Effects Of Symmetry On The Structural Controllability Of Neural Networks: A Perspective

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1 16 American Control Conference (ACC) Boston Marriott Copley Place Jly 6-8, 16. Boston, MA, USA Effects Of Symmetry On The Strctral Controllability Of Neral Networks: A Perspective Andrew J. Whalen 1, Sean N. Brennan 1, Timothy D. Saer, and Steven J. Schiff Abstract The controllability of a dynamical system or network describes whether a given set of control inpts can completely exert inflence in order to drive the system towards a desired state. Strctral controllability develops the canonical copling strctres in a network that lead to n-controllability, bt does not accont for the effects of explicit symmetries contained in a network. Recent work has made se of this framework to determine the minimm nmber and location of the optimal actators necessary to completely control complex networks. In systems or networks with strctral symmetries, grop representation theory provides the mechanisms for how the symmetry contained in a network will inflence its controllability, and ths affects the placement of these critical actators, which is a topic of broad interest in science from ecological, biological and man-made networks to engineering systems and design. I. INTRODUCTION Controllability is an essential concept to the design of feedback controllers for networked brain systems. For example, non-controllable mathematical models of real systems have sbspaces that inflence model behavior, bt cannot be controlled by an inpt. Sch sbspaces can be difficlt to determine in complex nonlinear brain networks. Recent advances in the theory of network control [1], [], demonstrate how a strctral controllability [] framework can be sed to search over the nodes of a network to find the minimm nmber and location of the optimal [] control points to obtain complete inflence over a given network. Since almost all of the present theory was developed for networks withot symmetries, here we present the detailed grop representation framework to re-envision strctral controllability to inclde systems possessing grop symmetries, which complements and expands Lin s seminal theorems on strctral controllability []. II. STURCTURAL CONTROLLABILITY Lin [] tilized the notion of strctre in a dynamic system to define two canonical sitations where a system wold This work was spported by grants from the National Academies - Keck Ftres Initiative, NSF grant DMS , Collaborative Research in Comptational Neroscience NIH grant 1R1EB14641, and NIH BRAIN Initiative grant 1R1EY A. J. Whalen and S. N. Brennan are with the Center for Neral Engineering, Department of Mechanical Engineering, Pennsylvania State University, University Park, PA 168, USA awhalen@ps.ed, sbrennan@ps.ed T. D. Saer, is with the Department of Mathematical Sciences, George Mason University, Fairfax, VA, USA tsaer@gm.ed S. J. Schiff is with the Center for Neral Engineering, Departments of Engineering Science and Mechanics, Nerosrgery and Physics, Pennsylvania State University, University Park, PA 168, USA sschiff@ps.ed be n-controllable, then proved how two basic controllable strctres cold be sed to evalate the controllability of a network by determining if they spanned the network. Recall that for an arbitrary linear time invariant (LTI) system ẋ(t) = Ax(t) + B(t), (1) with state variables x(t) R n, system matrix A R n n, inpt matrix B R n p and control inpt (t) R p n, the controllability matrix is defined as Q = [ B AB A B A (n 1) B ], () and when Q is fll rank (rank(q) = n), the system defined by the pair (A, B) is flly controllable. Lin spposed that the nonzero parameters of a real-world system (A, B) are only generally known within some measrement error, and accordingly, postlated that any specific system that was ncontrollable de to the choice of parameters was arbitrarily close to a parameter set that wold render the system flly controllable. From this view of a system, the position of the zeros in (A, B) are assmed to be fixed while the remaining system parameters are arbitrary. The strctre of the system (1) is hence defined by the zeros in (A, B), which we will denote: Definition 1. The set of matrix entries α ij in the pair (A, B) that are eqal to zero are called the