Edge-based Synchronization over Signed Digraphs with Multiple Leaders

1 Edge-based Synchronization o v er Signed Digr aphs with Multiple Leaders P elin Sekercioglu, Angela Fontan, Dimos V . Dimarogonas Abstract — W e address the edge-based synchr onization prob lem in first-or der m ulti-agent systems containing both cooperative and antagonistic interactions with one or m ulti- ple leader groups. The presence of m ultiple leader s and an- tagonistic interactions means that the multi-agent typically does not achieve consensus, unless specific conditions (on the number of leader s and on the signed graph) are met, in which case the agents reach a trivial form of consensus. In general, we show that the multi-a gent system exhibits a more general form of synchronization, including bipar- tite consensus and containment. Our approac h uses the signed edge-based agreement protocol f or signed netw orks described b y signed edge-Laplacian matrices. In particular , in this work, we present new spectral properties of signed edge-Laplacian matrices containing multiple zero eigen val- ues and establish global exponential stability of the syn- chr onization error s. Moreover , we compute the equilibrium to which all edge states con verge. Numerical simulations validate our theoretical results. Index T erms — Signed edge-based agreement protocol, synchr onization, signed networks. I . I N T R O D U C T I O N T HE coordination of multi-agent systems has been widely in vestigated, with numerous results addressing problems such as consensus under first-order , second-order , and general linear high-order dynamics [1]. When the interaction network includes a single leader , classical leader–follo wer consen- sus ensures that all follo wer agents asymptotically track the leader’ s state [2]. Howe v er , this frame work no longer applies when multiple leaders are present in the network. In such situations, where se veral possible agreement configurations can arise, the problem is better described as containment control [3]. The objectiv e of containment control is to drive the follo wers’ states to ward the containment set formed by the leaders’ initial states. Numerous works have in vestigated distributed containment control in v arious settings [4]–[6]. Howe ver , the majority of existing studies on consensus and containment in multi- agent systems focus on purely cooperative networks, that is, This work was suppor ted in par t by the W allenberg AI, Autonomous Systems and Software Prog ram (W ASP) funded by the Knut and Alice Wallenberg (KA W) F oundation, the Horizon Europe EIC project SymA ware (101070802), the ERC LEAFHOUND Project, the Swedish Research Council (VR), and Digital Futures . P . Sek ercioglu, A. Fontan, and D . V . Dimarogonas are with the Division of Decision and Control Systems, KTH Roy al In- stitute of T echnology , SE-100 44 Stockholm, Sweden (e-mail: { pelinse,angf on,dimos } @kth.se). scenarios in which coordination among agents is achie ved ex- clusiv ely through cooperativ e interactions. Howe v er , in many real-world scenarios, agents may exhibit antagonistic behav- iors. Examples include robotic applications [7], [8], and social networks where agents compete [9]–[11] and spread disinfor- mation [12], to name just a few . A representative example is found in opinion dynamics, where consumer behavior is shaped by both peer influence and external entities, such as competing marketers (leaders). These leaders, grouped into disjoint cooperati ve or competitive subsets, affect consumers. Consumers may oppose a disliked marketer by aligning with the other , or adopt a position between them if they trust both, fa voring the one with stronger influence. A common and widely adopted framew ork for represent- ing both cooperative and antagonistic relationships in multi- agent systems relies on signed graphs [9], [13]. W ithin this representation, cooperativ e interactions are encoded by positiv e edge weights, whereas antagonistic relationships are captured through negati ve edge weights. In such networks, rather than achieving standard consensus, the typical emergent synchronization behavior is bipartite consensus , where agents con ver ge to two symmetric equilibrium points, provided the signed graph is structurally balanced . A signed graph is said to be structurally balanced if the nodes can be divided into two disjoint subsets such that agents within the same subset coop- erate, while agents in different subsets interact antagonistically [9]. There are various studies on bipartite consensus control [9], [14] and on bipartite containment control [12], [15]–[17]. In this paper , we study the edge-based synchronization of signed multi-agent systems interconnected over directed signed graphs. W e recast the synchronization problem into one of stability of the appropriately defined synchronization errors. The main contributions are threefold: (i) we present new prop- erties of edge-Laplacian matrices (hereafter: edge Laplacians) containing multiple zero eigen values defining the signed multi- agent system (Lemma 7 and T able I); (ii) we characterize the eigenv ectors associated with the edge Laplacians within the signed edge-based network formulation (Equations (15) and (16)); and (iii) we establish exponential stability of the synchronization set and provide explicit estimates of the edge- based limit points of the agents in the signed edge-based network formulation (Theorem 3), using a L yapunov equation- based analysis pre viously proposed in [18]. In contrast to [12], [15], [17], which consider node-based formulations, we focus on edge-based synchronization. Compared to [19], which studies edge conv ergence in strongly connected signed 2 digraphs, and [20], which considers signed digraphs containing a spanning tree, we address the more general case of signed digraphs with multiple leaders and establish exponential sta- bility of the zero synchronization error set. From a technical viewpoint, our main results on exponential stability build on [12] and the framew ork introduced in [21], where the overall dynamics are decomposed into two coupled subsystems: the e volution of a weighted av erage and the synchronization errors relati ve to that average. W e generalize the latter to the setting of directed signed graphs, adopting an edge-based representation. In particular , we recast the synchronization problem as a stability problem for synchro- nization errors, defined as the difference between the edge states and