On Symmetry and Controllability of Multi-Agent Systems

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1 53rd IEEE Conference on Decision and Control December 15-17, Los Angeles, California, USA On Symmetry and Controllability of Multi-Agent Systems Airlie Chapman and Mehran Mesbahi Abstract This paper delves into the link between symmetry and controllability of networked systems. We examine symmetry results pertaining to the determining sets of a graph and eigenvalue multiplicity. These provide methods to reason how symmetry structure of the network informs the control input design. We generalize graph automorphisms, which represent graph symmetries, to signed fractional graph automorphisms via a semi-definite programming relaxation. Consequently, we extend existing results on the relationship between graph automorphisms and uncontrollability to signed fractional graph automorphisms, showing necessary and sufficient conditions for controllability. Index Terms Network controllability; Network observability; Graph symmetry; Signed fractional automorphisms; Coordination algorithms 1. INTRODUCTION Complex dynamic networks are an intrinsic part of the technological world including the internet, power grids, and communication networks, as well as in nature such as biological and chemical systems and social networks. An explosion of research in the area of network systems has resulted [1], [2], [3]. Network controllability and observability it of interest in situations where a networked system is influenced or observed by an external entity. Such scenarios include networked robotic systems, human-swarm interaction, and network security [3], [4], and in areas such as quantum networks [5]. Previous works on controllability have discovered a fundamental link between symmetry and controllability. This has been examined when the dynamics are driven by the adjacency matrix [5], the Laplacian matrix [6], and nonlinear systems [7]. Investigation has also focused on different classes of graphs such as circulants [8], grids [9], random [10], distance regular [11] and Cartesian product graphs [12]. Existing properties that relate the network s symmetry structure and injection points of control into the network provide sufficient conditions for uncontrollability. These conditions can often be reformulated as cardinality requirements on the number of inputs into the graph. We formalize this concept using the notion of the determining number of the graph [13] - the minimum number of nodes that must be fixed to break all graph symmetries. Furthermore, it has been shown that these symmetry properties are not necessary for uncontrollability. In this The research of the authors was supported by the U.S. Army Research Laboratory and the U.S. Army Research Office under contract number W911NF The authors are with the Department of Aeronautics and Astronautics, University of Washington, WA s: {airliec, mesbahi}@uw.edu. paper we relax the traditional notion of symmetry, defined by graph automorphisms, to fractional symmetries defined by fractional automorphisms. This is a concept introduced and explored in detail by Scheinerman and Ullman [14]. The notion of fractional automorphisms actually provides a necessary and sufficient condition for controllability. These results allow us to more easily reason about controllability of asymmetric graphs which only exhibit fractional automorphisms, such as almost all regular graphs. An attraction of the fractional automorphism formulation is that the link between controllability and symmetry no longer requires the NP-hard search for graph automorphisms. Instead only a convex optimization problem needs to be solved. The organization of the paper is as follows. In section 2, we begin by introducing relevant background material pertaining to graphs. We proceed in Section 3 to describe the dynamic problem that is the focus of this paper, the controlled consensus problem. Section 4 proceeds to describe some fundamental ties between breaking symmetry and controllability as well as the multiplicity of eigenvalues of the state matrix. Section 5 provides necessary and sufficient conditions for controllability for general input matrices. Section 6 provides an equivalent form of the results from Section 5 but in terms of a relaxed form of symmetry called signed fractional automorphisms. We finish with some concluding remarks in Section NOTATION AND BACKGROUND We provide the notation and a brief background on the constructs and models that will be used in this paper. For a column vector v R