strctre of the pair, and are defined as Ψ(A, B) : {α ij : α ij = } () where 1 i n, 1 j n+p, and where any two systems with eqal Ψ indicates both have zero entries in the same positions. Now, strctral controllability [] states for a controllable pair (A, B ) with strctre Ψ(A, B ), Ψ(A, B ) = Ψ(A, B ) = (A, B ) is S.C.; (4) i.e. if a pair (A, B ) is controllable, any other pair (A, B ) with the same strctre as (A, B ) is therefore strctrally controllable (S.C.). The major assmption of this prior work lies in the invariance of the arbitrary parameters (non-zero entires) in determining strctral controllability, which cold actally lead to a system being ncontrollable de to the specific non-zero entries in (A, B) /$1. 16 AACC 5785

2 α α α α 1 α (a) Dilation v α 11 α 1 (b) Isolation v α Fig. 1: The two canonical n-controllable strctres in [], with control inpt. v n (a) Bd (b) Stem Fig. : The two always controllable canonic strctres in []. v n A. Un-controllable Strctres: Dilation and Isolation The first n-controllable canonical strctre, is defined in [] as a dilation, and shown in Fig. 1a. It is easy to determine pon inspection of the system eqations in (5) that the controllability matrix Q, composed of the pair (A, B) from () will never be fll rank for n = with two colmns of zeros, ths even control inpt into all three nodes cannot independently control V 1, V, and V. α 14 A = α, B = α 4 (5) α α 4 Likewise, the second n-controllable strctre called an isolation, is exemplified in Fig. 1b; indeed by inspection of the system graph in the figre, it is readily apparent that the control inpt to node V can only inflence that node since there are no edges directed from V to either of the other two nodes, ths the control inflence of this node is isolated and controllability is lost when controlling only this isolated node. B. Controllable Strctres: Bd, Stem and Cacts In addition to the n-controllable canonical strctres, [] defined two strctres called a bd and a stem, which are always strctrally controllable. The first of the two, called a bd is shown in Fig. a, and the pair for (A, B) takes the form, α 1 α A = , B = α n 1 α n+1. α n where it is straightforward to determine by inspection that the controllability matrix Q of this pair (A, B) is fll rank for any n N. This elementary cycle is controllable from any node in the strctre, and when taken in combination with the second controllable strctre a directed path called a stem in Fig. b forms a more complex strctre that is completely controllable. Inherited from the stem strctre, the cacts is also completely controllable from the foremost pstream node in the chain as shown in Fig.. Lin proved (6) how a network that is spanned by a cacts is always strctrally controllable [], and these strctral definitions were the basis for finding the minimm reqired control inpts to completely control a linear network in [1], []. Fig. : The cacts is the nion of bd and stem strctres, which determines the strctrally controllability of a network that is spanned by a cacts []. III. GROUP REPRESENTATION THEORY For linear systems containing grop symmetries, Rbin and Meadows [4] sed a similarity transform T to change the coordinates of the n-dimensional system (1) to an orthogonal basis defined by the grop action of the symmetry on R n (called the symmetry basis). Frthermore, Ref. [4] demonstrated how grop representation theory [5] is sed to constrct the symmetry basis for a symmetric grop from the irredcible representations of the grop which transforms the system matrix A into block diagonal form. In some cases the type of symmetry wold case the network to be non-controllable de to symmetries (termed NCS), evident by inspection of the strctre of the transformed system. Likewise, this same type of similiarity transformation was shown to define the sbspaces of a network that synchronize [6], which also has deep implications for neral networks [7]. Essentially, this transformation may reveal a new strctre containing dilations or isolated nodes in which the transformed system is actally not controllable on the basis of that strctre. Strctral controllability [] did not explicitly cover symmetry, so for any strctrally controllable pair (A, B) that contains no dilations or isolated nodes, the presence of 5786