the weighted edge average. The analysis is carried out in signed edge-based coordinates, for which we inv estigate structural properties of the incidence and edge Laplacian matrices. As a key technical contribution, we provide new results that characterize the rank, null space, and eigen vector properties of signed edge Laplacians associated with directed signed graphs that contain one or multiple leaders. Then, we establish exponential stability of the synchronization set using L yapunov characterization for Laplacians with multiple zero eigen v alues in [18] and we provide the explicit edge limit values. T ogether , these results yield a complete characteriza- tion of edge synchronization ov er directed signed graphs with leaders. I I . P R E L I M I N A R I E S Notation: |·| denotes the absolute value for scalars, the Euclidean norm for vectors, and the spectral norm for matrices. card ( · ) indicates the cardinality of a set. diag ( z ) denotes a diagonal matrix whose diagonal elements are the entries of the vector z . N ( A ) denotes the null space of matrix A . R is the set of real numbers and R ≥ 0 the nonnegati ve orthant. A > 0 ( A ≥ 0 ) indicates that A is a positi ve definite (positive semidefinite) matrix. An M-matrix is a matrix whose of f- diagonal entries are nonpositive, and whose eigen v alues hav e nonnegati ve real parts. blkdiag ( A i ) indicates a block diagonal matrix formed by the matrices A i . Let G s = ( V , E ) be a signed graph, where V = { ν 1 , ν 2 , . . . , ν N } is the set of N nodes and E ⊆ V × V is the set of edges. Each edge in E has a sign, either positive or negati ve. If all edges ha ve a positi ve sign, the graph is called an unsigned graph (in standard graph theory). The graph is undirected if information flow between agents is bidirectional, meaning ( ν i , ν j ) = ( ν j , ν i ) ∈ E . Otherwise, the graph is directed and is commonly referred to as a digraph. The edge ε k = ( ν j , ν i ) ∈ E of a digraph denotes that the agent ν i , which is the terminal node (tail of the edge), can obtain information from the agent ν j , which is the initial node (head of the edge). The adjacency matrix of G s is A := [ a ij ] ∈ R N × N , where a ij  = 0 if and only if ( ν j , ν i ) ∈ E . a ij > 0 if and only if the edge ( ν j , ν i ) has a positi ve sign, indicating a cooperati ve relationship, and a ij < 0 if and only if the edge ( ν j , ν i ) has a negati ve sign, indicating an antagonistic relationship. In this work, we only consider unweighted digraphs, such that a ij = { 0 , 1 , − 1 } , without any self-loops. A signed digraph is ν 1 ν 2 ν 3 ν 4 ν 9 ν 5 ν 6 ν 7 ν 8 Fig. 1 : A signed digon sign-symmetric digraph containing 3 leader groups, where the black edges represent cooperativ e interactions and the dashed red edges represent antagonistic interactions. The first leader group is a SB-rooted SCC con- taining the leader nodes ν 1 , ν 2 , ν 3 , and ν 4 , the second leader group is a SUB-rooted SCC containing the leader nodes ν 5 , ν 6 , and ν 7 , and the third leader group is a root (leader) node, ν 8 . The node ν 9 is the follower node. said to be digon sign-symmetric if a ij a j i ≥ 0 . It means that the interaction between two interconnected agents alw ays has the same sign in both directions. Throughout this work, we make the following standing assumption: Assumption 1: The signed digraph is unweighted and digon sign-symmetric. A directed path is a sequence of distinct adjacent nodes in a digraph. When the nodes of the path are distinct except for its end vertices, the directed path is called a directed cycle. A directed spanning tree is a directed tree subgraph that includes all the nodes of the digraph. In this structure, ev ery agent (node) has a parent node, except for the r oot node , which has no incoming edges and is connected to ev ery other node via directed paths. Since each edge points from the root to other nodes, the tree contains no cycles. A digraph is said to be strongly connected if there exists a directed path between ev ery pair of nodes. In this work, we consider signed digraphs that may contain multiple leader nodes. A leader node is defined as either a root node or a node that is part of a r ooted str ongly-connected com- ponent (r ooted SCC) . A rooted SCC is a strongly connected subgraph without incoming edges. A leader group is either a single root node (representing a single-node leader group) or an entire rooted SCC (representing multiple leader nodes interconnected in a strongly connected subgraph). If the graph contains at least one leader group, the remaining nodes are referred to as followers.— See Figure 1. A signed graph is said to be structurally balanced (SB) if it can be split into two disjoint sets of vertices V 1 and V 2 , where V 1 ∪ V 2 = V , V 1 ∩ V 2 = ∅ such that for every ν i , ν j ∈ V p , p ∈ { 1 , 2 } , if a ij ≥ 0 , while for every ν i ∈ V p , ν j ∈ V q , with p, q ∈ { 1 , 2 } , p  = q , if a ij ≤ 0 . It is structurally unbalanced (SUB), otherwise [9]. The signed Laplacian matrix L s = [ ℓ s ij ] ∈ R N × N associated with G s is defined as ℓ s ij := P k ≤ N | a ik | , if i = j ; and ℓ s ij := − a ij , if i  = j [9], [13]. W e no w recall definitions of the signed incidence matrices of a signed digraph. Consider a signed graph G s containing N nodes and M edges. The signed incidence matrix E s ∈ R N × M of G s is defined as [ E s ] ik :=        +1 , if ε k = ( ν i , ν j ); − 1 , if ν i , ν j are cooperati ve and ε k = ( ν j , ν i ); +1 , if ν i , ν j are competiti ve and ε k = ( ν j , ν i ); 0 , otherwise , (1) SEKERCIOGLU et al. : EDGE-BASED SYNCHRONIZA TION OVER SIGNED DIGRAPHS WITH MUL TIPLE LEADERS 3 and the signed in-incidence matrix E s ⊙ ∈ R N × M of G s is defined as [ E s ⊙ ] ik :=    − 1 , if ν i , ν j are cooperati ve and ε k = ( ν j , ν i ); +1 , if ν i , ν j are competiti ve and ε k = ( ν j , ν i ); 0 , otherwise . (2) where ε k is the oriented edge interconnecting nodes ν i and ν j , k ≤ M , i, j ≤ N . The signed Laplacian L s ∈ R N × N and the signed edge Laplacian L e s ∈ R M × M of a signed digraph, they are giv en as L s = E s ⊙ E ⊤ s , L e s = E ⊤ s E s ⊙ . (3) The signed Laplacian of a signed digraph is not symmetric, and its eigenv alues all hav e nonnegati ve real parts. I I I . M O D E L A N D P R O B L E M F O R M U L A T I O N Consider a group of N first-order agents interconnected ov er a signed digraph G s with M cooperati ve and antagonistic edges. The dynamics of each agent are giv en by ˙ x i = u i , i ∈ { 1 , 2 , . . . , N } , (4) where x i ∈ R is the state of the i th agent, and u i ∈ R is the control input. The system (4) is interconnected with the control la w u i = − k 1 N X j =1 | a ij | [ x i − sign ( a ij ) x j ] , (5) where k 1 > 0 , and A = [ a ij ] is the adjacency matrix, with a ij ∈ { 0 , ± 1 } representing the adjacenc y weight between nodes ν j and ν i . It is well known that under the distributed control law (5), agents interconnected over a SB signed digraph achieve bipartite consensus if and only if the signed graph contains a directed spanning tree, while those over a SUB digraph reach trivial consensus provided that the graph contains a directed spanning tree and has no root node. Thus, we pose the following on the connectivity of the signed graph. Assumption 2: The signed digraph contains a directed span- ning tree. In this article, we analyze the behavior of the multi-agent system (4) in closed loop with the control law (5) and under the assumption that the edges interconnecting the agents are cooperativ e and antagonistic. The possible achie v able control objectiv es for the system (4)-(5) interconnected on a signed digraph depend on the structural balance property and the ov erall topology of the graph. In particular, the presence of rooted SCCs and root nodes plays a crucial role in determining the agents’ conv ergence behavior . On the one hand, a rooted SCC influences the con ver gence of the rest of the network, depending on whether it is SB or SUB. If the digraph contains a single root node, the system exhibits a leader-follo wing bipartite consensus, where all agents conv erge either to the leader’ s state or its opposite state if the digraph is SB. In such cases, the root node or the rooted SCC acts as a leader gr oup , influencing the remaining agents, referred to as followers . More formally , we define a leader group as either a single root node or a rooted SCC consisting of multiple nodes. On the other hand, if the digraph contains multiple root nodes or rooted SCCs, each of these leader groups dictates the behavior of the remaining agents. Instead of conv erging to a single or bipartite equilibrium, the agents achiev e bipartite containment, that is, they settle within a region defined by the states of the leader groups. In this particular case, the digraph no longer contains a directed spanning tree, as independent leader groups are not mutually reachable. Therefore, for signed digraphs with multiple leaders groups, we pose the following connectivity assumption. Assumption 3: The signed digraph contains m leader groups, where l 1 is the number of root nodes, l 2 S B is the number of SB-rooted SCCs, and l 2 S U B is the number of SUB- rooted SCCs with m = l 1 + l 2 > 1 and l 2 = l 2 S B + l 2 S U B ; L and F are the sets containing the index corresponding to the nodes in the leader groups and followers respectively . 1) The network contains k leader nodes, which can be organized into m leader groups of p i nodes included in a strongly connected subgraph (or p i = 1 if it is a single root node), where 1 < m ≤ k < N , i ≤ m , and P m i =1 p i = k . 2) Given each follower ν j , with j ∈ F , there exists at least one leader ν i , with i ∈ L , such that there exists at least one path from ν i to ν j . Remark 1: Assumption 3 is deri ved from the leader defini- tion in [15]. If we consider the signed digraph in Figure 1, the leader set is gi ven as L = { ν 1 , ν 2 , ν 3 , ν 4 , ν 5 , ν 6 , ν 7 , ν 8 } and the follower set is giv en as F = { ν 9 } . Moreover , l 1 = l 2 S B = l 2 S U B = 1 . Regarding Item 1 of Assumption 3, we have k = 8 leader nodes organized into m = 3 leader groups. The first leader group contains p 1 = 4 nodes, the second leader group contains p 2 = 3 nodes, and the third leader group contains p 3 = 1 node, where P m i =1 p i = p 1 + p 2 + p 3 = 8 . The achiev able control objectiv e for (4) in closed loop with the distributed control law (5) and interconnected over a SB digraph: • containing a directed spanning tree (Assumption 2) is to ensure agents achiev e bipartite consensus , where agents con ver ge to the same value in modulus but not in sign, that is, lim t →∞ [ x i ( t ) − sign ( a ij ) x j ( t )] = 0 , ∀ i, j ≤ N . (6) • containing m leader groups, under Assumption 3, is to ensure agents achieve bipartite containment , that is, lim t →∞ [ x j ( t ) − max i ∈L ( s i x i ( t ))][ x j ( t ) − min i ∈L ( s i x i ( t ))] ≤ 0 , (7) for each j ∈ F , where s i = 1 if agent ν j is cooperativ e with leader ν i , and s i = − 1 otherwise. The achiev able control objectiv e for (4) in closed loop with the distributed control law (5) interconnected ov er a SUB digraph: • containing a directed spanning tree (Assumption 2) and a SUB-rooted SCC, without a root (leader) node, is to ensure agents achiev e trivial consensus , where all agents con ver ge to zero, that is, lim t →∞ x i ( t ) = 0 , ∀ i ≤ N ϕ . (8) 4 • containing a directed spanning tree (Assumption 2) and SB-rooted SCC or a root node, is to ensure agents achie ve interval bipartite consensus , that is, lim t →∞ x i ( t ) ∈ [ − θ , θ ] , ∀ i ≤ N , (9) where θ > 0 . • containing m leader groups, under Assumption 3, is to ensure agents achieve bipartite containment , that is, lim t →∞  | x j ( t ) | − max i ∈L | x i ( t ) |  ≤ 0 , j ∈ F . (10) Remark 2: If the considered digraph is SUB and contains multiple leader groups, where all rooted SCCs are SUB and the digraph contains a root node, the bipartite containment objectiv e (10) boils down to interval bipartite consensus (9). If the digraph contains only SUB-rooted SCCs, the bipartite containment objective (10) boils down to tri vial consensus (8). Remark 3: The bipartite containment objective of (7), (10) can be interpreted as follo ws. When the digraph is SB (7), for each follower i ∈ F , the achie vable con ver gence zone is determined by a containment set defined by the absolute value of the leaders’ states, taken with the sign that captures the type of interaction between the follower and leader , positiv e if cooperativ e, negati ve if antagonistic. In particular, this leads to two symmetric con ver gence zones, determined by the gauge permutation capturing the SB property of the signed digraph (Lemma 2 in Section IV). Instead, when the digraph is SUB (10), the containment set is defined only by the absolute value of the leaders’ states, leading to a unique achiev able con ver gence zone. For more details, simulation examples, and illustration of conv ergence