Controllability of complex networksYang-Yu Liu1,2, Jean-Jacques Slotine3,4 & Albert-Laszlo Barabasi1,2,5
The ultimate proof of our understanding of natural or technological systems is reflected in our ability to control them.Although control theory offersmathematical tools for steering engineered and natural systems towards a desired state, aframework to control complex self-organized systems is lacking. Here we develop analytical tools to study thecontrollability of an arbitrary complex directed network, identifying the set of driver nodes with time-dependentcontrol that can guide the system’s entire dynamics. We apply these tools to several real networks, finding that thenumber of driver nodes is determined mainly by the network’s degree distribution. We show that sparseinhomogeneous networks, which emerge in many real complex systems, are the most difficult to control, but thatdense and homogeneous networks can be controlled using a few driver nodes. Counterintuitively, we find that inboth model and real systems the driver nodes tend to avoid the high-degree nodes.
According to control theory, a dynamical system is controllable if, with asuitable choice of inputs, it can be driven from any initial state to anydesired final state within finite time1–3. This definition agrees with ourintuitive notion of control, capturing an ability to guide a system’sbehaviour towards adesired state through the appropriatemanipulationof a few input variables, like a driver prompting a car to move with thedesired speed and in the desired direction by manipulating the pedalsand the steering wheel. Although control theory is a mathematicallyhighly developed branch of engineering with applications to electriccircuits, manufacturing processes, communication systems4–6, aircraft,spacecraft and robots2,3, fundamental questions pertaining to the con-trollability of complex systems emerging in nature and engineering haveresisted advances. The difficulty is rooted in the fact that two independ-ent factors contribute to controllability, each with its own layer ofunknown: (1) the system’s architecture, represented by the networkencapsulating which components interact with each other; and (2) thedynamical rules that capture the time-dependent interactions betweenthe components. Thus, progress has beenpossible only in systemswhereboth layers are well mapped, such as the control of synchronized net-works7–10, small biological circuits11 and rate control for communica-tion networks4–6. Recent advances towards quantifying the topologicalcharacteristics of complex networks12–16 have shed light on factor (1),prompting us to wonder whether some networks are easier to controlthan others and how network topology affects a system’s controllability.Despite some pioneering conceptual work17–23 (SupplementaryInformation, section II), we continue to lack general answers to thesequestions for large weighted and directed networks, which most com-monly emerge in complex systems.
Network controllabilityMost real systems are driven by nonlinear processes, but the controll-ability of nonlinear systems is in many aspects structurally similar tothat of linear systems3, prompting us to start our study using thecanonical linear, time-invariant dynamics
dx(t)dt
~Ax(t)zBu(t) !1"
where the vector x(t)5 (x1(t), …, xN(t))T captures the state of a
system ofN nodes at time t. For example, xi(t) can denote the amount
of traffic that passes through a node i in a communication network24
or transcription factor concentration in a gene regulatory network25.The N3N matrix A describes the system’s wiring diagram and theinteraction strength between the components, for example the trafficon individual communication links or the strength of a regulatoryinteraction. Finally, B is the N3M input matrix (M#N) that iden-tifies the nodes controlled by an outside controller. The system iscontrolled using the time-dependent input vector u(t)5 (u1(t), …,uM(t))
T imposed by the controller (Fig. 1a), where in general the samesignal ui(t) can drivemultiple nodes. If wewish to control a system, wefirst need to identify the set of nodes that, if driven by different signals,can offer full control over the network. We will call these ‘drivernodes’. We are particularly interested in identifying the minimumnumber of driver nodes, denoted by ND, whose control is sufficientto fully control the system’s dynamics.The system described by equation (1) is said to be controllable if it
can be driven from any initial state to any desired final state in finitetime, which is possible if and only if theN3NM controllability matrix
C~(B,AB,A2B, . . . ,AN{1B) !2"
has full rank, that is
rank(C)~N !3"
This represents the mathematical condition for controllability, and iscalled Kalman’s controllability rank condition1,2 (Fig. 1a). In practicalterms, controllability canbe alsoposed as follows. Identify theminimumnumber of driver nodes such that equation (3) is satisfied. For example,equation (3) predicts that controlling node x1 in Fig. 1b with the inputsignalu1 offers full controlover the system, as the states of nodesx1,x2,x3and x4 are uniquely determined by the signal u1(t) (Fig. 1c). In contrast,controlling the top node in Fig. 1e is not sufficient for full control, as thedifference a31x2(t)2 a21x3(t) (where aij are the elements of A) is notuniquely determined by u1(t) (see Fig. 1f and SupplementaryInformation section III.A). To gain full control, wemust simultaneouslycontrol node x1 and any two nodes among {x2, x3, x4} (see Fig. 1h, i for amore complex example).To apply equations (2) and (3) to an arbitrary network, we need to
know the weight of each link (that is, the aij), which for most real
1Center for Complex Network Research and Departments of Physics, Computer Science and Biology, Northeastern University, Boston, Massachusetts 02115, USA. 2Center for Cancer Systems Biology,Dana-Farber Cancer Institute, Boston, Massachusetts 02115, USA. 3Nonlinear Systems Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. 4Department ofMechanical EngineeringandDepartmentofBrain andCognitiveSciences,Massachusetts Institute of Technology, Cambridge,Massachusetts02139,USA. 5DepartmentofMedicine,BrighamandWomen’sHospital, Harvard Medical School, Boston, Massachusetts 02115, USA.
