Physical controllability of complex networks

A challenging problem in network science is to control complex networks. In existing frameworks of structural or exact controllability, the ability to steer a complex network toward any desired state is measured by the minimum number of required driver nodes. However, if we implement actual control by imposing input signals on the minimum set of driver nodes, an unexpected phenomenon arises: due to computational or experimental error there is a great probability that convergence to the final state cannot be achieved. In fact, the associated control cost can become unbearably large, effectively preventing actual control from being realized physically. The difficulty is particularly severe when the network is deemed controllable with a small number of drivers. Here we develop a physical controllability framework based on the probability of achieving actual control. Using a recently identified fundamental chain structure underlying the control energy, we offer strategies to turn physically uncontrollable networks into physically controllable ones by imposing slightly augmented set of input signals on properly chosen nodes. Our findings indicate that, although full control can be theoretically guaranteed by the prevailing structural controllability theory, it is necessary to balance the number of driver nodes and control cost to achieve physical control.

arbitrarily initial state to an arbitrarily final state. We believe that this difficulty is fundamental, as we were not able to remove or even mitigate the problem of divergence despite extensive and systematic computational efforts in implementing various ways to optimize the numerical algorithm. This difficulty in realizing actual control persists for a large number of model and real world networks. While somewhat unsettling, the issue prompts us to hypothesize that the existing controllability frameworks are merely mathematical, as the implementation of actual control would often require infinite precision computations and, more seriously, an infinite amount of energy. To make the notion of controllability of complex networks meaningful, the issue of physical controllability must be addressed.
In this paper, we develop a physical controllability framework for complex networks to address whether actual control can be achieved in an experimentally or computationally feasible way. Given a complex network, we first use the structural controllability theory 4 to determine N D and a set of unmatched nodes to which control signals are to be applied. Then, with a given pair of arbitrarily initial and final states as well as a finite control time, we calculate the optimal control signals 36 and evolve the whole networked system, which is essentially a linear dynamical system under external driving, to determine whether the system can be driven from the initial state to the final state in the given amount of time. During this process, the energy required for control can be calculated through the standard formula in linear systems theory 36,37 , which expresses the energy as the integral of the product of a number of matrices, including the inverse of the positive-definite, symmetric Gramian matrix. Freedom in choosing the initial and final states and independent network realizations render feasible a statistical analysis of the control process. We find that, typically, there are two cases, depending on whether the network can be physically controlled. For the physically controllable case, the whole system, starting from the chosen initial condition, can actually converge to the final state in the prespecified time within a predefined precision. In this case, the Gramian matrix is well-behaved, meaning that both its condition number and the energy are not unrealistically large. For the physically uncontrollable case, the system cannot reach the final state within the predefined precision in the given time. In such a case, the Gramian matrix is singular in the sense that its condition number can be arbitrarily large, so is the corresponding energy. Increasing the precision of the computation, e.g., by using special simulation packages with round-off error orders of magnitude smaller than that associated with the conventional double-precision computation, would convert a few uncontrollable cases into controllable ones, but vast majority of the uncontrollable cases remain unchanged.
The main result of this paper is a proposal of a general, probabilistic measure to characterize the physical controllability for complex networks of arbitrary topology. For physically uncontrollable networks, it is important to develop effective strategies to make them physically controllable. To accomplish this goal, we gain insights by calculating the control energy for a bidirectional 1D chain and obtaining an analytical relation between energy E and chain length L. We then apply the result to general networked systems based on the idea of longest control chain (LCC) 27 . Optimization strategies can be derived to decrease the control energy drastically. In fact, if the system is physically uncontrollable, a viable way to make it controllable is to increase the number of control signals beyond N D . Our framework of physical controllability thus contains the following essential ingredients: (1) N D , the minimum number of control signals determined by the existing mathematical controllability frameworks, (2) a measure of physical controllability, (3) control energy E determined by the Gramian matrix, and (4) augmentation of N D for physically uncontrollable networks. The existing mathematical controllability theories 4,10 thus provide a base for our physical controllability framework. The quantity N D , on which the mathematical controllability theories focus, can effectively be regarded as the lower bound of the actual number of control signals required. To realize physical control, depending on the specific system and control settings, either N D control signals suffice or substantially more signals are needed.