p, both v i and [v] i denote its ith element. For matrix M R p q, [M] ij denotes the element in its ith row and jth column. The identity matrix is denoted I and e i is the column vector with all zero entries except [e i ] i = 1. The column vector of all ones is denoted as 1. The cardinality of a set S is denoted as S. An undirected graph G = (V, E) is characterized by a node set V with cardinality n, an edge set E comprised of unordered pairs of nodes with cardinality m. An edge exists between two neighboring nodes i to j if {i, j} E. The adjacency matrix of G, denoted A(G), is an n n matrix with [A(G)] ij = [A(G)] ji = 1 when {i, j} E and 0 otherwise. The degree matrix (G) R n n is a diagonal matrix with the number of edges incident to node i at position (i, i). The Laplacian matrix L(G) = (G) A(G), with 1 T L(G) = 1 T, L(G)1 = 1, and eigenvalues 0 = λ 1 (G) λ 2 (G) λ n (G). The distance between nodes i and j in a graph, denoted as d(i, j), is the length of the shortest path consisting of edges between nodes i and j /14/$ IEEE 625

2 There are many effective methods by which larger graphs can be synthesized from a set of smaller graphs [15]. The Cartesian product is one such method and is defined for a pair of factor graphs G 1 = (V 1, E 1 ) and G 2 = (V 2, E 2 ) and denoted by G = G 1 G 2. The product graph G has the node set V 1 V 2 and there is an edge from node (i, p) to (j, q) in V 1 V 2 if and only if either i = j and (p, q) is an edge of E 2, or p = q and (i, j) is an edge of E 1. A graph is called prime if it cannot be decomposed into the product of non-trivial graphs, otherwise a graph is referred to as composite. All connected graphs have a unique prime factor decomposition of the form G k1 1 Gkm m, where G i is prime for all i, and G ki i denotes k i Cartesian products of G i. Further, this decomposition can be found in polynomial time [15]. 3. PROBLEM SETUP The focus of this paper is the popular controlled consensus problem [3] defined on a graph G = (V, E), where x i (t) R represents the state of node i V at time t with interaction between nodes defined over the edges of the graph E. An external control vector u(t) R q at time t is applied to node i through some state matrix B i R q. An individual node s dynamics is given by ẋ i (t) = (x i x j ) + B T i u(t). {i,j} E In addition, the dynamics at time t is observed by a vector y(t) R p through a matrix C R p n. The full system dynamics is then ẋ(t) = L(G)x(t) + Bu(t) (3.1) y(t) = Cx(t), where B = [ B 1, B 2,..., B n ] T R n q. It is often of interest where the inputs and outputs of system (3.1) are in terms of the nodes of the graph G. If the set of input nodes in the n node graph is S = {i 1, i 2,..., i q } for i 1 < i 2 < < i q, the corresponding input matrix is B = [ e i1, e i2,..., e iq ] R n q. We uniquely denote the input matrices of this form as B n (S). Similarly, the output matrices are defined as C n (S) := B n (S) T. If it is clear from the context, we remove the subscript n for brevity. The dynamics (3.1) are referred to as controllable if the pair ( L(G), B) is controllable, and similarly for observability if the pair ( L(G), C) is observable. Due to the duality between controllability and observability, results pertaining to controllability of the pair ( L(G), B) are equally applicable to observability of the pair ( L(G), B T ). 4. SYMMETRY The automorphisms of a graph describe its symmetries and have been previously shown to play an important role in the controllability of dynamics similar to (3.1) [6]. A graph automorphism is a permutation σ of the node set such that G contains an edge {i, j} if and only if it also contains an edge {σ(i), σ(j)}. The set of automorphisms, which forms a group, is denoted as Aut(G). A graph with only a trivial automorphism group is referred to as asymmetric. Figure 1: An asymmetric graph pertaining to Example 2 with ( L(G), B(S)) uncontrollable, where S = {1}. Every graph automorphism can be represented uniquely as a permutation matrix J which commutes with the adjacency matrix, i.e., JA(G) = A(G)J. Further, the Laplacian matrix similarly commutes with J, i.e., JL(G) = L(G)J. An automorphism σ fixes the node set S if σ(i) = i for all i S. Equivalently, the permutation matrix J associated with σ fixes the matrix B(S), i.e., JB(S) = B(S). The following is an adaption of the results found in [6] to dynamics (3.1), and so is quoted here without proof. Proposition 1. A graph G is uncontrollable from any pair ( L(G), B(S)), if there exists a nontrivial automorphism of G which fixes all inputs in