zones, see [17, Section V]. In this article, we show that, under Assumptions 1–3, depending on whether the associated signed digraph is SB or SUB and contains multiple leader groups or not, synchro- nization in the signed OMAS leads to bipartite consensus (6), trivial consensus (8), interval bipartite consensus (9) or bipartite containment (10). In the case agents achieve trivial consensus or bipartite consensus, their edges states, defined as e k = x i − sign ( a ij ) x j , ε k = ( ν j , ν i ) ∈ E , (11) where k ≤ M denotes the index of the interconnection between the j th and i th agents, conv erge to zero. Howe ver , as the networks considered here, in addition to signs on the edges, contain a priori, rooted SCCs or multiple root nodes, the resulting Laplacians can also have multiple zero eigen values in some cases. This also results, in general, in multiple con ver gence points for agents and their edge states, which means that edge states do not always con ver ge to zero. Then, following the frame work laid in [21] and extending it to signed networks with associated Laplacians containing multiple zero eigen v alues and to the signed edge-based formulation, we define the weighted av erage system for the edge states. Let ξ be the number of zero eigen values of L e s . Then, we define the weighted edge av erage state as e m := ξ X i =1 v r i v ⊤ l i e, (12) where v r i and v l i are the right and left eigen vectors as- sociated with the zero eigen values of the edge Laplacian, e := [ e 1 e 2 . . . e M ] ⊤ , and e k is defined in (11). W e define the synchronization errors as ¯ e = e − e m = [ I − ξ X i =1 v r i v ⊤ l i ] e, (13) where ¯ e := [ ¯ e 1 ¯ e 2 . . . ¯ e M ] ⊤ . Then, the control objectiv e is equiv alent to making the synchronization errors con verge to zero, that is, lim t →∞ ¯ e k ( t ) = 0 , ∀ k ≤ M . (14) Before introducing our main results in Section V, we remind from [18, Section IV] in Section IV -C some properties of the signed edge Laplacians that are useful for establishing our main results, using notions on signed graphs and the signed edge-based formulation from [9] and [20], respectiv ely . I V . S I G N E D E D G E - B A S E D F O R M U L A T I O N A. Spectral proper ties of signed edge Laplacians For a general digraph, the following properties hold regard- ing similarity transformations, the positi ve (semi-)definiteness of its Laplacians, and the dimension of their null space [9], [19], [20]. W e first start by the rank properties of the in- incidence matrix and then consider SB and SUB digraphs containing a directed spanning tree, respecti vely . Similar prop- erties for undirected signed graphs can be found in [22]. Lemma 1: For a signed digraph containing a directed span- ning tree, rank ( E s ⊙ ) = N − 1 if the digraph contains a root node, and rank ( E s ⊙ ) = N , otherwise. Pr oof: A directed spanning tree of N agents contains exactly N − 1 independent edges. If the digraph contains a root node, this root node has no incoming edges, which implies that the corresponding row in the in-incidence matrix is filled by zeros. In contrast, all other nodes have at least one incoming edge, resulting in at least one nonzero entry (either 1 or − 1 ) in their respective rows of the in-incidence matrix. Consequently , rank ( E s ⊙ ) = N − 1 . Con versely , in the case the digraph does not contain a root node, we have rank ( E s ⊙ ) = N . W e can transform the Laplacian and incidence matrices of SB signed graphs into matrices corresponding to unsigned graphs using some similarity (gauge) transformations. In the case the graph is SUB, these transformations cannot be used. Lemma 2: (Gauge transformation [9]) For a SB signed graph, there exists a diagonal matrix D ∈ D , where D = { D = diag ( d ) | d = [ d 1 d 2 . . . d N ] , d i ∈ {− 1 , 1 } , i ≤ N } , such that all off-diagonal elements of D L s D are non-positive, i.e., D L s D is an M-matrix. Lemma 3: (Edge-gauge transformation [19, Lemma 4]) For a SB signed graph, there exist diagonal matrices D = diag( d ) , from Lemma 2, and D e = diag ( d e ) , where d e = [ d e 1 . . . d e M ] , i ≤ M , with d e i = 1 if ν i ∈ V 1 , and d e i = − 1 if ν i ∈ V 2 , where ν i is the initial node of the edge, such that E = D E s D e has the structure of an incidence matrix corresponding to an unsigned graph. W e now analyze the rank properties of the signed Laplacian and the signed edge Laplacian when the digraph contains a SEKERCIOGLU et al. : EDGE-BASED SYNCHRONIZA TION OVER SIGNED DIGRAPHS WITH MUL TIPLE LEADERS 5 directed spanning tree, considering both the SB (Lemma 4) and SUB (Lemma 5) cases. Then, Lemma 6 studies the null space of these matrices. Finally , Lemma 7 addresses the case where the digraph lacks a spanning tree and multiple leader groups are present. For SB signed digraphs, we ha ve the follo wing regarding the rank properties of the Laplacian and incidence matrices. Lemma 4: For a SB digraph containing a directed spanning tree, the following holds. (i) L s has a simple zero eigenv alue and its other eigenv alues hav e positi ve real parts. (ii) rank ( E s ) = N − 1 . (iii) rank ( L s ) = rank ( L e s ) = N − 1 . Pr oof: (i) By assumption, the digraph is SB and contains a directed spanning tree, so we apply the gauge transformation (Lemma 2) to L s , and the statement follows directly from [23, Lemma 3.3]. (ii) By assumption, the digraph is SB and contains a directed spanning tree, so we apply the edge-gauge transformation (Lemma 3) to E s . The statement follows directly from [24]. (iii) W e follow the steps in the proof of [25, Lemma 2]. Suppose λ  = 0 is an eigen v alue of L s , which is associated with a nonzero eigen vector v r , such that L s v r = E s ⊙ E ⊤ s v r = λv r  = 0 . It is clear that E ⊤ s v r  = 0 . Let ¯ v r = E ⊤ s v r . Then, by left-multiplying E ⊤ s on both sides, we obtain E ⊤ s E s ⊙ E ⊤ s v r = E ⊤ s λv r . By replacing (3) in the latter , we obtain L e s ¯ v r = λ ¯ v r , which means λ is also an eigenv alue of L e s . Consequently , the nonzero eigen v alues of L s and L e s are equal to each other . Then, according to Item (ii), prov en abov e, and from Lemma 1, we ha ve rank ( L s ) ≤ min { rank ( E s ⊙ ) , rank ( E ⊤ s ) } = N − 1 and rank ( L e s ) ≤ min { rank ( E ⊤ s ) , rank ( E s ⊙ ) } = N − 1 . Since they both hav e N − 1 nonzero eigen values, the statement follows. Unlike the SB case, when the digraph is SUB and contains a directed spanning tree, then its structure falls into one of three distinct