Results
Definition of physical controllability. We consider the standard setting of a linear dynamical system subject to control input 1,4,10 : T is a vector defining the set of control input signals, A = {a ij } N×N is the adjacency matrix with N being the number of nodes in the network, and = is control input matrix specifying the set of N D "driver" nodes 4 , each receiving a control signal that corresponds to one component of the control vector u. From the linear systems theory, optimal control of a linear dynamical network in the sense of minimized energy cost can be achieved when the input control signals u t are chosen as 36,37 is the Gramian matrix, a positive-definite and symmetric matrix 36 , which serves as the base to determine quantitatively if a system is actually controllable. In particular, the system is controllable only when W is nonsingular (invertible) for given control precision 36,37 . With the control input signal u, the energy cost is 36 where control is initiated at t = 0 and ended at t = t f . To present concrete evidence for the existence of physically uncontrollable networks, we use the Erdos-Renyi (ER) type of directed random networks 38  a single parameter P b . The meaning of P b is the following. Given a pair of linked nodes, i and j, the probability that the link points from the smaller-degree node to the larger-degree one is P b , and the probability in the opposite direction is 1 − P b . The link direction is chosen randomly if i and j have the same degree. To determine the set of driver nodes, we use the maximum-matching algorithm 30 , which gives the control matrix B. For each combination of A and B, we first randomly choose the initial and final states. We then calculate the corresponding Gramian matrix W, its condition number, the input signal u t , the actual final states x * t f , and finally the control energy E(t f ). Repeating this process for each and every independent network realization in the ensemble enables an extensive statistical analysis of the control process.
Mathematically, if the Gramian matrix W is singular, the energy diverges. Through extensive and systematic numerical computations, we find that, even when W is non-singular in the mathematical sense, for typical complex networks its condition number can be enormously large, making it effectively singular as any physical measurement or actual computation must be associated with a finite precision. Say in an experiment the precision of measurement is ε. In a computational implementation of control, ε is effectively the computer round-off error. Consider the solution vector X of the linear equation: W · X = Y, where Y is a known vector. Let C W be the condition number of W. The accuracy of the numerical solution of X, denoted by e X = 10 −k (k is a positive integer), is bounded by the product between C W and ε 40 . We see that, if C W is larger than ε ≡ − C 10 / k W , it is not possible to bring the system to within 10 −k of the final state at finite control cost, so physically control cannot be achieved in finite time.
For a large number of networks drawn from an ensemble of networks with a pre-defined topology, the condition numbers of their Gramian matrices are often orders of magnitude larger than C W . Figure 1 shows the correlation between the condition number C W and the control error e X . We observe that, within a certain range of C W , an approximate scaling relation exists between C W and e X , as shown in panels (a, c, e, g). However, the scaling disappears outside the range where the Gramian matrix W is ill conditioned, leading to considerable errors when computing the matrix inverse. In principle, the scaling regime can be extended with improved computational precision, but not indefinitely. For the networks with an ill conditioned Gramian matrix, not only is the control vector unable to drive the system to the target state, but the associated energy can be extremely large. These observations suggest the following criterion to define physical controllability in terms of the control energy cost: a network is controllable with respect to a specific control setting if and only if the condition number of its Gramian matrix is less than C W , a critical number determined by both the measurement or computational error and the required precision of control. For a given set of network parameters (hence a given network ensemble) and control setting, the probability that the condition number of the Gramian matrix is less than C W , P C ( ) W , can effectively serve as a quantitative measure of physical controllability. Increasing the precision of the computation, e.g., by using special simulation packages with round-off error orders of magnitude smaller than that associated with the conventional double-precision computation, would convert a few uncontrollable cases into controllable ones, but vast majority of the uncontrollable cases remain unchanged.