the set S. If S satisfies Proposition 1 it is referred to as leader symmetric [6], moreover leader symmetry is not sufficient for controllability. The naming convention stems from considering input nodes as leaders in the network. Rahmani et al. [6] presented the following asymmetric graph example to illustrate this. Example 2. Consider Figure 1, an asymmetric graph on 7 nodes. Consider the input set S = {1}, as L(G) has a left eigenvector v = [0, 0.51, 0.30, 0.05, 0.40, 0.67, 0.22] T then v T B(S) = 0 and so by the Popov-Belevitch-Hautus test (PBH) ( L(G), B(S)) is uncontrollable [16]. As L(G) is asymmetric then S is not leader symmetric, illustrating that Proposition 1 is not necessary. A. Breaking Symmetry Proposition 1 highlights the requirement that a controllable input set must break the symmetry structure of G. Determining sets are a useful construct to describe this process. Definition 3. A subset S of the vertices of a graph G is called a determining set if whenever g, h Aut(G) so that g(s) = h(s) for all s S, then g = h. The determining number of a graph G, denoted Det(G), is the smallest integer r so that G has a determining set of size r. Another term for a determining set S is the fixing set, due to the fact that no non-trivial automorphism of G fixes all members in S. Formally, a set S V (G) is a determining set if and only if the stabilizing set of S, defined 626

3 (a) (b) Figure 2: (a) The Petersen graph. The black nodes form a 3 node resolving set. (b) The asymmetric 3-regular Frucht graph. as Stab(S) := {g Aut(G) σ(v) = v, for all σ S}, only contains the trivial automorphism. Directly from Proposition 1 and the definition of determining sets we have the following corollary. Corollary 4. A necessary condition for controllability of the pair (A(G), B(S)) is that S is a determining set. Hence, S Det(G). When known, the determining number of a graph is subsequently a simple cardinality check for uncontrollability of an input set S. Determining sets can be small in comparison to the graph size or automorphism group size. For example, the hypercube Q n has 2 n vertices and 2 n n! automorphisms, but has a determining set of cardinality only log 2 n + 1. In contrast, the complete graph has n vertices and n! automorphisms, but requires fully n 1 vertices in a determining set [17]. From Corollary 4, lower bounds on the determining number of a graph provide a minimum cardinality on a controllable set S. The following are two such known bounds: Proposition 5. [17] The determining number is bounded as (n Det(G))! n! Aut(G) and Det(G) log n Aut(G). The following demonstrates the applicability of Corollary 4 and Proposition 5. Example 6. Consider the 10 node Petersen graph G, shown in Figure 2a. Its well known automorphism group is the symmetric group of order 5 with cardinality 5!. From either bound of Proposition 5, Det(G) 3, hence at least three inputs are required to render the graph controllable. As calculating the automorphism group of a graph is NPhard, it is helpful to consider more computationally efficient methods to form or bound the cardinality of determining sets. One such method is via the graph s prime factor decomposition defined in Section 2. The automorphism group of a graph is intimately linked to the automorphisms of its prime factors. This link translates through to the determining set of the graph summarized in the following theorem. Theorem 7. [13] Let G = G k1 1 Gkm m be the prime factor decomposition for a connected graph G. Then Det(G) = max{det(g ki i )}. A consequence of this observation is that only the determining set of a graph s factors need to be directly found. The computational reduction will be proportional to the number of prime factors. Another alternative to calculating the determining set directly is to find a set of nodes S with the property that every node in the graph can be uniquely identified by its distance from the nodes in S. Such a set is known as a resolving, reference, or locating set [13], [?] and is formally defined as follows. Definition 8. A subset S of the nodes of a connected graph G is called a resolving set if for every i, j V (G) there is a s S so that d(s, i) d(s, j). Since every node i is uniquely defined by its distance to a resolving set S, and an automorphism σ preserved distances, σ(i) is uniquely determined by its distance from σ(s). Hence, a nontrivial σ can not fix every element in S, and so every resolving set S is a determining set. By examining all pairwise graph distances between nodes in G, it is simple to form a greedy