cases: the presence of a SUB-rooted SCC, a SB-rooted SCC, or a root node, as detailed in the following lemma. Lemma 5: For a SUB digraph containing a directed span- ning tree, the following holds. (i) L s has only eigen values with positive real parts if and only if the digraph contains a SUB-rooted SCC. (ii) L s has a simple zero eigenv alue and its other eigenv alues hav e positiv e real parts if and only if the digraph contains a SB-rooted SCC or a root node. (iii) rank ( E s ) = N . (iv) rank ( L s ) = rank ( L e s ) if and only if the digraph contains a SUB-rooted SCC or a root node. Moreov er , rank ( L s ) = ( N , if G s contains SUB-rooted SCC N − 1 , if G s contains a root node. (v) N − 1 = rank ( L s ) < rank ( L e s ) = N if and only if the digraph contains a SB-rooted SCC. Pr oof: (i) - (ii) follow from [20, Lemma 5]. (iii) Since, SB and SUB are mutually exclusiv e properties, the statement follo ws ( [20, Lemma 2]). (iv), “ ⇐ ”: Assume that the digraph contains a SUB-rooted SCC or a root node. W e separate the two cases. When a SUB digraph contains a SUB-rooted SCC, from Item (i), we know that L s has N nonzero eigen val- ues. On the other hand, from Lemma 1, rank ( E s ⊙ ) = N since the digraph does not contain a root node. So, rank ( L s ) ≤ min { rank ( E s ⊙ ) , rank ( E ⊤ s ) } = N and rank ( L e s ) ≤ min { rank ( E ⊤ s ) , rank ( E s ⊙ ) } = N . Since, from the proof of Item (iii) of Lemma 4, L s and L e s hav e the same nonzero eigenv alues, meaning that they both have N nonzero eigen values, the statement rank ( L s ) = rank ( L e s ) = N follows. Instead, when a SUB digraph contains a root node, from Item (ii), we know that L s has a unique zero eigenv alue and N − 1 nonzero eigen v alues. Moreover , from Lemma 1, rank ( E s ⊙ ) = N − 1 since the digraph contains a root node. Then, the statement rank ( L s ) = rank ( L e s ) = N − 1 follo ws. (iv), “ ⇒ ”: Assume that rank ( L s ) = rank ( L e s ) . Moreover , assume (by contradiction) that the digraph contains an SB- rooted SCC (i.e., it does not contain a SUB-rooted SCC or a root node). Using Item (ii), this implies rank ( L s ) < N . Moreov er , from [26, Section 0.4.5], we know that rank ( L e s ) ≥ rank ( E ⊤ s ) + rank ( E s ⊙ ) − N . Then, using Item (iii) and Lemma 1, we conclude that rank ( L e s ) = N  = N − 1 = rank ( L s ) , obtaining a contradiction. (v) follo ws from the proof of Item (iv). Lemma 6: ( [20, Lemma 7]) For a SB digraph containing a directed spanning tree, N ( L ⊤ e s ) = N ( E s ) holds. For a SUB digraph containing a directed spanning tree: (i) if the signed digraph does not contain a root node, then N ( L ⊤ e s ) = N ( E s ) holds. (ii) if the signed digraph contains a root node, then N ( L ⊤ e s )  = N ( E s ) holds. Pr oof: The statement follows directly from the definition of L e s in (3), the rank properties of E s and L e s in Lemmata 4 and 5, and the rank properties of E s ⊙ of in Lemma 1. Now , we consider signed digraphs containing multiple leader groups, meaning that the signed digraph does not contain a directed spanning tree, since root nodes and rooted SCCs have no incoming edges— See Figure 1. The follo wing lemma is an original contribution of this paper . Lemma 7: Consider a signed digraph containing m leader groups, where m > 1 . Let l 1 ≥ 0 be the number of root nodes, l 2 S B ≥ 0 be the number of SB-rooted SCCs, and l 2 S U B ≥ 0 be the number of SUB-rooted SCCs, where m = l 1 + l 2 and l 2 = l 2 S B + l 2 S U B . Assume that gi ven each follower ν j , there exists at least one leader ν i , such that there exists at least one path from ν i to ν j . Then, the following holds. (i) L s has l 1 + l 2 S B zero eigenv alues and its other eigen v al- ues ha ve positive real parts. (ii) rank ( E s ⊙ ) = N − l 1 . (iii) rank ( E s ) = N − 1 if the digraph is SB, and rank ( E s ) = N otherwise. (iv) rank ( L s ) = N − l 1 − l 2 S B . (v) N − l 1 − 1 ≤ rank ( L e s ) ≤ N − l 1 . Moreov er , if the digraph is SB, (a) rank ( L e s ) = N − 1 , if l 1 = 0 , (b) rank ( L e s ) = N − l 1 , if l 2 S B = 0 . Otherwise, rank ( L e s ) = N − l 1 . 6 (vi) N ( L ⊤ e s ) = N ( E s ) , if all leader groups are rooted SCCs and N ( L ⊤ e s )  = N ( E s ) , if the digraph contains at least one root node. Pr oof: (i) From [9], we hav e that a strongly-connected component adds a zero eigen v alue to the signed Laplacian if and only if it is SB. This, together with [12, Lemma 1] proves that L s has l 1 + l 2 S B zero eigen values. (ii) If the digraph has l 1 root nodes, they have no incoming edges, meaning the rows associated with the roots in the in- incidence matrix have only zeros, while all other nodes hav e at least one incoming edge. As a result, rank ( E s ⊙ ) = N − l 1 . If l 1 = 0 , then, rank ( E s ⊙ ) = N . (iii) The statement follo ws from Item (ii) of Lemma 4 and Item (iii) of Lemma 5. (iv) From [26, 0.4.5], it follows that N − ( l 1 + l 2 S B ) ≤ rank ( L s ) ≤ min { rank ( E s ⊙ ) , rank ( E ⊤ s ) } . From (i), L s has exactly N − ( l 1 + l 2 S B ) nonzero eigen values. Moreover , from [12, Lemma 1] we ha ve that for L s , the zero eigen v alue is semisimple, meaning its algebraic multiplicity equals its geometric multiplicity . Therefore the statement follo ws. (v) First, from the proof of Item (iii) of Lemma 4, we kno w that L s and L e s hav e the same nonzero eigen v alues, so N − ( l 1 + l 2 S B ) ≤ rank ( L e s ) . In addition, from [26, 0.4.5], we have rank ( E ⊤ s ) + rank ( E s ⊙ ) − N ≤ rank ( L e s ) ≤ min { rank ( E ⊤ s ) , rank ( E s ⊙ ) } . Then, for the case of SB digraphs, from (ii) and (iii), we hav e rank ( E s ⊙ ) = N − l 1 and rank ( E s ) = N − 1 . W e also hav e that l 2 S U B = 0 . For l 1 = 0 , we hav e N − 1 ≤ rank ( L e s ) ≤ N − 1 , and (a) follows. For l 2 S B = 0 , we have N − l 1 ≤ rank ( L e s ) ≤ N − l 1 , and (b) follows. For the case of SUB digraphs, from (ii) and (iii), we have rank ( E s ⊙ ) = N − l 1 and rank ( E s ) = N . Then, we hav e N − l 1 ≤ rank ( L e s ) ≤ N − l 1 , so rank ( L e s ) = N − l 1 , and if l 1 = 0 , rank ( L e s ) = N . Thus the statement follows. (vi) In the case l 1 = 0 , we have rank ( E s ) = rank ( L e s ) = N − 1 if the digraph is SB, and rank ( E s ) = rank ( L e s ) = N , otherwise. Since rank ( E s ⊙ ) = N , we can deduce that N ( L ⊤ e s ) = N ( E s ) . In the case l 1 ≥ 1 , on the one hand, for SB digraphs, we ha ve N − l 1 − 1 ≤ rank ( L e s ) ≤ N − l 1 . For SUB digraphs, we ha ve rank ( L e s ) = N − l 1 . On the other hand, rank ( E s ) = N − 