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networks are either unknown (for example regulatory networks) orare known only approximately and are time dependent (for exampleInternet traffic). Even if all weights are known, a brute-force searchrequires us to compute the rank of C for 2N2 1 distinct combina-tions, which is a computationally prohibitive task for large networks.To bypass the need to measure the link weights, we note that thesystem (A, B) is ‘structurally controllable’26 if it is possible to choosethe non-zero weights inA and B such that the system satisfies equation(3). A structurally controllable system can be shown to be controllablefor almost all weight combinations, except for some pathological caseswith zero measure that occur when the system parameters satisfycertain accidental constraints26,27. Thus, structural controllability helpsus to overcome our inherently incomplete knowledge of the linkweights in A. Furthermore, because structural controllability impliescontrollability of a continuum of linearized systems28, our results canalso provide a sufficient condition for controllability for most non-linear systems3 (Supplementary Information, section III.A).To avoid the brute-force search for driver nodes, we proved that the
minimum number of inputs or driver nodes needed to maintain fullcontrol of the network is determined by the ‘maximum matching’ inthe network, that is, the maximum set of links that do not share startor end nodes (Fig. 1c, f, i). A node is said to be matched if a link in themaximum matching points at it; otherwise it is unmatched. As weshow in the Supplementary Information, the structural controllability
problem maps into an equivalent geometrical problem on a network:we can gain full control over a directed network if and only if wedirectly control each unmatched node and there are directed pathsfrom the input signals to all matched nodes29. The possibility ofdetermining ND, using this mapping, is our first main result. As themaximum matching in directed networks can be identified numer-ically in atmostO(N1/2L) steps30, where L denotes the number of links,the mapping offers an efficient method to determine the driver nodesfor an arbitrary directed network.
Controllability of real networksWe used the tools developed above to explore the controllability ofseveral real networks. The networks were chosen for their diversity:for example, the purpose of the gene regulatory network is to controlthe dynamics of cellular processes, so it is expected to evolve towards astructure that is efficient from a control perspective, potentiallyimplying a small number of driver nodes (that is, small nD;ND/N). In contrast, for the World Wide Web or citation networks con-trollability has no known role, making it difficult even to guess nD.Finally, it might be argued that social networks, given their perceivedneutrality (or even resistance) to control, should have a high nD, as it isnecessary to control most individuals separately to control the wholesystem. We used the mapping into maximummatching to determinetheminimum set of driver nodes (ND) for the networks in Table 1, the
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Figure 1 | Controlling a simple network. a, The small network can becontrolled by an input vector u5 (u1(t), u2(t))
T (left), allowing us to move itfrom its initial state to some desired final state in the state space (right).Equation (2) provides the controllability matrix (C), which in this case has fullrank, indicating that the system is controllable. b, Simple model network: adirected path. c, Maximummatching of the directed path. Matching edges areshown in purple,matchednodes are green andunmatchednodes arewhite. Theuniquemaximummatching includes all links, as none of them share a commonstarting or endingnode.Only the topnode is unmatched, so controlling it yieldsfull control of the directed path (ND5 1). d, In the directed path shown in b, alllinks are critical, that is, their removal eliminates our ability to control thenetwork. e, Small model network: the directed star. f, Maximum matchings of
the directed star. Only one link can be part of the maximum matching, whichyields three unmatched nodes (ND5 3). The three different maximummatchings indicate that three distinct node configurations can exert fullcontrol. g, In a directed star, all links are ordinary, that is, their removal caneliminate some control configurations but the network could be controlled intheir absence with the same number of driver nodes ND. h, Small examplenetwork. i, Only two links can be part of a maximummatching for the networkin h, yielding four unmatched nodes (ND5 4). There are all together fourdifferent maximummatchings for this network. j, The network has one criticallink, one redundant link (which can be removed without affecting any controlconfiguration) and four ordinary links.