3 α 11 α x σ y y α z α Fig. 4: An example -node network with S symmetry when = α, and α 11 = α. symmetry cold still case the network to be NCS (as shown in [4]), as the act of transforming the network to the symmetry basis wold redefine the strctre to one that is n-controllable. These two theorems together paint a more complete pictre of controllability than either alone as shown in [8], where both are sed in concert to explain and nderstand why certain neral networks were not controllable from particlar inpts. Inclding symmetry constraints makes strctral controllability a more general concept, as it does not depend on the explicit non-zero entries of the system pair (A, B) (necessary, bt not sfficient), while a network that has the NCS property possesses specific sets of the non-zero entries in (A, B) that define the symmetry contained by the system. A. The Symmetry Grop and Basis Symmetry present in a network is defined by the set of symmetry operations R that transform the system into itself. Formally, this set of network permtations forms an algebraic grop called the symmetric grop on q elements, from which matrix representations of the grop elements D(R) can be constrcted from monomial matrices (those with only one non-zero entry per colmn that describe how each operation R permtes the state variables of the network). For example, a network with S symmetry will have the form given in Fig. 4. The S symmetric grop contains two elements or symmetry operations R : {E, σ y }, the identity E, and a reflection across the y axis σ y swap x with z. The cardinality g of the set of symmetry operations R for the symmetric grop gives the order of the symmetric grop S q as g = q!. For the network in Fig. 4, n =, q =, g =, and we can constrct matrix representations D(R) of the elements of S as, 1 1 E = 1 1 and σ y = 1 1, (7) where these matrix representations carry the grop action of S to the linear vector space on R for the system pair: A = α 11 α, B = α 4 (8) α α defined in (1). In [4] the similarity transformation that takes the system into the coordinates of the symmetry basis is defined from the irredcible representations Γ (p) (R) of the grop symmetry present in the network. The nmber of irredcible representations p = 1... k can be determined from the character of the representation χ(r). Defined by taking the trace of the matrix representations χ(r) = Tr[D(R)], (9) the vale of the character determines the conjgacy class for each grop element R and the total nmber of conjgacy classes (distinct character vales) eqals the nmber of irredcible representations. The conjgacy class of each grop element refers to the type of symmetry operation of that grop element, so that rotations, reflections, etc. have the same conjgacy class, and matrix representations of the grop elements D(R) have the same trace. Compting the character (9) of the matrix representations D(R) in (7) for each grop element in S yields χ(e) =, and χ(σ y ) = 1, indicating two conjgacy classes which implies there are two irredcible representations of S on R. Frthermore, the irredcible representations form an orthogonal basis in the g- dimensional space of the grop, and following from Schr s lemma on orthogonality [5], the dimensionality theorem provides the relation between the nmber of irredcible representations and their dimensionality as k lp = g, (1) p=1 where the sm is taken over all irredcible representations of the grop k, and l p is the dimension of the pth irredcible representation. Ths, sing the character of the representations (9) to find the total nmber of irredcible representations along with the dimensionality eqation in (1), we can find all of the possible irredcible representations that span the grop [5]. So for or network in Fig. 4 with S symmetry and irredcible representations, the dimensionality eqation (1) allows s to conclde that their are a total of, 1- dimensional irredcible representations of S. The 1-dimensional irredcible representations of S can be fond from the identity representation (every grop element represented by 1) and the alternating representation A n, defined by A n = det[d(r)] which yields the table of irredcible representations Γ (p), with p = {1, } from (1) : R E σ Γ (1) (R) 1 1 Γ () (R) 1-1 (11) Finally, the similarity transform T that takes the system (1) into the coordinates of the symmetry basis is defined in [4] as a projection of the irredcible representations onto the space of the system (1) on R n : G (p) i = R Γ (p) (R) iid(r) (1) 5787

