k i,j p,p = 0 when p ∈ T × . Then, for all the topics p ∈ T ◦ , we need to ha ve σ ( x j,p − x i,p ) = 0 to make ˙ V = 0 . Then, to make ψ = 0 , it is required to k i,j p,q σ ( x j,p − x i,p ) σ ( x j,q − x i,q ) = 0 when p ∈ T ◦ and q ∈ T × , or k i,j p,q σ ( x j,p − x i,p ) σ ( x j,q − x i,q ) = 0 when p, q ∈ T × . For the former case, since σ ( x j,p − x i,p ) = 0 , it does not need to hav e σ ( x j,q − x i,q ) = 0 . Thus, for the topics q ∈ T × , a consensus may not be achiev ed. For the latter case, due to the same reason as the proof of Claim 1 , there will be some topics that do not reach a consensus. Theor em 4 and Observation 2 lead a conclusion that each topic needs to be p -coupled to hav e a complete consensus. Howe ver , as remarked in Remark 4 , it is not argued that the p - coupling for all topics, i.e., all-topic coupled, is the necessary and sufficient condition for a complete opinion consensus. From the equation (26), we can infer that the interdependent couplings between topics are required to speed up the opinion consensus. So, to have an opinion consensus on a topic, the agents of the society need to discuss directly on the same topic. But, if they have some opinion couplings with other topics, the consensus of the topic may be achieved more quickly . Next, let us consider the proportional feedbacks modeled by (4) and (5). For the proportional feedbacks, using the same L yapunov candidate V = 1 2 k x k 2 , we can obtain the deriv ative of V as: ˙ V = − X ( i,j ) ∈E d X p =1 d X q =1 k i,j p,q × σ ( x j,p − x i,p ) σ ( x j,q − x i,q ) ( c 1 k x j,p − x i,p k + c 0 )( c 1 k x j,q − x i,q k + c 0 ) ≤ 0 (28) Since the denominator of the right-hand side of (28) is always positiv e, the equilibrium set for ˙ V = 0 is decided if and only if σ ( x j,p − x i,p ) σ ( x j,q − x i,q ) = 0 for all p, q ∈ T . Consequently , we have the same results as the inv erse-proportional feedback couplings. Observation 3. Let us consider gener al heter ogeneous- coupling network, i.e., G i 1 ,j 1 6 = G i 2 ,j 2 for some edges Agent 1 Agent 2 Agent 3 Agent 4 T opic 1 T opic 2 T opic 3 Fig. 4. A network composed of four agents with three topics. ( i 1 , j 1 ) 6 = ( i 2 , j 2 ) . If some topics ar e not p -coupled, then a complete opinion consensus is not ensured. Pr oof. Let us suppose that there is no direct coupling between agents ¯ j and ¯ i , on a specific topic ¯ p . Then, in φ of (26), the term ( x ¯ j, ¯ p − x ¯ i, ¯ p ) 2 is missed. But, the term σ ( x ¯ j, ¯ p − x ¯ i, ¯ p ) may be included in ψ in the form of σ ( x ¯ j, ¯ p − x ¯ i, ¯ p ) σ ( x ¯ j,p − x ¯ i,p ) if there are cross couplings between the topic ¯ p and any other topics p . If there is a direct coupling on the topic p between agents ¯ j and ¯ i , then the term σ ( x ¯ j,p − x ¯ i,p ) will be zero; thus, σ ( x ¯ j, ¯ p − x ¯ i, ¯ p ) does not need to be zero to make ˙ V zero. Or , if there is no direct coupling on the topic p between agents ¯ j and ¯ i , still either σ ( x ¯ j, ¯ p − x ¯ i, ¯ p ) or σ ( x ¯ j,p − x ¯ i,p ) does not need to be zero also. Thus, a complete opinion consensus is not ensured. The results of Observation 2 and Observation 3 leave a question about the clustered opinions. Let us consider a network depicted in Fig. 4. From the term φ in (26), all the topics between agents 1 and 2 , and all the topics between agents 3 and 4 reach an opinion consensus. Due to the interdependent couplings between agents 2 and 3 , we hav e the interdependency terms as ψ = k 2 , 3 1 , 2 σ ( x 2 , 1 − x 3 , 1 ) σ ( x 2 , 2 − x 3 , 2 ) + k 3 , 2 2 , 1 σ ( x 2 , 2 − x 3 , 2 ) σ ( x 2 , 1 − x 3 , 1 ) + k 2 , 3 2 , 3 σ ( x 2 , 2 − x 3 , 2 ) σ ( x 2 , 3 − x 3 , 3 ) + k 3 , 2 3 , 2 σ ( x 2 , 3 − x 3 , 3 ) σ ( x 2 , 2 − x 3 , 2 ) . Thus, by Barbalat’ s lemma, to make ˙ V zero, we need to hav e ψ = 0 . From the above equation, for example, if σ ( x 2 , 2 − x 3 , 2 ) = 0 , then ψ becomes zero. The largest in variant set for having ˙ V = 0 is obtained as D = D d ∪ D u , where the desired set is giv en D d = { x : x 1 = x 2 = x 3 = x 4 } and undesired set is given as D u = { x : x 1 = x 2 , x 3 = x 4 , x 2 6 = x 3 } In the undesired set, the opinions of agents 2 and 3 may be related as (i) x 2 , 2 = x 3 , 2 , but x 2 , 1 6 = x 3 , 1 and x 2 , 3 6 = x 3 , 