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 sucient 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 eort 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 dierent amounts of energy. For example, to move ˙ = + the weighted network of Fig. 1a to the three dierent 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]

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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 dierent 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

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Figure 2 | Controlling a network through all nodes (N N). Distribution p( ) of eigen-energies required to control several model and real systems. D = E a, Scale-free model network without degree–degree correlation. b, Scale-free model networks with positive (r 0.25) or negative (r 0.24) = = degree–degree correlation. c, Erdős–Rényi model network. d, Airline transportation network. e, Internet AS-level network. f, US power grid network. g, Israeli social network. h, User-interaction network of an online forum. i, Interlocking network of Norwegian companies. j, Human protein–protein interaction network. k, Human heterogeneous network. l, Functional coactivation network of the human brain. The straight lines show prediction (4) and the error bars represent standard deviations. For model networks the edges’ weights A are uniformly drawn from [0,1]. The self-loops A A , ij ii = j ij where 0.25, representing a small perturbation to diagonal entries, to ensure that the network is stable. The data sources and basic characteristics of = P these networks are discussed in Supplementary Section VI A. requiring dierent energies, we explore the eigen-space of G. Denote undirected networks all eigenvalues of A are negative, thus we by the eigen-energies (that is, the minimum energy required denote the eigenvalues by so that the absolute eigenvalues are Ei i to drive the network to the eigen-directions of G). According to i >0 for all i. We sort the absolute eigenvalues in ascending order equation (2) 1/µ ,withµ corresponding to the eigenvalues of 0< < < < , finding that (Supplementary Section I) Ei = i i 1 2 ··· N G. Generally, the energy surface for a network with N nodes is a super-ellipsoid spanned by the N eigen-energies of G. To determine G V (V TBBTV ) C V T (3) the distribution of these eigen-energies we decompose the adjacency = [ ] matrix as A V 3V T, where V represents the eigenvectors of A where denotes Hadamard product, defined as (X Y ) X Y , = ij = ij ij and 3 diag , ,..., are the eigenvalues. For stable and C is a matrix with entries C 1/( ). For a given = { 1 2 N } ij = i + j

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Biological networks 0.2 0.2 0.2 Gene_coexpress Mutualism_net Cat_cortex Prediction Prediction Prediction 0.0 0.0 0.0 100 1050 10100 10150 100 1040 1080 10120 100 1050 10100 10150

Figure 3 | Controlling a network through a single node (N 1). Panels show the complementary cumulative distribution p>( ) of eigen-energies required D = E to control several model and real systems. a, Scale-free model network without degree–degree correlation. b, Scale-free model networks with positive (r 0.34) or negative (r 0.36) degree–degree correlation. c, Erdős–Rényi model network. d, Sampled airline network. e, Internet AS-level network in = = 1997. f, IEEE power grid test network. g, Human-contact network of the ACM Hypertext 2009 conference. h, Email interaction network. i, Phone call network between dierent countries. j, Connected component of the human gene-coexpression network. k, Mutualism ecological network in Mauritius. l, Inter-region network of cat cortex. The insets are log–log plots of probability distributions p( ) with logarithmic binning3. The straight lines show E prediction (5) and the error bars represent standard deviations. The data sources and basic characteristics of these networks are discussed in Supplementary Section VI A. network, (3) captures the impact of the input matrix B on the control eigenvectors. Thus 2 and p( ) (1/2)p(), meaning the Ei = i E = properties of the system, allowing us to analyse the distribution of distribution of eigen-energies is proportional to the distribution eigen-energies for dierent numbers of driver nodes and determine of the network’s absolute eigenvalues. We add self-loops as N the required energy for each direction. Aii ( j 1 Aij), where > 0 is a small perturbation to ensure = + = that all eigenvalues of A are negative. This scheme has been widely Controlling a system through all nodes used in previousP studies on dynamical processes taking place 29 38 If we can control all nodes (that is, ND N), B becomes a unit on networks, such as opinion dynamics , synchronization and = T 16 1–3 diagonal matrix. In this case G V diag 1/2 V and the eigen- control . For networks with degree distribution p(k) k the = { i} ⇠ directions of the controlled system are the same as the network’s distribution of the absolute eigenvalues of A also obeys a power

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max max max 40 50 15 log( log( 30 log( log( 20 10 5 0 0 10 20 30 40 50 0 5 10 15 20 0 10 20 30

N/ND N/ND N/ND

Figure 4 | Controlling a network through a ﬁnite fraction of its nodes. a, Multi-peak distributions p[log( )] for 1