Note that, physical controllability is characterized by the condition number of the Gramian matrix W, which is defined by the adjacency matrix A, the control matrix B, and the control time from t 0 to t f . The adjacency matrix A totally defines the structure of the underlying network and, in the absence of control, solely determines the evolution of the system from an initial state. The purpose of control is to design the control matrix B so that the Gramian matrix is numerically to ensure that the system is physically controllable, which can be accomplished regardless of whether the matrix A is stable or unstable.
We also note that, in a linear dynamical system, the Gramian matrix W is determined by the network structure, the control configuration, and the control time; it does not depend on the dynamical trajectory. As a result, additive noise of reasonable amplitude does not affect the physical controllability of the network.
Structural controllability does not imply physical controllability. We present evidence that structural and physical controllabilities are not necessarily compatible with each other. Figure 2(a,b) show the percentage of driver nodes, n D ≡ N D /N, versus the directional link probability P b . We see that n D is minimized for P b = 0.5, indicating a maximal (optimal) level of structural controllability because only a few control signals are needed to control the whole network 4 . But can physical controllability be achieved in the same parameter regime where structural controllability is optimized? Figure 2(c,d) show the corresponding physical controllability P C ( ) W versus the network parameter P b . We see that, in both regimes of small and large P b values where structural controllability is weak [corresponding to relatively high values of n D in Fig. 2(a,b)], the physical controllability is relatively strong. In the regime of small P b values, most directed links in the network point from small to large degree nodes. In this case, the network is more physically controllable, in agreement with intuition. The striking result is that, in the regime of intermediate P b values (e.g., P b around 0.5) where the number of driver nodes to control the whole network is minimized so that structural controllability is regarded the strongest, the physical controllability is in fact the weakest, as the probability of the condition number P C ( ) W being small is close to zero. For example, for the random networks in Fig. 2(c), for 〈 k〉 = 4, the minimum value of P C ( ) W is only about 0.1 for P b ≈ 0.6, while for 〈 k〉 = 6 and 〈 k〉 = 8, the minimum values are essentially zero. Surprisingly, near zero values of P C ( ) W occur in a wide range of the parameter P b , e.g., [0.3, 0.8] and [0.2, 0.9] for 〈 k〉 = 6 and 〈 k〉 = 8, respectively, as shown in Fig. 2(c). This indicates that the network is physically uncontrollable for most cases where structural controllability is deemed to be strong. BA scale-free networks behave similarly, as illustrated in Fig. 2(d). Another finding from Fig. 2 is that N D is symmetric about P b = 0.5, but the symmetry is broken for P C ( ) W , indicating that there is no simple linear correlation between N D and P C ( ) W . It is thus necessary to find the fundamental structural properties responsible for the smallness of P C ( ) W . Through a detailed analysis of the energy cost associated with controlling a simple one-dimensional chain and a double chain network ( Methods) and of the energy scaling 27 , we identify the longest control chains (LCCs), the shortest paths through which the control energy is "flowed" to all nodes in the network, as the fundamental structural component responsible for the control energy. The longer the LCCs, the more singular the Gramian matrix, and the smaller the probability P C ( ) W . The maximal LCC is effectively the control diameter of the network 27 .