1 for SB digraphs and rank ( E s ) = N for SUB digraphs, which indicates N ( L ⊤ e s )  = N ( E s ) . Thus the statement follows. B. On the eigenv ectors of signed edge Laplacians From Lemmata 4 and 5, we know that for signed digraphs containing a directed spanning tree, and from Lemma 7 for signed digraphs with multiple leader groups, the signed edge Laplacian generally has multiple zero eigen v alues. Conse- quently , there exist multiple right and left eigenv ectors associ- ated with each zero eigen v alue of L e s . Moreover , depending on the structure of the signed digraph, these zero eigenv alues can hav e different geometric ( γ ) and algebraic ( ξ ) multiplic- ities. This affects both the Jordan decomposition of L e s and the properties of its associated eigenv ectors. The algebraic and geometric multiplicity properties of signed edge Laplacians of signed graphs containing a directed spanning tree and a root node or a rooted SCC, were already presented in [20]. In T able I, we summarize the geometric and algebraic multiplicities of the zero eigenv alues for each class of signed digraph, including the ones containing multiple leader groups. These properties are crucial for defining the edge states and the av erage edge system, which in turn are used to characterize synchronization errors and analyze the conv ergence beha vior of agents in the OMAS. Based on the information in T able I and following the framew ork in [26], the Jordan canonical form of L e s can be expressed as L e s = U J U − 1 , where U = [ v r 1 v r 2 . . . v r M ] ∈ C M × M contains the right eigen vectors and U − 1 = [ v ⊤ l 1 v ⊤ l 2 . . . v ⊤ l M ] ⊤ ∈ C M × M contains the left eigen vectors. The matrix J = blkdiag [ J 0 , ¯ J ] , where ¯ J corresponds to the Jordan blocks associated with eigen v alues having positiv e real parts, and J 0 contains the Jordan blocks corresponding to the zero eigenv alues, gi ven as J 0 = blkdiag       0 1 0 0  , . . . ,  0 1 0 0  | {z } ξ − γ Jordan blocks , 0 , . . . , 0 | {z } 2 γ − ξ zeros      . (15) Then, from (15), the right and left eigen vectors associated with the zero eigen values satisfy the following. L e s v r 1 = 0 v ⊤ l 1 L e s = v ⊤ l 2 L e s v r 2 = v r 1 v ⊤ l 2 L e s = 0 . . . . . . L e s v r 2( ξ − γ ) − 1 = 0 v ⊤ l 2( ξ − γ ) − 1 L e s = v ⊤ l 2( ξ − γ ) L e s v r 2( ξ − γ ) = v r 2( ξ − γ ) − 1 v ⊤ l 2( ξ − γ ) L e s = 0 L e s v r 2( ξ − γ )+1 = 0 v ⊤ l 2( ξ − γ )+1 L e s = 0 . . . . . . L e s v r ξ = 0 v ⊤ l ξ L e s = 0 . (16) From T able I, it is clear that whenev er the graph contains at least one SB SCC in the graph, the geometric and algebraic multiplicities of the zero eigen v alue differ . As a consequence, for the first 2( ξ − γ ) left and right eigen vectors, the product with L e s is nonzero for ev ery other pair , while the rest yield zero. On the other hand, the product in volving the remaining 2 γ − ξ eigen vectors is zero. Remark 4: Throughout the paper , the left and right eigen- vectors associated with the zero eigen v alues of L e s are nor- malized such that v ⊤ l i v r i = 1 for all i ≤ ξ . Remark 5: Recall from (3) that the signed Laplacian and edge-Laplacian can be written as L e s = E ⊤ s E s ⊙ ∈ R M × M and L s = E s ⊙ E ⊤ s ∈ R N × N , respectiv ely . From the proof of Item (iii) of Lemma 4, we know that L e s and L s hav e the same nonzero eigen v alues. From [12, Lemma 1] we hav e that for SEKERCIOGLU et al. : EDGE-BASED SYNCHRONIZA TION OVER SIGNED DIGRAPHS WITH MUL TIPLE LEADERS 7 T ABLE I : Geometric ( γ ) and algebraic ( ξ ) multiplicities of the zero eigen values of the directed signed edge Laplacian L e s . Note that γ = dim(k er( L e s )) = M − rank( L e s ) . T ype Leader nodes γ ξ Proof SB l 1 = l 2 S B = 0 , or l 1 = 1 , or l 2 S B = 1 M − N + 1 M − N + 1 Item (iii) of Lemma 4 l 1 > 1 , l 2 S B = 0 M − N + l 1 M − N + l 1 Items (iv)-(v) of Lemma 7 l 1 = 0 , l 2 S B > 1 M − N + 1 M − N + l 2 S B Items (iv)-(v) of Lemma 7 l 1 ≥ 1 , l 2 S B ≥ 1 M − N + l 1 M − N + l 1 + l 2 S B Items (iv)-(v) of Lemma 7 SUB l 1 = 1 , l 2 S B = l 2 S U B = 0 M − N + 1 M − N + 1 Item (iv) of Lemma 5 l 2 S B = 1 , l 1 = l 2 S U B = 0 M − N M − N + 1 Item (v) of Lemma 5 l 2 S U B = 1 , l 1 = l 2 S B = 0 M − N M − N Item (iv) of Lemma 5 l 2 S B = 0 , l 1 , l 2 S U B ≥ 0 M − N + l 1 M − N + l 1 Items (iv)-(v) of Lemma 7 l 2 S B > 0 , l 1 , l 2 S U B ≥ 0 M − N + l 1 M − N + l 1 + l 2 S B Items (iv)-(v) of Lemma 7 L s , the zero eigenv alue is semisimple, meaning its algebraic multiplicity equals its geometric multiplicity . Then, the size of ev ery Jordan block corresponding to the zero eigenv alue is exactly of size 1 × 1 . Let A = E ⊤ s and B = E s ⊙ . It is well known that the sizes of the Jordan blocks associated with the zero eigen value of AB and B A may dif fer by at most one [27]. Then, this is also the case for L e s and L s . Consequently , the nilpotency index of the zero eigenv alue of L e s is at most two, and the Jordan blocks associated with λ = 0 are necessarily of size 1 × 1 or 2 × 2 . This justifies the Jordan structure giv en in (16). C . Ly apunov Equation f or Directed Signed Edge Laplacians T o establish synchronization of multi-agent system over signed digraphs, we pro ve the exponential tability of the set { ¯ e = 0 } . In particular, we remind the results on how to construct strict L yapunov functions, in the space of ¯ e , referring to functions expressed in terms of ¯ e , presented in [18, Theorems 2–4]. For a signed digraph containing a directed spanning tree, we ha ve the following. Theor em 1: ( [18, Theorem 3]) Let G s be a signed digraph of N agents interconnected by M edges. Let L e s be the associated directed edge Laplacian, which contains ξ zero eigen v alues. Then, the following are equiv alent: (i) G s contains a directed spanning tree, (ii) for any Q ∈ R M × M , Q = Q ⊤ > 0 and for any { α 1 , α 2 , . . . , α ξ } with α i > 0 , there e xists a matrix P ( α i ) ∈ R M × M , P = P ⊤ > 0 such that P L e s + L ⊤ e s P = Q − ξ X i =1 α i  P v r i v ⊤ l i + v l i v ⊤ r i P  , (17) where v r i , v l i ∈ R M are, respectively , the right and left eigen vectors of L e s associated with the i th 0 eigen v alue. ξ is given in T able I, satisfies ξ = M − N + 1 if the signed digraph is SB, and also if it is SUB with a root node or SB- rooted SCC; otherwise, ξ = M − N . Pr oof: See proof of [18, Theorem 3]. Cor ollary 1: ( [18, Theorem 2]) Let G s be a signed directed spanning tree. Then the associated edge Laplacian L e s has no zero eigenv alues, i.e., ξ = 0 , and for any Q ∈ R ( N − 1) × ( N − 1) , Q = Q ⊤ > 0 , there exists a matrix P ∈ R ( N − 1) × ( N − 1) , P = P ⊤ > 0 , such that P L e s + L ⊤ e s P = Q. (18) Pr oof: See proof of [18, Theorem 2]. Theor em 2: ( [18, Theorem 4]) Let G s be a signed digraph of N agents interconnected by M edges, containing m leader groups. Let l 1 be the number of root nodes, l 2 S B be the number of SB-rooted SCCs, and l 2 S U B be the number of SUB-rooted SCCs, where m = l 1 + l 2 and l 2 = l 2 S B + l 2 S U B . Let L e s be the associated directed edge Laplacian. Then the following are equi valent: (i) the graph has m leader groups, as defined in Assumption 3, and gi ven each follower ν j , ∀ j ∈ F , there exists at least one leader ν i , ∀ i ∈ L , such that there exists at least one path from ν i to ν j , (ii) for any Q ∈ R M × M , Q = Q ⊤ > 0 and for any { α 1 , α 2 , . . . , α ξ } with α i > 0 , there e xists a matrix P ( α i ) ∈ R M × M , P = P ⊤ > 0 such that (17) holds, where v ri , v li ∈ R M are the right and left eigen vectors of L e s associated with the i th 0 eigenv alue. Moreov er , ξ satisfies ξ = M − N + l 1 + l 2 S B whether the signed digraph is SB or SUB. Pr oof: See proof of [18, Theorem 4]. V . E X P O N E N T I A L S T A B I L I T Y In this section, we present our main result. W e consider the synchronization problem of multi-agent systems modeled by (4)-(5) ov er directed signed graphs, and establish the exponential stability of the synchronization errors. Next, we deriv e the synchronization errors using the edge-Laplacian notation introduced in Section II, and then reformulate the control problem accordingly . Using the definition of the incidence matrix in (1), we may express the edge states in (11) in v ector form e = E ⊤ s x, (19) where E s is the incidence matrix corresponding to the graph. For signed digraphs, from the definition of the directed Lapla- cian in (3), we write the control law in (5) in vector form as u = − k 1 E s ⊙ e. (20) 8 Differentiating the edge states (19) yields ˙ e = − k 1 E ⊤ s E s ⊙ e = − k 1 L e s e. (21) Similarly , dif ferentiating the synchronization errors (13), ˙ ¯ e = [ I − ξ X i =1 v r i v ⊤ l i ] ˙ e = − k 1 [ I − ξ X i =1 v r i v ⊤ l i ] L e s e, (22) we obtain ˙ ¯ e = − k 1 L e s ¯ e. (23) Remark 6: Observe that, to obtain (23), we distinguish two cases based on T able I and the relations in (16): the first case where ξ = γ , and the second where ξ > γ . Here, γ and ξ denote the geometric and algebraic multiplicities of the zero eigen v alue. • Case 1 ( ξ = γ ): In this case, all left eigen vectors satisfy v ⊤ l i L e s = 0 . Consequently , we have ˙ ¯ e = − k 1 L e s ¯ e. Additionally , since L e s v r i = 0 , it also holds that ˙ ¯ e = − k 1 L e s [ I − P ξ i =1 v r i v ⊤ l i ] e. Finally , substituting (13) in the latter , we obtain (23). • Case 2 ( ξ > γ ): F or i ∈ { 1 , 2 , . . . , ξ − γ } , we have the relations v ⊤ l 2 i − 1 L e s = v ⊤ l 2 i , v ⊤ l 2 i E ⊤ s = 0 , and L e s v r 2 i = v r 2 i − 1 , E s v r 2 i − 1 = 0 . The remaining 2 γ − ξ right and left eigen vectors associated with zero eigenv alues satisfy L e s v r i = 0 and v ⊤ l i L e s = 0 . Thus, for this case as well, ˙ ¯ e = − k 1 L e s [ I − P ξ i =1 v r i v ⊤ l i ] e holds, and substituting (13) in the latter , Eq. (23) follows. W e are now ready to present our main result, global expo- nential stability of the origin of the set { ¯ e = 0 } . Theor em 3: Consider the signed multi-agent system (4) in closed loop with the control law (20). Under Assumptions 1 and 2 or 1 and 3, for any Q = Q ⊤ > 0 there exists P = P ⊤ such that V ( ¯ e ) := 1 2 ¯ e ⊤ P ¯ e, ˙ V ( ¯ e ) = − 1 2 k 1 ¯ e ⊤ Q ¯ e. Then, the set { ¯ e = 0 } is globally exponentially stable. Furthermore, for a signed digraph, under Assumption 2: (i) If G s is SB, then agents achieve bipartite consensus, that is, the inequality (6) holds. (ii) If G s is SUB and contains a SUB-rooted SCC without a root node, then agents achiev e trivial consensus, that is, the inequality (8) holds. (iii) If G s is SUB and contains either a SB-rooted SCC or a root node, then agents achiev e interval bipartite consensus, that is, the inequality (9) holds. On the other hand, for a signed digraph containing m leader groups, under Assumption 3: (iv) If G s contains at least one root node or a SB-rooted SCC, then agents achieve bipartite containment; that is, if G s is SB, then inequality (7) holds, whereas if G s is SUB, inequality (10) holds. (v) If G s contains only SUB-rooted SCCs, then agents achiev e trivial consensus, that is, the inequality (8) holds. Pr oof: Let Q = Q ⊤ > 0 and α > 0 be arbitrarily fixed. By Assumption 2 and Theorem 1, there exists a symmetric positiv e definite matrix P = P ⊤ > 0 such that (17) holds. In the case where the considered graph is a spanning tree, by Corollary 1, there e xists a P = P ⊤ > 0 such that (18) holds. In the case the digraph contains multiple leader groups, by Assumption 3 and Theorem 2, there exists a P = P ⊤ > 0 such that (17) holds. Then, consider the L yapunov function candidate V ( ¯ e ) := 1 2 ¯ e ⊤ P ¯ e. Its total time deriv ativ e along the trajectories of (23) yields ˙ V ( ¯ e ) = − k 1 ¯ e ⊤ P L e s ¯ e. If the underlying signed digraph is a directed spanning tree, using (18), we obtain ˙ V ( ¯ e ) = − 1 2 k 1 ¯ e ⊤ Q ¯ e. If the underlying signed digraph contains a directed span- ning tree or multiple leader groups, using (17), we obtain ˙ V ( ¯ e ) = − 1 2 k 1 ¯ e ⊤ Q ¯ e + 1 2 ¯ e ⊤  P ξ i =1 α i ( P v r i v ⊤ l i + v l i v ⊤ r i P )  ¯ e. From the definition of the synchronization errors (13) and using the identity v ⊤ l i v r i = 1 , i ≤ ξ (Re- mark 4), we have P ξ i =1 α i P v r i v ⊤ l i [ I − P ξ i =1 v r i v ⊤ l i ] e = P ξ i =1 α i P v r i v ⊤ l i e − P ξ i =1 α i P v r i v ⊤ l i e = 0 and ¯ e ⊤ [ I − P ξ i =1 v r i v ⊤ l i ] ⊤ P ξ i =1 α i v l i v ⊤ r i P = P ξ i =1 e ⊤ α i v l i v ⊤ r i P − P ξ i =1 e ⊤ α i v l i v ⊤ r i P = 0 . Thus, ˙ V ( ¯ e ) = − 1 2 k 1 ¯ e ⊤ Q ¯ e < − 1 2 k 1 λ min ( Q ) | ¯ e | 2 < 0 , in both cases, where λ min ( Q ) > 0 is the smallest eigenv alue of Q . The first statement of the proposition follo ws. Then, dif ferentiating the weighted a verage system (12), we obtain the dynamical equation ˙ e m = P ξ i =1 v r i v ⊤ l i ˙ e = − k 1 P ξ i =1 v r i v ⊤ l i L e s e = 0 (See Remark 6). Its solution giv es e m (0) = e m ( t ) . Moreov er , from the previous re- sult, we have lim t →∞ ¯ e ( t ) = 0 , which giv es, from (13), lim t →∞ e ( t ) = e m ( t ) = e