algorithm to form a resolving set of G. This is a straight forward method to guarantee all symmetries in a graph are broken, and in doing so guarantee that Corollary 4 is satisfied. Example 9. Re-examining the Petersen graph G in Figure 2a, the set S = {1, 3, 9} is a resolving set. Further, from Example 6, a lower bound on the determining number is 3. Hence, S is a minimal cardinality determining set and Det(G) = 3. Real-world graphs often evolve, or are designed to exhibit, a large number of symmetries making the results of this section pertinent. Unfortunately, over all graphs, almost all finite graphs are asymmetric [18]. We proceed to examine another uncontrollability promoting feature. B. Eigenvalue Multiplicity Like the determining number, the multiplicity of the eigenvalues of L(G) provide a lower bound on the number of inputs required to render a system controllable. The following lemma pertains to this result. Lemma 10. [19] If (A, B) is controllable and rank (B) = q, the geometric multiplicity of each eigenvalue of A is at most q. For the input matrix B(S), as S is composed of a distinct set of nodes, rank(b(s)) = S. Hence there must be at least as many inputs S as the largest geometric multiplicity of the eigenvalues of L(G) for the pair ( L(G), B(S)) to be controllable. The uniqueness of the eigenvalues of A(G), has been previously correlated with the symmetry structure of G. The following results are two such examples. Proposition 11. [20] If all eigenvalues of A(G) are simple, then every automorphism of G has order 1 or 2, i.e., σ(σ(i)) = i for all i V. Proposition 12. [12] If all eigenvalues of A(G) are simple, then G = G 1 G m is the prime factor decomposition of 627

4 G with prime factors G i. Further, Aut(G) = Aut(G 1 ) Aut(G m ). Given these results and the correlation between uncontrollability and eigenvalue multiplicity and uncontrollability and symmetry, there is a tendency to infer that eigenvalue multiplicity is strongly coupled to symmetry. This is not the case. A family of graphs contradicting this assumption is the strongly k-regular graphs. A strongly k-regular graph is a graph where every node has k neighbors. In addition for every pair of nodes {i, j}, if i and j are neighbors they have λ common neighbors else they have µ common neighbors. A strongly k-regular graph has exactly three eigenvalues {k, 1 2 (λ µ)± α}, with multiplicity {1, 1 2 (n 1± 1 α (2k+ (n 1)(λ µ)))}, where α = (λ µ) 2 +4(k µ) [21]. Further, almost all regular graphs and strongly regular graphs have a trivial automorphism group, i.e., no nontrivial symmetry [18]. Consequently, strongly k-regular graphs are asymmetric with high eigenvalue multiplicity. Section 6 helps to fill the gap between uncontrollability from graph regularity and uncontrollability from symmetry. 5. CONTROLLABILITY AND OBSERVABILITY The following section examines conditions on controllability of the general linear dynamics ẋ = Ax + Bu (5.2) y = Cx as well as specific dynamics (3.1). This first result follows the same form as the proof of Proposition 1. Proposition 13. For diagonalizable A, if P I with P B = B and AP = P A = (A, B) is uncontrollable Proof: For every left eigenvector v associated with eigenvalue λ of A then P T v is also an eigenvector associated with λ as v T A = λv T implies that (P T v) T A = v T P A = v T AP = λv T P = λ(p T v) T. As A is diagonalizable and so has a spanning set of eigenvectors, for some v, v T P v T hence v T P v T 0 is also an eigenvector associated with the eigenvalue λ. Further, ( v T P v T ) B = v T P B v T B = v T B v T B = 0. Therefore by the PBH test (A, B) is uncontrollable [16]. Interestingly, when the input matrix is a column vector, denoted as B, then Proposition 13 can be reversed. This is summarized in the following theorem. Proposition 14. For diagonalizable A, (A, B) is uncontrollable = P I with P B = B and AP = P A. Proof: By the PBH test, the pair (A, B) is uncontrollable if there exists a left eigenvector v of A associated with eigenvalue λ such that v T B = 0 [16]. Consider P = I wv T I where w is the right eigenvector of A associated with λ then AP P A = A(I wv T ) (I wv T )A = A Awv T + A + wv T A = λwv T + λwv T = 0, and P B B = ( I wv T ) B B = B wv T B B = 0. From Proposition 13 and 14 we have the following necessary and sufficient condition for controllability. Theorem 15. For diagonalizable A, (A, B) is uncontrollable P I with P B = B and AP = P A. This result specific to dynamics (3.1) is summarized in the following corollary. Corollary 16. ( L(G), B) is uncontrollable