3 , (ii) x 2 , 2 6 = x 3 , 2 , but x 2 , 1 = x 3 , 1 and x 2 , 3 = x 3 , 3 , (iii) x 2 , 3 6 = x 3 , 3 , but x 2 , 1 = x 3 , 1 and x 2 , 2 = x 3 , 2 , or (iv) x 2 , 1 6 = x 3 , 1 , but x 2 , 2 = x 3 , 2 and x 2 , 3 = x 3 , 3 . Thus, a part of opinions reaches a consensus, while a part of opinions may reach clustered consensus. It is clear that if there are some topics that are p -coupled, then a complete clustered consensus cannot take place. Also, ev en though the network is not p -coupled for all p , if the network is connected, then a complete clustered consensus is not ensured since the connected neighboring topics would 1 2 3 4 5 Fig. 5. Underlying topology for numerical simulations. reach a consensus. Thus, a complete opinion consensus rarely occurs as f ar as the network is connected. But, a partial opinion consensus would occur easily if it is not all-topic coupled. In fact, if the network is not all-topic coupled, the network would have opinion-based clustered consensus. It means that if agents of network are connected, some opinions would be agreed among agents, b ut some opinions would be di vided into clusters. Or, most of opinions would be clustered, depending on interaction network topology G and the topic topologies G p . Observation 4. Suppose that a network is connected. Even though φ = 0 , a complete cluster ed consensus is not ensured. Pr oof. Due to the term ψ including σ ( x j,p − x i,p ) σ ( x j,q − x i,q ) , at least one of the topics p and q needs to be agreed. Thus, a complete clustered opinion consensus does not occur . Now , with the statements of Observation 2 , Observation 3 and Observation 4 , we can see that if agents of a society are not all-topic coupled, but just connected in the sense of interaction graph G , then both a complete opinion consensus and complete clustered consensus are not ensured. I V . S I M U L A T I O N S A. Case of P ositive Semidefinite Laplacian Let us consider five agents with the underlying interaction network topology as depicted in Fig. 5. The initial opinions of agents are giv en as x 1 = (1 , 2 , 3) T , x 2 = (2 , 4 , 4) T , x 3 = (3 , 1 , 5) T , x 4 = (4 , 3 , 2) T , x 5 = (5 , 6 , 1) T . The initial opinions of agents for the three topics are different each other . T o verify the results of Section III-A, the follo wing coupling matrices are considered. K 1 , 2 =   1 1 0 1 1 0 0 0 0   ; K 1 , 3 =   1 0 0 0 1 1 0 1 1   K 2 , 3 =   2 0 1 0 2 1 1 1 2   ; K 3 , 4 =   1 1 1 1 1 1 1 1 1   K 4 , 5 =   1 0 1 0 1 0 1 0 1   (29) which are all positiv e semidefinite. From the abov e coupling matrices, it is shown that the topic consensus graph G p,con for all p = 1 , 2 , 3 is connected. Thus, as e xpected from Theor em 3 , a consensus for all topics is achieved. Fig. 6 sho ws that all the values of the topics of agents reach a consensus as time passes. Next, let us change K 1 , 3 and K 3 , 4 as K 1 , 3 =   0 0 0 0 1 1 0 1 1   ; K 3 , 4 =   0 0 0 0 1 1 0 1 1   which are still positive semidefinite. Ho wev er, due to the new K 3 , 4 , there is a disconnection in topic 1 between agents 3 and 4 . Thus, the topic 1 is not connected in the topic consensus graph G 1 ,con . As expected from Theorem 1 , there will be two clusters. Fig. 7 sho ws that the topic 1 does not reach a consensus; there are two clusters (one cluster with agents 1 , 2 , and 3 , and another cluster with agents 4 and 5 ). B. General Cases Let the coupling topologies for each edge be given as: K 1 , 2 =   1 1 0 1 1 1 0 1 1   ; K 1 , 3 =   1 1 0 1 1 1 0 1 1   K 2 , 3 =   1 1 0 1 1 1 0 1 1   ; K 3 , 4 =   1 1 0 1 1 1 0 1 1   K 4 , 5 =   1 1 0 1 1 1 0 1 1   which are indefinite matrices. Since all the topics are p - coupled, it is an all-topic coupled network. Also, since the coupling matrices for all edges are equiv alent, it is a homoge- neous network. With the above coupling matrices, as expected from Theorem 4 , the topics of agents reach a complete opinion consensus. Next, let us change the matrix K 3 , 4 as K 3 , 4 =   0 1 0 1 0 1 0 1 1   (30) In this case, the topic 1 and 2 are not p -coupled, although the underlying interaction network is connected. As observed in Observation 3 , Fig. 8 shows that the topic 1 does not reach a consensus, while the topic 2 still reaches a consensus. In the topic 