Physical controllability of an electrical circuit network and a strategy to balance control energy and extra inputs. To further illustrate the concept of physical controllability, we consider a real one-dimensional cascade parallel RC circuit network, as schematically illustrated in Fig. 3(a). The network can be represented by a bidirectional 1D chain with self-loops for all the nodes, as shown in Fig. 3 The network size can be enlarged, say by one unit, by attaching an additional branch of resistor and capacitor at the right end of the circuit. The state u i (t) of node i at time t is the voltage of capacitor i, and the input voltage u(t) represents the control signal. The purpose of control is to drive the voltages of the capacitors from a set of values to another within time t f through the input voltage u(t). The control energy can then be calculated by Eq. (3). The actual energy dissipated in the circuit during the control process is given by t real 0 f where U(t) ≡ u(t) and I(t) are the input voltage and current at time t, respectively, and E real is in units of Joule. By making the circuit equivalent to a 1D chain network, we have three types of energy: the control energy of the actual circuit calculated from Eq. (3), the dissipated energy of the circuit from Eq. (4), and the control energy of the 1D equivalent network. Figure 3(c) shows that the control energy and the dissipated energy of the circuit do not differ substantially from the energy calculated from the unidirectional 1D chain. Among the three types of energy, the energy cost associated with the control process calculated from Eq. (4) is maximal.
Our extensive computations reveal that many structurally controllable networks are not physically controllable due to a combination of the ill-conditioned Gramian matrix and the finite computational or experimental error. Our analysis of the chain model (Methods) suggests a simple but effective strategy to reduce the energy significantly so as to enhance the physical controllability of the network: to place extra control signals along the LCCs to break the chains into shorter subchains. (In Methods, we show how the redundant control input can be planted in a circuit network.) To be illustrative, we consider a unidirectional 1D chain and add an extra control input at the ith node. As shown in Fig. 3(d), the magnitude of the control energy is reduced dramatically. The optimal location to place the extra control should be near in the middle of the chain so as to minimize the length of the LCC using a minimal number of extra control signals. In Fig. 3(d), the red circles represent a 1D chain and indicate that this simple strategy of adding one redundant control signal near the middle can reduce ( ) W for ER random and BA scalefree networks of size N = 100, respectively, where P C ( ) W is the probability that the condition number of the Gramian matrix is less than some physically reasonable threshold value. Comparing (a) with (c), or (b) with (d), we observe the striking phenomenon that, in the parameter regime where the number of driver nodes is minimized so that the corresponding networks are deemed to be most structurally controllable, they are physically uncontrollable. The phenomenon persists regardless of the network size and type. All nodes are selfloop free. The qualitative behavior is robust against variations in the value of C W .
Scientific RepoRts | 7:40198 | DOI: 10.1038/srep40198 dramatically the required energy. For the circuit network in Fig. 3, the redundant control input can be realized by inducing external current input into a capacitor. In Fig. 3(d), the real energy is represented by green triangles, which reaches the minimum when the extra input is putting around the middle. Applying a single redundant control input can thus be an extremely efficient strategy to make the one-dimensional chain network physically controllable.
Control energy optimization of complex networks. For a complex network, there often exist multiple LCCs, requiring multiple redundant control inputs. With insights from the RC circuit example, we see that a strategy is to place one redundant control input at the middle of each LCC. In this case, each LCC in the network . An extra external current input i e (t) serves as a redundant control input injected into the capacitor C 3 , where i 3 and i 4 denote the currents through resistors R 3 and R 4 , respectively. In absence of the extra current input, i 3 (t) − i 4 (t) is the current through the branch of C 3 . (b) Network representation of the circuit in (a) as a bidirectional 1D chain network of seven nodes, where the external voltage input u(t) is injected into node 1 (yellow driver node, the controller). The dynamical state of node i is described by the voltage of its capacitor, u i (t). Links (blue) between nodes are bidirectional and have uniform weight 1 in either direction. Each node has a self-link (red) of weight − 2, except the ending node (node 7) whose self-link has weight − 1. The extra external current input i e (t) serves as a redundant control input injected into node 3 of the network in (b). Now there are two driver nodes (yellow) in the network, nodes 1 and 3. (c) Energy required for controlling a unidirectional chain (red circle) and the corresponding circuit (blue square) as well as the dissipated energy (green triangle) of the circuit calculated from Eq. (4) versus chain length L. (d) Control and dissipated energies in presence of a redundant control signal to node i (i > 1), which breaks the chain into two subchains of lengths i and L − i, respectively. is broken into two subchains. Figure 4(a) shows the effect of this optimization strategy on the energy distribution. For comparison, the same number of redundant control inputs are also applied randomly throughout the network. The reduction ratio between the control energy under optimization strategy, Δ E, and the original control energy E characterizes the effectiveness of the optimization process. In particular, if the distribution of Δ E/E is concentrated on large values of Δ E/E, then the corresponding optimization strategy can be deemed to be effective. As shown in Fig. 4(a), for relatively large Δ E/E values, P(Δ E/E) as a result of optimization has values that are systematically larger than those under random control signal augmentation, while the opposite situation is observed for regions with relatively smaller Δ E/E. Thus, our optimization strategy outperforms the random strategy. The networks requiring proper optimization to be physically controlled are typically those with large control diameters. Figure 4(b) show that this is indeed the case: for networks with larger values of D C , the performance of our optimization strategy is significantly better than that with random placement of extra controllers.