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obtained trend defying our expectations: as a group, gene regulatorynetworks display high nD (,0.8), indicating that it is necessary toindependently control about 80% of nodes to control them fully. Incontrast, several social networks are characterized by some of thesmallest nD values, suggesting that a few individuals could in principlecontrol the whole system.Given the important role hubs (nodes with high degree) have in
maintaining the structural integrity of networks against failures andattacks31,32, in spreading phenomena32,33 and in synchronization8,34, itis natural to expect that control of the hubs is essential to control anetwork. To test the validity of this hypothesis, we divided the nodesinto three groups of equal size according to their degree, k (low,mediumand high). As Fig. 2a, b shows for two canonical network models(Erdos–Renyi35,36 and scale-free15,37–39), the fraction of driver nodes issignificantly higher among low-knodes than among the hubs. InFig. 2c,we plot the mean degree of the driver nodes, ÆkDæ, as a function of themean degree, Ækæ, of each network in Table 1 and several networkmodels. In all cases, ÆkDæ is either significantly smaller than or compar-able to Ækæ, indicating that in both real and model systems the drivernodes tend to avoid the hubs.To identify the topological features that determine network con-
trollability, we randomized each real network using a full randomiza-tion procedure (rand-ER) that turns the network into a directedErdos–Renyi random network with N and L unchanged. For several
networks there is no correlation between the ND of the original net-work and the ND of its randomized counterpart (Fig. 2d), indicatingthat full randomization eliminates the topological characteristics thatinfluence controllability. We also applied a degree-preserving rando-mization40,41 (rand-Degree), which keeps the in-degree, kin, and out-degree, kout, of each node unchanged but selects randomly the nodesthat link to each other. We find that this procedure does not alter ND
significantly, despite the observed differences in ND of six orders ofmagnitude (Fig. 2e). Thus, a system’s controllability is to a great extentencoded by the underlying network’s degree distribution, P(kin, kout),which is our second and most important finding. It indicates that ND
is determined mainly by the number of incoming and outgoing linkseach node has and is independent of where those links point.
An analytical approach to controllabilityThe importance of the degree distribution allows us to determine ND
analytically for a network with an arbitrary P(kin, kout). Using thecavity method42–44, we derived a set of self-consistent equations(Supplementary Information, section IV) whose input is the degreedistribution and whose solution is the average nD (or ND) over allnetwork realizations compatible with P(kin, kout), which is our thirdkey result. As Fig. 2f shows, the analytically predicted ND agreesperfectly with ND
rand-Degree (and hence is in good agreement withthe exact value, ND
real), offering an effective analytical tool to study
Table 1 | The characteristics of the real networks analysed in the paperType Name N L nDreal nDrand-Degree nDrand-ER
For each network, we show its type and name; number of nodes (N) and edges (L); and density of driver nodes calculated in the real network (nDreal), after degree-preserved randomization (nDrand-Degree) and afterfull randomization (nDrand-ER). For data sources and references, see Supplementary Information, section VI.