1 , agents 3 , 4 and 5 reach a consensus, while agents 4 and 5 reach a consensus. But, when the matrix A 3 , 4 is changed again as K 3 , 4 =   1 1 0 1 0 1 0 1 1   (31) all the topics hav e reached a consensus although it is not all- topic coupled. Let us change the weight matrices K 2 , 3 and K 1 , 3 as K 1 , 3 =   0 1 0 1 1 1 0 1 0   ; K 2 , 3 =   0 1 0 1 1 1 0 1 0   (32) In this case, the network is not all-topic coupled. As shown in Fig. 9, the topics 1 and 3 do not reach a consensus, while the topic 2 reaches a consensus. Next, let us consider φ = 0 and G i,j ∀ ( i, j ) ∈ E are complete graphs. Fig. 10 shows the simulation result. All the topics do not reach a consensus. V alues of topic 1 V alues of topic 2 V alues of top ic 3 Fig. 6. Consensus under positiv e semidefinite Laplacian: Left - T opic 1 (i.e., x i, 1 , i = 1 , . . . , 5 ). Center - T opic 2 (i.e., x i, 2 , i = 1 , . . . , 5 ). Right - T opic 3 (i.e., x i, 3 , i = 1 , . . . , 5 ). Fig. 7. Partial opinion consensus with a disconnected topic consensus graph. 0123456789 1 0 Tim e 10 4 1 1.5 2 2.5 3 3.5 4 4.5 5 V alues of topic 1 012345678 9 1 0 Time 10 4 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 V alues of topic 2 0123456789 1 0 Tim e 10 4 1 1.5 2 2.5 3 3.5 4 4.5 5 V alues of topic 3 Fig. 8. The topics 1 and 2 are not p -connected, due to zero diagonal terms in K 3 , 4 : Left - T opic 1 (i.e., x i, 1 , i = 1 , . . . , 5 ). The agents 4 and 5 reach a consensus, and agents 1 , 2 and 3 reach a consensus for the topic 1 . Center - T opic 2 (i.e., x i, 2 , i = 1 , . . . , 5 ). Right - T opic 3 (i.e., x i, 3 , i = 1 , . . . , 5 ). 0123456789 1 0 Tim e 10 4 1 1.5 2 2.5 3 3.5 4 4.5 5 V alues of topic 1 012345678 9 1 0 Time 10 4 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 V alues of topic 2 0123456789 1 0 Tim e 10 4 1 1.5 2 2.5 3 3.5 4 4.5 5 V alues of topic 3 Fig. 9. Not all-topic connected, with zero diagonal terms in K 1 , 3 and K 2 , 3 ; only the topic 2 is p -connected. The topics 1 and 3 do not reach a consensus (clustered), while the topic 2 reaches a consensus. For both the topics 1 and 3 , agents 1 and 2 reach a consensus, and agents 3 , 4 , and 5 reach a consensus. 0123456789 1 0 Tim e 10 4 1 1.5 2 2.5 3 3.5 4 4.5 5 V alues of topic 1 012345678 9 1 0 Time 10 4 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 V alues of topic 2 0123456789 1 0 Tim e 10 4 1 1.5 2 2.5 3 3.5 4 4.5 5 V alues of topic 3 Fig. 10. φ = 0 with complete interdependency graphs. The agents do not reach a consensus ev en for a topic. V . C O N C L U S I O N The cooperative opinion dynamics on multiple interdepen- dent topics may be considered as a consensus problem of multi-layer networks. Each topic can be considered as a basic layer and the term a i,j p,q may describe a cross-layer connection between the layer p and layer q , and between agent i and agent j . The basic layer is the direct connections that are essential for achieving a consensus on this layer . This paper shows that the opinion dynamics with multiple interdependent topics, which is the consensus dynamics in multi-layer networks, possesses some ne w properties different from the usual consensus in one layer . Clustering phenomenon occurs quite often, e ven though the number of connections between agents is large. In general, adding a direct connection a i,j p,p forces a consensus between agents i and j on the topic p . On the other hand, adding a set of cross-layer connections { a i,j p,q } q 6 = p,q =1 ,...,d may not so significantly helpful for the agents i and j to reach a consensus on topic p . But, from simulations, it is sho wn that the cross-layer connections are still beneficial for a consensus on the topics. Of course, as analyzed in the case of positiv e semidefinite Laplacian matrices, the cross couplings are also very helpful for a consensus. In our future efforts, we would like to ev aluate the polarization phenomenon of bipartite graphs under the setup of multiple interdependent couplings, which may be a general one of [9], [10] in multidimensional spaces. It is also interesting to change the overall formulation in discrete-time cases; then the discontinuity arising in the sign functions can be handled more easily . 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