Discussion
As stated in ref. 4, the ultimate proof that one understands a complex network completely lies in one's ability to control it. However, we find that strong structural controllability is no guarantee that the network can be physically controlled. To resolve this paradox, We develop a physical controllability framework in terms of the control energy cost and the number of external input signals. To illustrate the framework, we focus on the situation where the structural controllability theory yields a minimum number of external input signals required for full control of the network, and determine whether in these situations the control energy is affordable so as to realize actual control. Our systematic computations and analysis reveal a rather unexpected phenomenon: due to the singular nature of the control Gramian matrix, in the parameter regimes where optimal structural controllability is achieved in the sense that the number of driver nodes is minimized, energy cost can be physically impossible to accommodate. To obtain a systematic understanding, we focus on a bidirectional 1D chain and study the relationship between energy and chain length. We then apply the 1D chain model to complex networks based on the idea of LCCs. In fact, the simple chain model captures the scaling behavior of energy distribution found in random networks 23,27 . The chain model also provides a guiding principle to articulate optimization strategies to reduce the control energy, which are tested using a RC circuit network and model complex networks.
An intuitive picture of the interrelation between mathematical controllability 4,10 and our physical controllability is the following. In a fictitious world where the Gramian matrix is not singular (regardless of its condition number) and the computer round off or experimental errors are absolutely zero, using N D controllers as determined by the structural controllability theory can bring the networked system from any initial state to any final state in a given time. However, in the physical world, the inevitable measurement or computational errors will have a devastating consequence in the execution of actual control as the Gramian matrix is typically effectively singular with an arbitrarily large condition number. The dynamical interplay between the error and the singular Gramian matrix makes the system uncontrollable in the sense that it cannot be driven to the final state in finite time within the desirable precision and the energy required in the process diverges. Often, to realize physical control, many more control signals than those determined by the structural controllability theory are needed.
Our work indicates the difficulty of achieving actual control of complex networks associated with even linear dynamics. Although the mathematical controllability theories 4,10 offer theoretically justified frameworks to guide us to apply external inputs on a minimum set of driver nodes, when we implement control to steer a system to a desired state, the energy consumption is likely to be too large to be affordable. For nonlinear dynamical networks, we continue to lack a general controllability framework and an understanding of required control energy, although progress has been made 22,[41][42][43][44][45][46] , in spite of the fact that for specific types of systems, e.g., gene regulatory networks, controllability can be defined in terms of the coexisting attractors (final destinations) of the system 45 . Unlike linear networked systems, controllability of a nonlinear network depends on both the network structure and the system dynamics. We speculate that the physical controllability of a nonlinear dynamical network, if it can indeed be defined, would depend on both the structural controllability and the system dynamics. At the present we still know very little about controlling complex networks hosting nonlinear dynamics, and further effort is needed to address this challenging but greatly important problem shared by a wide range of fields.