4 α C y α 1 α v α 1 Fig. 5: An example -node bd network which is strctrally controllable and has C symmetry when = α = α 1. α 11 x z α Fig. 6: The network in Fig. 4 transformed into a new strctre defined by the S symmetry basis. where G (p) i generates basis vectors on R n from the pth irredcible representation Γ (p), * indicates the complex conjgate, and i indicates the (i, i)-th entry of the irredcible representation for i = 1... l p. Once G (p) i is compted for each p and i, the similarity transform is constrcted from the normalized linearly independent vectors that span R n. For or example network in Fig. 4, we can compte G for each irredcible representation, G (1) 1 = Γ (1) (E) 11D(E) + Γ (1) (σ y ) 11D(σ y ) = =, (1) where each linearly independent colmn of G forms a colmn of T. After normalizing we have 1,, 1 = τ 11 τ 1 τ 1 τ, (14) normalize 1 τ 1 τ which defines the first and second colmns of T. Compting G for the second irredcible representation we have, G () 1 = Γ () (E) 11D(E) + Γ () (σ y ) 11D(σ y ) = =, (15) which after normalization yields the final colmn of T 1 τ 1 = τ. (16) normalize 1 1 τ and the similarity transformation matrix T is given as B. Symmetry in the Strctre 1 T = 1. (17) 1 As mentioned previosly in Sec. III-A, when a system or network contains an explicit grop symmetry the generic entries of the pair (A, B) become constrained, and in fact once the pair (A, B) is transformed by T into the coordinates of the symmetry basis, the strctre of the transformed pair (Â, ˆB) cold be different from (A, B) which cold also alter whether or not (A, B) is strctrally controllable. As an example, take the network in Fig. 4 which has the pair (8). The nconstrained network is easily shown to be strctrally controllable as the controllability matrix Q compted from (8) is fll rank for this choice of the non-zero entries. However, if we constrain α 11 = α and = α, the network has S symmetry and transforming (A, B) in (8) into the symmetry basis we have  = T AT, ˆB = T B which yields  = α 11 α, α ˆB = α 4 (18) where is the complex conjgate transpose, and the transformed pair (Â, ˆB) has a new strctre in Fig. 6 that is easily seen to contain an isolated node and therefore not strctrally controllable as well as NCS. While this reslt exemplifies how symmetry in a network can alter its apparent strctre, let s also demonstrate how certain symmetries leave the network strctre invariant. Consider again Lin s bd strctre for n = in Fig. 5 with the pair A = α, B = α 1 α 4 (19) For C symmetry on R we have matrix representations D(R) constrcted as 1 1 C = 1, C = () and the identity element E as before in (7). Next, the table of irredcible representations of C cyclic symmetry is given in [8] as R E C C Γ (1) (R) Γ () (R) 1 ω ω Γ () (R) 1 ω ω (1) where ω = e πi, C is a rotation by π/ and C by 4π/. Now generating the symmetry basis via (1) we have for the 5788

5 first irredcible representation, G (1) 1 =Γ (1) (E) 11D(E) + Γ (1) (C ) 11D(C )... + Γ (1) (C) 11D(C) = = normalize τ 11 = τ 1. τ 1 Next compte G for the second irredcible representation G () 1 =Γ () (E) 11D(E) + Γ () (C ) 11D(C )... + Γ () (C) 11D(C) =1 1 + ω 1 + ω = 1 ω ω ω 1 ω ω ω ω normalize = τ 1 τ, 1 τ and lastly for the third irredcible representation G () 1 =Γ () (E) 11D(E) + Γ () (C ) 11D(C )... + Γ () (C) 11D(C) =1 1 + ω 1 + ω = 1 ω ω ω 1 ω ω = τ 1 τ. ω normalize ω 1 τ ω ω () () (4) Ths or similarity transformation T for C symmetry is constrcted as T = e 4πi e πi, (5) 1 e πi e 4πi and the transformed system (Â, ˆB) defined by  = T AT, ˆB = T B becomes: α 11 α  = α 14, ˆB = α α 4 (6) α 4 which is readily seen to be flly controllable, and hence strctrally controllable in the presence of C symmetry, which agrees exactly with the presentation of the bd strctre in []. IV. DISCUSSION Here we have shown the details of the application of grop representation theory in order to determine the effects of symmetry on strctrally controllable networks. While [] identified the basic canonical strctres that define how a control inpt can reach throgh all nodes of a generic network, the connection between strctral