39,40 law p() (see Supplementary Section II A). Consequently, network (Fig. 2g), the user-interaction network of an online ⇠ forum (Fig. 2h), the human protein–protein interaction network p( ) (4) (Fig. 2j) and the human heterogeneous network (Fig. 2l), in line E ⇠E with the prediction (4). In contrast, for several networks with indicating that the system can be easily driven in most directions bounded degree distribution, such as the ErdÆs–Rényi random of the state space, requiring a small . A few directions require network (Fig. 2c), the US power grid network (Fig. 2f), the E considerable energy and the most dicult direction needs41 interlocking network of Norwegian companies (Fig. 2i) and the 1/( 1) N . The fact that is sublinear in N for > 2 functional coactivation network of the human brain (Fig. 2l), Emax ⇠ Emax indicates that, when N N, the energy density /N remains p( ) is also bounded, as predicted by in (4). Such D = E D E !1 bounded. In Fig. 2 we test the prediction (4) for several model, networks require even less energy to control their progress in infrastructural, social and biological networks. We find that p( ) their most dicult direction. Taken together, we find that for E follows a power law for uncorrelated or correlated scale-free N N the distribution of eigen-energies is uniquely determined D = model networks (Fig. 2a,b), the airline transportation network by network topology, and we lack significant energetic barriers (Fig. 2d), the Internet AS-level network (Fig. 2e), an Israeli social for control.

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Controlling a system through a single node Table 1 | Controlling complex networks with dierent numbers If there is not degeneracy in A’s eigenvalues we can control an of driver nodes. undirected network by driving only a single node11,42. In this case V TBBTV V V (1/N), where h is the index of the ={ ih jh} ⇠ O Number of Distribution of eigen- Maximum control chosen driver node. Thus V TBBTV can be viewed as a small driver nodes energies energy 1/( 1) perturbation to the matrix C in (3). The statistical behaviour of ND Np( ) max N = E ⇠E E ⇠ N/ND the eigenvalues of G is mainly determined by the eigenvalues of C, 1( ) E p( 0)d 0. Based on (5) we w(t), which we assume to be a Gaussian white noise with E = E E E obtain p>( ) (ln max ln ), decreasing linearly with ln .We zero mean and variance one. We aim to estimatex ˆ o of the E ⇠ E E R E ⌧ 2 test our prediction on several network models (Fig. 3a–c) and real initial state xo while minimizing the dierence y(t) yˆ(t) dt 0 k k networks (Fig. 3d–l), finding that the corresponding eigen-energies between the output y(t) that is actually observed and the output At R span over a hundred orders of magnitude and this exceptional range yˆ(t) Ce xˆ o that would be observed in the absence of noise. With = 45 of variations is reasonably well approximated by (5) for both p>( ) the maximum-likelihood approximation , the expectation xˆ o xo E 45 T 1 h i= and p( ). Taken together, if we attempt to control a network from and the covariance matrix xx Go (⌧), where x xˆ o xo is the E h˜ ˜ ⌧i=ATt T At ˜ ⌘ a single node (N 1), the required energy varies enormously for estimation error and Go(⌧) e C Ce dt is the observability D = = 0 dierent directions, almost independently of the network structure, Gramian. Therefore, the variance 2 of the approximation in R making some directions prohibitively expensive energetically. direction x is ˜ 2 T 1 Controlling a system through a ﬁnite fraction of its nodes (⌧) x Go (⌧)x (6) (1 ) =˜ ˜ When p( ) , the distribution p( ) e Eˆ , where ln . E ⇠ E Eˆ ⇠ Eˆ ⌘ E Thus, if ND N, p( ) is an exponential (one-peak) distribution indicating that the estimation uncertainty varies with the direction = Eˆ 1 for > 2 in (4); if N 1, as p( ) in (5), p( ) is a uniform of the state space. To illustrate this, consider the network in Fig. 1e D = E ⇠ E Eˆ distribution. To understand the transition from (4) for N N to that moves along the trajectory of Fig. 1g, while we measure the D = (5) for N 1, we investigate the distribution p( ) when 1

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© 2015 Macmillan Publishers Limited. All rights reserved NATURE PHYSICS DOI: 10.1038/NPHYS3422 ARTICLES degree correlations47 and community structure48 (Supplementary References Table 1). To assess the impact of these topological characteristics 1. Albert, R. & Barabási, A.-L. Statistical mechanics of complex networks. Rev. we perform the degree-preserved randomization49 on each network, Mod. Phys. 74, 47–97 (2002). eliminating local clustering, degree correlations and modularity. 2. Cohen, R. & Havlin, S. Complex Networks: Structure, Robustness and Function (Cambridge Univ. Press, 2010). We find that the distribution of eigen-energies required to drive 3. Newman, M. E. J. Networks: An Introduction (Oxford Univ. Press, 2010). each randomized network follows the predictions (4) and (5) 4. Boccaletti, S., Latora, V., Moreno, Y., Chavez, M. & Hwang, D. Complex (Supplementary Figs 6 and 7), indicating that degree distribution networks: Structure and dynamics. Phys. 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