P I with P B = B and L(G)P = P L(G). Further, if 1 T B 0 then, ( L(G), B) is uncontrollable P I, with P B = B and L(G)P = P L(G), P 1 = 1 and 1 T P = 1 T. Proof: The first statement follows from Theorem 15, noting that as L(G) is symmetric it is diagonalizable, and for each eigenvector its left and right eigenvectors are the same. From the proof of Proposition 13 and 14, as B is not orthogonal to the left eigenvector 1 of L(G) then the pair ( L(G), B) is uncontrollable if and only if P = I vv T, where v 1 and as all eigenvectors are orthogonal, i.e., v T 1 = 0. Hence, P 1 = (I vv T )1 = 1 and 1 T P = 1 T (I vv T ) = 1 T, as required. Corollary 16 is the main mechanism for the extension of the previous symmetry results on controllability and will be explored in the following section. As an aside the following corollary is an equivalent formulation of Theorem 15 requiring a nonzero Q as a certificate of uncontrollability, rather than a nonidentity P. Corollary 17. For diagonalizable A, (A, B) is uncontrollable Q 0 with P Q = 0 and AQ = QA. Proof: Let Q = I P in Theorem 15 and the result follows. Interestingly, the right-hand side of Theorem 15 and Corollary 17 can be restricted to the family of idempotent matrices, i.e., matrices M such that M 2 = M. This follows from the proof of Proposition 14 with P = I wv T and selecting eigenvectors v and w such that v T w = 1. Some properties of idempotent matrix M 0 is that it is diagonalizable; its eigenvalues are either 0 or 1; the rank of the matrix is its trace; and M 1 for any matrix norm [22]. The exploration of this idempotent property is left as future work. 6. FRACTIONAL AUTOMORPHISMS Proposition 1 provides a sufficient leader symmetry condition on S for uncontrollability of the pair ( L(G), B(S)). This section relaxes the definition of automorphisms to acquire both a symmetry-based necessary and sufficient condition for controllability facilitated through Corollary 16. The algebraic condition for P to represent an automorphism of the graph G is that A(G)P = P A(G) 1 T P = 1 T P 1 = 1 P ij {0, 1}. (6.3) 628

5 Scheinerman and Ullman [14] proposed an integer relaxation of condition (6.3) to P ij 0, making P a doubly stochastic matrix instead of a permutation matrix P. The authors subsequently describes what has been dubbed as a fractional automorphism. We define a further relaxation to an signed fractional automorphism by removing the nonnegative constraint on P ij. Another interpretation suggested by Scheinerman and Ullman [14] of the fractional automorphism can be formulated by examining a perfect matching between nodes in G = (V, E) and G = (V, E ) where G is a copy of G. Consider the graph H on V V corresponding to a permutation σ with edges (i, σ(i)), which we will call links, as the links represent a perfect matching H is a 1-regular graph. Now form the graph H + by adding the edges E and E to H. Examining all the two-step paths from i E to j E, if one exists, it is either composed of an edge in E and link in H (an edge-link path) or a link in H and an edge in E (a link-edge path). The condition of an automorphism is for any i, j V if {i, j} E if and only if {σ(i), σ(j)} E is therefore equivalent to the condition that for all i V and j V there is an edge-link path from i to j if and only if there is a link-edge path from i to j. Figure 3c and 3d provides an example of the graph H + for two automorphisms on G in Figure 3a. For fractional automorphisms, the same approach to the formation of H can be used but with a weighted links between V and V when P ij 0 and with magnitude P ij. This forms a weighted bipartite graph H. Similarly, there will be weighted two-step paths in H + from i E to j E. The analogous definition of fractional automorphism is that for all i V and j V the sum of all links along edge-link paths from i to j equals the sum of all links along link-edge paths from i to j. The signed fractional automorphism graph H + follows similarly but with links allowed to take negative values. A fractional automorphism of graph G in Figure 3a is P = , with corresponding H + graph displayed in Figure 3b. We note that, unlike permutation matrices, P A(G) = A(G)P does not necessarily imply that P L(G) = L(G)P and so we define leader symmetry with respect to L(G). A relaxed version of leader symmetry can also be formulated for fractional automorphisms specifically. Definition 18. An input set S is (signed) fractionally leader symmetric with respect to L(G) if there exists a nontrivial (signed) fractional automorphism on L(G) which fixes