controllability and networks containing symmetries was established in [8] and demonstrated here. Additionally, it is worth noting that the dilation n-controllable strctre in Fig. 1a from [] is a pecliar one in that the downstream nodes V 1 and V are absent any self-connections, which case degeneracy and hence n-controllability in the form of two colmns of zeros in the system pair (A, B) as shown in (5). The importance of these self-connections in defining the strctral controllability is given treatment in [9], and inclding the intition that the controllability of nonlinear networks depends on the system trajectory [8], calls into qestion the tility of the dilation canonical strctre in determining strctral controllability 5789

6 for real world networks which have nodal dynamics at each node. Grop elements that belong to the same conjgacy class have the same character, which categorically describes the type of symmetry operations and provides frther insight into how certain types of symmetry inflence controllability (degeneracy) of a network. For example, a class of grop elements that all commte with one another (called Abelian) are the rotational symmetry operations and these are defined by: A n = E, (7) where n is the order of the generating element, and E is the identity. The intition here comes from the fact that symmetry operations that commte do not introdce degeneracy into the network, hence if the grop symmetry is comprised of only rotations (and the identity element which is jst A 1 = E) as in Lin s bd strctre, or the example C symmetry network in Fig. 5, no degeneracy is introdced by the symmetry to the network, which will allow controllability. Frthermore, if the grop symmetry of the network contains non-commting elements (symmetry operations), then a degeneracy may be inherent in the network casing it to be NCS, as in the example S symmetry network in Fig. 4, which contains a non-commting reflection reslting in n-controllability. Strctral controllability is indeed affected by certain types of symmetries present in the network which manifests as change in the strctre of the transformed system, while other types of symmetries leave the strctre invariant. The link between symmetries and synchrony has been established in previos work [1], and the fact that synchronos network activity in living neral networks sggests sch symmetries are inherent to brains. These reslts offer new insights into the strategic strctral control design of the class of networks containing symmetries, and demonstrates the tility of grop representation theory as applied to sch networks which extends to any natral or man-made network. REFERENCES [1] Y. Li, J. Slotine, and A. Barabási, Controllability of complex networks, Natre, pp. 1 7, 11. [Online]. Available: [] S. Peqito, S. Kar, and A. Agiar, A framework for strctral inpt/otpt and control configration selection in large-scale systems, Atomatic Control, IEEE Transactions on, vol. PP, no. 99, pp. 1 1, May 15. [] C.-T. Lin, Strctral controllability, Atomatic Control, IEEE Transactions on, vol. 19, pp. 1 8, [4] H. Rbin and H. Meadows, Controllability and Observability in Linear TimeVariable Networks With Arbitrary Symmetry Grops, Bell System Technical Jornal, 197. [Online]. Available: [5] M. Tinkham, Grop Theory And Qantm Mechanics. McGraw-Hill Inc., San Francisco, [6] L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Mrphy, and R. Roy, Clster synchronization and isolated desynchronization in complex networks with symmetries. Natre commnications, vol. 5, no. May, p. 479, Jan. 14. [7] P. J. Uhlhaas and W. Singer, Neral synchrony in brain disorders: relevance for cognitive dysfnctions and pathophysiology. Neron, vol. 5, no. 1, pp , Oct. 6. [8] A. J. Whalen, S. N. Brennan, T. D. Saer, and S. J. Schiff, Observability and controllability of nonlinear networks: The role of symmetry, Phys. Rev. X, vol. 5, p. 115, Jan 15. [Online]. Available: [9] N. J. Cowan, E. J. Chastain, D. a. Vilhena, J. S. Fredenberg, and C. T. Bergstrom, Nodal dynamics, not degree distribtions, determine the strctral controllability of complex networks. PloS one, vol. 7, no. 6, pp. 1 5, Jan. 1. [1] M. Golbitsky, D. Romano, and Y. Wang, Network periodic soltions: patterns of phase-shift synchrony, Nonlinearity, vol. 5, no. 4, pp , Apr