Methods
Control energy of one-dimensional chain model. To gain insights into how a network's structure affects the control energy, we rewrite Eq.
A t T f f . Since H is positive definite and symmetric like W, its inverse H −1 can be decomposed in terms of its eigenvectors as H −1 = Q · Λ · Q T , where Q = [q 1 , q 2 , … , q N ] is composed of the orthonormal eigenvectors that satisfy Q · Q T = Q T · Q = I, and Λ = diag{λ 1 ,λ 2 , … , λ N } is the diagonal eigenvalue matrix of H −1 in a descending order. Numerically, we find that λ 1 is typically much larger than other eigenvalues. We thus have In an undirected network, the adjacency matrix A is positive definite and symmetric. We can decompose A into the form A = V · S · V T , where the columns of V constitute the orthonormal eigenvectors of A and S = diag{s 1 , s 2 , … , s N } is the diagonal eigenvalue matrix of A in a descending order. We thus have be the eigenvalue matrix of H in a descending order. The energy can be expressed as We consider a bidiretional 1D chain network and provide an analytical calculation of the relationship between control energy and chain length L. In the undirected chain, the adjacency matrix is defined as Substituting this in Eqs (6) and (7), after some algebraic manipulation, we obtain with cos( ) and performing a Taylor expansion on b j t around t = 0, we obtain . Consequently, the numerator in the Rayleigh quotient can be expressed as Since y = [y 1 , y 2 , … , y L ] T is an arbitrary nonzero column vector, for each L and T, we can choose y = y m insofar as K 1 and K 2 are relatively small compared with K 3 . We can normalize ⋅ y y m T to arrive at where λ P L is the smallest eigenvalue of P. Since P is symmetric and positive definite, using Cholesky decomposition we can obtain its factorization 40 as P = U T · U, where U is the upper triangular matrix with its diagonal being the square roots of eigenvalues of P. Equation (8) can then be written as = ⋅ ⋅ ⋅ ⋅ ⋅ . Since orthonormal transform does not change the eigenvalues of a matrix, H has the same eigenvalues as is the diagonal eigenvalue matrix of P in a descending order. We then have where j and k run from 1 to L. For arbitrary but fixed x 0 , the control energy E(t f ) can be approximated as where we see that E(t f ) increases faster than exponential with L. As shown in Fig. 5(a), the energy required to control a unidirectional 1D chain nearly overlaps with that of a bidirectional one with identical weights. From Fig. 5(b-d) we see that Eq. (12) provides a reasonably accurate estimate of the control energy. Furthermore, we find numerically that Eq. (5) holds for random and scale-free networks. As shown in Fig. 6, there is a strong correlation between the average network control energy, 〈 E〉 , and the smallest eigenvalue of the H-matrix, λ − H 1 N , for ER random and BA scale-free networks, indicating that the network control energy is essentially determined by the smallest eigenvalue of its H-matrix. Network representation of a circuit system. We consider a cascade parallel R-C circuit consisting of three identical resistors and capacitors as an example to illustrate how the circuit can be abstracted into a directed network, as shown in Fig. 7. For convenience, we set R 1 = R 2 = R 3 = R and C 1 = C 2 = C 3 = C, and denote the currents through R 1 , R 2 , and R 3 as i 1 (t), i 2 (t), and i 3 (t), respectively. The equations of the circuit are   After some algebraic manipulation, we have which can be written as     It is necessary to keep all other nodes unaffected while introducing exactly one extra control input into the circuit. However, any additional voltage change in any part of the circuit can lead to voltage changes on all the capacitors. A change in the current through a capacitor will not affect the currents in other components of the network, since only the time derivative of its voltage is affected. Thus, a meaningful way to introduce an extra control signal input to one node of a circuit's network is to inject current into one particular capacitor in the circuit.