S, i.e., L(G)P = P L(G), P 1 = 1, 1 T P = 1 and P B(S) = B(S) for P I. We slightly relax the definition by referring to an input matrix B as (signed) fractionally leader symmetric if the P representing the nontrivial (signed) fractional automorphism (a) (c) (d) Figure 3: (a) Original graph G (b) Graph H + corresponding to a fractional automorphism (c) Graph H + corresponding to the automorphism σ (1, 2, 3, 4, 5) = (4, 2, 3, 1, 5) (d) Graph H + corresponding to the automorphism σ (1, 2, 3, 4, 5) = (1, 3, 2, 4, 5). on L(G) satisfies in P B = B. Directly from Corollary 16 the next result follows. Theorem 19. Let 1 T B 0. The pair ( L(G), B) is uncontrollable B is signed fractionally leader symmetric with respect to L(G). Further, S is an uncontrollable input set S is signed fractionally leader symmetric. Proof: The first statement applies Definition 18 directly to Corollary 16. The second statement follows from the property that B(S) is a nonnegative matrix and so 1 T B(S) 0. An attraction of Theorem 19 is finding a P corresponding to an signed fractional automorphism can be formulated as a convex optimization problem. To prevent the trivial automorphism P = I an objective function is added to guarantee nontrivial P solutions when they exist. The optimization formulation is as follows: min P R n n (b) n P ii (6.4) i=1 L(G)P = P L(G) 1 T P = 1 T P 1 = 1 P B = B. Hence assuming 1 T B 0, ( L(G), B) is controllable the solution to problem (6.4) is P = I. We proceed to re-examine Example 2 in the context of Theorem 19 and optimization problem (6.4). Example 20. Revisiting the graph G in Figure 1. For S = {1}, solving the optimization problem (6.4) with B(S) and 629

6 L(G) gives [ 1 0 P = ( 11 T I ) ]. Hence as P is doubly stochastic, S is fractionally leader symmetric, hence ( L(G), B(S)) is uncontrollable. For all other S = {i} where i 2,..., 7, there does not exist a P I that solves problem (6.4) and so ( L(G), B(S)) is controllable for all these single inputs. As mentioned in Section 4, regular graphs are almost always asymmetric. Interestingly, this is never the case for fractional automorphisms on regular graphs. In fact every k-regular graph on n nodes shares a common nontrivial fractional automorphism. Proposition 21. Every k-regular graph G on n nodes has a fractional automorphism represented by P = 1 n 11T. Further, P A(G) = A(G)P P L(G) = L(G)P. Proof: The first result follows from the fact that every the row and column sum of A(G) are the same. The second follows from the property that L(G) = ki A(G) and P (ki) = (ki)p for all P. A consequence of this result is the following. Proposition 22. Let G be a k-regular graph. (a) The pair ( L(G), B) is controllable if and only if (A(G), B). (b) The pair ( L(G), B) is uncontrollable when B = α1 for all α R. Proof: (a) Follows directly from the second statement in Proposition 21. (b) As P = 1 n 11T represents a fractional automorphism of L(G) then for B = α1, P B = α n 11T 1 = α1 = B and so the result follows from Theorem 19. It is particularly fruitful to exercise Theorem 19 on asymmetric k-regular graphs as they exhibit no pure automorphisms. Hence, any uncontrollability properties were previously inaccessible to Proposition 1. For example Section 4(b) due to its eigenvalue multiplicity we know that every asymmetric strongly k-regular graph G is uncontrollable from any single input node. Therefore, for every node in G there exists an signed fractional automorphism that fixes it. With respect to Theorem 19, a natural question is whether fractional leader symmetry is necessary and sufficient for uncontrollability. The answer to this question is negative and can be illustrated by examining the famous asymmetric 3- regular Frucht graph. Example 23. Let G be the Frucht graph displayed in Figure 2b. Exercising the optimization problem (6.4) on every single input of G we find that single input nodes T = {1, 2, 6, 10, 11, 12} are controllable while the remaining single input nodes in the set T = {3, 4, 5, 7, 8, 9} are uncontrollable. Further, every signed fractional automorphism that fixes each node in T is not a fractional automorphism, i.e., it is necessary to check signed fractional automorphisms to establish uncontrollability of the nodes in T. 7. CONCLUSION This paper presents a discussion on the links between symmetry and controllability of networked dynamics. The notion of determining sets and eigenvalue multiplicity were explored which provide lower bounds on the cardinality of a controllable input set. 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