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Spectrum of Controlling and Observing Complex Networks

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Spectrum of Controlling and Observing Complex Networks ARTICLES PUBLISHED ONLINE: 10 AUGUST 2015 | DOI: 10.1038/NPHYS3422 Spectrum of controlling and observing complex networks Gang Yan1†, Georgios Tsekenis1†, Baruch Barzel2, Jean-Jacques Slotine3,4, Yang-Yu Liu5,6 and Albert-László Barabási1,6,7,8* Recent studies have made important advances in identifying sensor or driver nodes, through which we can observe or control a complex system. But the observational uncertainty induced by measurement noise and the energy required for control continue to be significant challenges in practical applications. Here we show that the variability of control energy and observational uncertainty for dierent directions of the state space depend strongly on the number of driver nodes. In particular, we find that if all nodes are directly driven, control is energetically feasible, as the maximum energy increases sublinearly with the system size. If, however, we aim to control a system through a single node, control in some directions is energetically prohibitive, increasing exponentially with the system size. For the cases in between, the maximum energy decays exponentially when the number of driver nodes increases. We validate our findings in several model and real networks, arriving at a series of fundamental laws to describe the control energy that together deepen our understanding of complex systems. any natural and man-made systems can be represented as concentration of a metabolite in a metabolic network35,the networks1–3, where nodes are the system’s components and geometric state of a chromosome in a chromosomal interaction links describe the interactions between them. Thanks to network14, or the belief of an individual in opinion dynamics29,36. M T these interactions, perturbations of one node can alter the states The vector u(t) u (t),u (t),...,u (t) represents the external =[ 1 2 ND ] of the other nodes4–6. This property has been exploited to control control inputs, and B is the input matrix, with B 1 if control ij = a network—that is, to move it from an initial state to a desired input uj(t) is imposed on node i. The adjacency matrix A captures final state7–9—by manipulating the state variables of only a subset the interactions between the nodes, including the possibility of self- 10,11 10–26 of its nodes . Such control processes play an important role in loops Aii representing the self-regulation of node i. the regulation of protein expression27, the coordination of moving robots28, and the inhibition of undesirable social contagions29.Atthe Control energy same time the interdependence between nodes means that the states The system (1) can be driven from an initial state xo to any desired of a small number of sensor nodes contain sufficient information final state x within the time t 0, ⌧ using an infinite number d 2 [ ] about the rest of the network, so that we can reconstruct the system’s of possible control inputs u(t). The optimal input vector aims to ⌧ full internal state by accessing only a few outputs30. This can be minimize the control energy7 u(t) 2dt, which captures the 0 k k utilized for biomarker design in cellular networks, or to monitor in energy of electronic and mechanical systems or the amount of effort R real time the state and functionality of infrastructural31 and social– required to control biological and social systems. If at t 0the 32 33 = ecological systems for early warning of failures or disasters . system is in state xo 0, the minimum energy required to move the = 7,16–18 Although recent advances in driver and sensor node system to point xd in the state space can be shown to be identification constitute unavoidable steps towards controlling and T 1 observing real networks, in practice we continue to face significant (⌧) x G− (⌧)x (2) E = d c d challenges: the control of a large network may require a vast amount ⌧ T of energy16–18, and measurement noise34 causes uncertainties in where G (⌧) eAt BBTeA t dt is the symmetric controllability c = 0 the observation process. To quantify these issues we formalize the Gramian. When the system is controllable all eigenvalues of G (⌧) R c dynamics of a controlled network with N nodes and ND external are positive. Equation (2) indicates that for a network A and an input control inputs as7–10 matrix B the control energy (⌧) also depends on the desired state E xd. Consequently, driving a network to various directions in the state x(t) Ax(t) Bu(t) (1) space requires different amounts of energy. For example, to move ˙ = + the weighted network of Fig. 1a to the three different final states xd where the vector x(t) x (t), x (t), ... , x (t) T describes the with x 1, we inject the optimal signals u(t) shown in Fig. 1b =[ 1 2 N ] k dk = states of the N nodes at time t and xi(t) can represent the onto node 1, steering the system along the trajectories shown in 1Center for Complex Network Research and Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA. 2Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel. 3Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. 4Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. 5Channing Division of Network Medicine, Brigham and Women’s Hospital, Harvard Medical School, Boston, Massachusetts 02115, USA. 6Center for Cancer Systems Biology, Dana Farber Cancer Institute, Boston, Massachusetts 02115, USA. 7Department of Medicine, Brigham and Women’s Hospital, Harvard Medical School, Boston, Massachusetts 02115, USA. 8Center for Network Science, Central European University, H-1051 Budapest, Hungary. †These authors contributed equally to this work. *e-mail: [email protected] NATURE PHYSICS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturephysics 1 © 2015 Macmillan Publishers Limited. All rights reserved ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS3422 a Controlling e Observing u(t) y(t) −3.2 1 −3.2 1 1.3 1.0 1.3 1.0 0.7 0.7 −2.7 2 3 −2.2 −2.7 2 3 −2.2 x˙(t) = Ax(t) + Bu(t)˙x(t) = Ax(t) y(t) = Cx(t) + w(t) bf150 0.25 100 0.00 50 ) ) t t ( ( u 0 y −0.25 −50 −100 −0.50 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0123 t t cg x 8 o 1.0 xd x 6 o 4 3 3 x 0.5 x 2 0 0.0 0.2 6 4 15 0.5 2 5 0.1 −5 0.0 x 0 x x2 x 2 −2 −15 1 0.0 −0.5 1 dh 1 4 0.25 2 0 2 0.00 1 −1 −0.25 2 0 0 0.5 0 −2 −1 0.0 x∼ ≡ x^ − x −0.5 −2 o o −2 −4 1234 (log-scaled) log( ) Control energy Observational uncertainty Figure 1 | Controlling and observing a network. a, The control of a three-node weighted network with one external signal u(t) that is injected to the red driver node. Hence the input matrix is B 1,0,0 T. The nodes have negative self-loops that make all eigenvalues of the adjacency matrix A negative. =[ ] b, Optimal control signals that minimize the energies required to move the network from the initial state x 0,0,0 T to three dierent desired states x o =[ ] d with x 1 in the given time interval t 0,3 . c, Trajectories of the network state x(t) driven respectively by the control signals in b. d, Control energy k dk= 2[ ] surface, showing the amount of energy required to move the network by one unit distance (that is, x 1) in dierent directions. The surface is an k dk= ellipsoid spanned by the eigen-energies for the controllability Gramian’s three eigen-directions (arrows). The squares correspond to the three final points used in b and c. e, Observing the network with one output y(t). Node 1 is selected as the sensor (green), thus the output matrix is C 1,0,0 . The =[ ] measurement noise w(t) is assumed as Gaussian white noise with zero mean and variance one. f, A typical output y(t) that is used to approximate the initial state x . g, A typical trajectory of the system state x(t). h, Estimation errorx ˜ xˆ x , wherex ˆ is the maximum-likelihood estimator of the initial o = o− o o state. Starting from the same initial state we ran the system 5,000 times independently, each dot representing the estimation error of one run. The uncertainty ellipsoid (black) corresponds to the standard deviation ofx ˜ in any direction. Fig. 1c. The corresponding minimum energies are shown in Fig. 1d. decays quickly to a nonzero stationary value when the control The control energy surface for all normalized desired states is an time ⌧ increases16. Henceforth we focus on the control energy ellipsoid, implying that the required energy varies drastically as we (⌧ ) and the controllability Gramian G G (⌧ ). E ⌘E !1 ⌘ c !1 move the system in different directions. Given a network A and an input matrix B, the controllability As real systems normally function near a stable state—that Gramian G is unique, embodying all properties related to the is, all eigenvalues of A are negative37—the control energy (⌧) control of the system. To uncover the directions of the state space E 2 NATURE PHYSICS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturephysics © 2015 Macmillan Publishers Limited. All rights reserved NATURE PHYSICS DOI: 10.1038/NPHYS3422 ARTICLES a 100 bc100 0.15 Erdős−Rényi 10−2 10−2 0.10 ( ) ( ) ( ) p p SF_assortative p 10−4 10−4 (r = 0.25) 0.05 Model networks SF_disassortative Scale-free (r = −0.24) Prediction Prediction 0.00 10−6 10−6 100 101 102 100 101 102 0 5 10 15 20 25 30 de100 100 f0.5 Power grid (US) Exponential fit 0.4 10−2 −2 10 0.3 −4 ( ) ( ) 10 ( ) p p p 0.2 10−4 10−6 0.1 Open_flights AS_CAIDA Infrastructural networks Prediction Prediction 0.0 10−6 10−8 100 101 102 103 100 101 102 103 104 0 5 10 15 20 gh10−1 100 i0.8 Inter-companies Exponential fit −1 10 0.6 10−3 10−2 0.4 ( ) ( ) ( ) p p p 10−3 10−5 Social networks 0.2 −4 Social net (Israel) 10 Online forum Prediction Prediction 0.0 10−7 10−5 101 102 103 104 100 101 102 103 02468 jk100 100 l Brain coactivation 0.3 Exponential fit 10−1 10−2 10−2 0.2 ( ) ( ) ( ) p p p 10−3 10−4 0.1 Hetero.
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