m (0) . Then, from (12) and (19), we obtain lim ¯ t →∞ e ( t ) = h v r 1 v ⊤ l 1 + · · · + v r ξ v ⊤ l ξ i e (0) = h v r 1 v ⊤ l 1 + · · · + v r ξ v ⊤ l ξ i E ⊤ s x (0) , where v r i and v l i are the right and left eigen vectors associated with the ξ zero eigen- values of L e s . (1) F or a signed digraph containing a spanning tree: (i) In the case the signed digraph is SB, ξ = M − N + 1 , and since, from Lemma 6, v ⊤ l 1 E ⊤ s = v ⊤ l 2 E ⊤ s = · · · = v ⊤ l ξ E ⊤ s = 0 , it leads to lim t →∞ E ⊤ s x ( t ) = 0 . Then, using the edge-gauge transformation from Lemma 3, we have that lim t →∞ E ⊤ s x ( t ) = lim t →∞ D e E ⊤ D x ( t ) = 0 . From [28, Theorem 3.4], the null space of E ⊤ is spanned by 1 N , so lim t →∞ D x ( t ) = c 1 N , where c ∈ R . Therefore, we can deduce that lim t →∞ x ( t ) = αD 1 N , where α ∈ R is a constant determined by the initial conditions of the agents and the signed interactions. This implies that the system achieves bipartite consensus. (ii) In the case that the signed graph is SUB and contains a SUB-rooted SCC without a root node, from Item (i) of Lemma 6, we hav e that v ⊤ l 1 E ⊤ s = v ⊤ l 2 E ⊤ s = · · · = v ⊤ l ξ E ⊤ s = 0 , it leads to lim t →∞ E ⊤ s x ( t ) = 0 . Since from Item (iii) of Lemma 5, rank ( E s ) = N , the only solution is lim t →∞ x ( t ) = 0 , which implies that the system achie ves trivial consensus. (iii) In the case that the signed graph is SUB and contains a SB-rooted SCC, from Item (i) of Lemma 6, we have that N ( L ⊤ e s ) = N ( E s ) . On the other hand, from T able I, the algebraic multiplicity of the zero eigen v alue is M − N + 1 , while its geometric multiplicity is M − N . Then, from Eq. (16), we hav e L e s v r 2 = v r 1 and v ⊤ l 1 L e s = v ⊤ l 2 . Thus, v ⊤ l 1 E ⊤ s  = 0 , and lim t →∞ e ( t ) = v r 1 v ⊤ l 1 E ⊤ s x (0) . In the case that the signed SEKERCIOGLU et al. : EDGE-BASED SYNCHRONIZA TION OVER SIGNED DIGRAPHS WITH MUL TIPLE LEADERS 9 ν 1 ν 2 ν 3 ν 4 ν 5 e 1 e 2 e 3 e 4 e 5 (a) G 1 ν 1 ν 2 ν 3 ν 4 ν 5 ν 6 ν 7 ν 8 ν 9 e 1 e 2 e 3 e 4 e 5 e 6 e 8 e 9 e 7 e 10 (b) G 2 ν 1 ν 2 ν 3 ν 4 ν 5 ν 6 ν 7 ν 8 ν 9 e 1 e 2 e 3 e 4 e 5 e 6 e 8 e 9 e 7 e 10 (c) G 3 ν 1 ν 2 ν 3 ν 4 ν 5 ν 6 ν 7 ν 8 ν 9 e 1 e 2 e 3 e 4 e 5 e 6 e 8 e 9 e 7 e 10 (d) G 4 Fig. 2 : The black edges represent cooperative interactions, and the dashed red edges represent antagonistic interactions. (a) G 1 is SUB and has ν 5 as a root node. (b) G 2 is SB ( V 1 = { ν 1 , ν 3 , ν 5 , ν 9 } and V 2 = { ν 2 , ν 4 , ν 6 , ν 7 , ν 8 } ) and has ν 7 and ν 9 as root nodes. (c) G 3 is SB ( V 1 = { ν 1 , ν 3 , ν 5 , ν 7 , ν 8 , ν 9 } and V 2 = { ν 2 , ν 4 , ν 6 } ) and contains ν 9 as a root node and a SB-rooted SCC formed by ν 1 , ν 8 , and ν 9 . (d) G 3 is SUB and contains ν 9 as a root node and a SUB-rooted SCC formed by ν 1 , ν 8 , and ν 9 . graph is SUB and contains a root node, from Item (ii) of Lemma 6, we hav e that N ( L ⊤ e s )  = N ( E s ) so v ⊤ l k E ⊤ s  = 0 for all k ≤ ξ . Moreover , we also have from Item (iii) of Lemma 5 that rank ( E s ) = N so lim t →∞ e ( t )  = 0 . Then, lim t →∞ e ( t ) = P ξ i =1 v r i v ⊤ l i E ⊤ s x (0) . Thus, the only achie vable objectiv e is the interval bipartite consensus. (2) For a signed digraph containing multiple leaders, L e s contains M − N + l 1 + l 2 S B zero eigen values. (iv) First, suppose that the signed digraph contains at least one root node. From Item (vi) of Lemma 7, we hav e N ( L ⊤ e s )  = N ( E s ) . Let γ denote the geometric multiplicity of the zero eigen value. Then, from T able I and the relations in (16), we obtain lim t →∞ e ( t ) = h P γ − ξ − 1 k =0 v r 2 k +1 v ⊤ l 2 k +1 + P ξ i =2( γ − ξ ) v r i v ⊤ l i i E ⊤ s x (0) . No w , suppose that all leader groups are rooted SCCs. In this case, L e s has M − N + l 2 S B zero eigenv alues. On the one hand, Lemma 7 implies N ( L ⊤ e s ) = N ( E s ) . On the other hand, from T able I: (a) If the graph is SB, then the algebraic multiplicity of the zero eigen v alue is M − N + l 2 S B , while its geometric multiplicity is M − N + 1 . (b) If the graph is SUB, but contains at least one SB-rooted SCC, then the algebraic and geometric multiplicities are M − N + l 2 S B and M − N , respectively . Consequently , from (15) and (16), lim t →∞ e ( t ) = P γ − ξ − 1 k =0 v r 2 k +1 v ⊤ l 2 k +1 E ⊤ s x (0) . Thus, following the results in [17], the only achiev able objectiv e for the agents is bipartite containment. (v) Finally , consider the case in which the signed digraph contains only SUB-rooted SCCs. Then, both the algebraic and geometric multiplicities of the zero eigen v alue are equal to M − N , and v ⊤ l 1 E ⊤ s = v ⊤ l 2 E ⊤ s = · · · = v ⊤ l ξ E ⊤ s = 0 . Hence, lim t →∞ E ⊤ s x ( t ) = 0 . Since, from Item (iii) of Lemma 7, rank ( E s ) = N , the only possible solution is lim t →∞ x ( t ) = 0 , which implies that the system achieves trivial consensus. V I . N U M E R I C A L E X A M P L E W e illustrate our theoretical findings by considering a signed multi-agent systems ev olving according to (4)-(5), represented by four dif ferent signed digraphs depicted in Figure 2. Let k 1 = 4 and P generated by (18) or (17), depending on the considered graph structure, with Q = I M × M , α k = 1 , k ≤ 9 . The initial conditions of the agents for G 1 are [3 . 5 , 4 , − 2 , − 6 . 5 , 5 . 5] and for G 2 , G 3 , and G 4 are [3 . 5 , 4 , − 2 , − 6 . 5 , 5 . 5 , − 10 . 5 , 3 . 5 , 12 , 5 . 5] . The trajectory ev olution of the states of the agents (4)-(5), edges (21) and synchronization errors (23) is sho wn in Figure 3a–3d for each graph in Figure 2, respectively . It is clear that in Figure 3a, agents achieve interval bipartite consensus since G 1 is SUB and contains a root node, and in Figures 3b–3d, agents achiev e bipartite containment since G 2 , G 3 and G 4 contain multiple leader groups. The edge states do not con v erge to zero and all synchronization errors conv erge to zero. V I I . C O N C L U S I O N In this paper , we addressed the edge-based synchronization control problem of signed multi-agent systems interconnected ov er signed digraphs which contain one or multiple leader groups. First, we presented new spectral properties of signed edge Laplacians containing multiple zero eigenv alues and eigen vectors associated with each zero eigen value. Then, we established exponential stability of the synchronization set using the L yapunov equation-based characterization of the signed edge Laplacians with multiple zero eigenv alues. 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