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the impact of various network parameters onND. Although the cavitymethod does not offer a closed-form solution, we can derive thedependence of nD on key network parameters in the thermodynamiclimit (NR‘). We find, for example, that for a directed Erdos–Renyinetwork nD decays as
nD<e{ kh i=2 !4"for large Ækæ. For scale-free networks with degree exponent cin5cout5 c in the large-Ækæ limit38, we have
nD<exp {12
1{1
c{1
! "kh i
# $!5"
which has the same Ækæ dependence as equation (4) in the cR‘ limit.Equation (5) predicts that cc5 2 is a critical exponent for the con-trollability of an infinite scale-free network, as only for c. cc can wecontrol the full system through a finite subset of nodes (that is,nD, 1). For c# cc in the thermodynamic limit, all nodes must beindividually controlled (that is, nD5 1). We note that cc is differentfrom c5 3, which is the critical exponent for a number of networkphenomena driven by the divergence of Æk2æ, fromnetwork robustnessto epidemic spreading31–33,45. To check the validity of the analyticalpredictions, we determined the Ækæ dependence of nD numerically forboth Erdos–Renyi and scale-free networks, confirming the asymp-totic exponential dependence of nD on Ækæ, as predicted by equa-tions (4) and (5). Furthermore, the predicted nD value is in excellent
agreement with the numerical results for c. 3 (Fig. 3d, e). Near c5 2,however, nD as predicted by the cavitymethod deviates from the exactnD value owing to degree correlations that are prominent at cc5 2 andcan be eliminated by imposing a degree cut-off in constructing thescale-free networks39,46 (Supplementary Information, section IV.B).Equation (5) also shows that nD decreases as c increases (for fixed
Ækæ), indicating that nD is affected by degree heterogeneity, repre-senting the spread between the less connected and the more con-nected nodes. We defined the degree heterogeneity as H5D/Ækæ,where D5
PiP
jjki2 kjjP(ki)P(kj) is the average absolute degree dif-ference of all pairs of nodes (i and j) drawn from the degree distri-bution P(k). The degree heterogeneity is zero (H5 0) for networks inwhich all nodes have the same degree, such as the random regulardigraph (Fig. 3a), in which the in- and out-degrees of the nodes arefixed to Ækæ/2 but the nodes are connected randomly. For Ækæ$ 2, thisgraph always has a perfect matching47, which means that a singledriver node can control the whole system (Supplementary Informa-tion, section IV.B1). The degree heterogeneity increases as we movefrom the random regular digraph to an Erdos–Renyi network (Fig. 3b)and eventually to scale-free networks with decreasing c (Fig. 3c).Overall, the fraction of driver nodes, nD, increasesmonotonically withH, whether we keep c (Fig. 3f) or Ækæ (Fig. 3g) constant.Taking these results together, we find that the denser a network is,
the fewer driver nodes are needed to control it, and that small changesin the average degree induce orders-of-magnitude variations in nD.
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Figure 2 | Characterizing and predicting the driver nodes (ND). a, b, Role ofthe hubs in model networks. The bars show the fractions of driver nodes, fD,among the low-, medium- and high-degree nodes in two network models,Erdos–Renyi (a) and scale-free (b), withN5 104 and Ækæ5 3 (c5 3), indicatingthat the driver nodes tend to avoid the hubs. Both the Erdos–Renyi and thescale-free networks are generated from the static model38 and the results areaveraged over 100 realizations. The error bars (s.e.m.), shown in the figure, aresmaller than the symbols. c, Mean degree of driver nodes compared with themean degree of all nodes in real and model networks, indicating that in real
systems the hubs are avoided by the driver nodes. d, Number of driver nodes,ND
rand-ER, obtained for the fully randomized version of the networks listed inTable 1, compared with the exact value, ND
real. e, Number of driver nodes,ND
rand-Degree, obtained for the degree-preserving randomized version of thenetworks shown inTable 1, comparedwithND
real. f, The analytically predicatedND
analytic calculated using the cavity method, compared with NDrand-Degree. In
d–f, data points and error bars (s.e.m.) were determined from 1,000 realizationsof the randomized networks.
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Furthermore, the larger are the differences between node degrees, themore driver nodes are needed to control the system. Overall, networksthat are sparse and heterogeneous, which are precisely the character-istics often seen in complex systems like the cell or the Internet13–16,require the most driver nodes, underscoring that such systems aredifficult to control.
Robustness of controlTo see how robust is our ability to control a network under unavoid-able link failure, we classify each link into one of the following threecategories (Fig. 1d, g, j): ‘critical’ if in its absence we need to increasethe number of driver nodes to maintain full control; ‘redundant’ if itcan be removed without affecting the current set of driver nodes; or‘ordinary’ if it is neither critical nor redundant. Figure 4 shows thedensities of critical (lc5 Lc/L), redundant (lr5 Lr/L) and ordinary(lo5 Lo/L) links for each real network, and indicates that most net-works have few or no critical links. Most links are ordinary, meaningthat they have a role in some control configurations but that thenetwork can still be controlled in their absence.
To understand the factors that determine lc, lr and lo, in Fig. 5a, c, fwe show their Ækæ dependence for model systems. The behaviour of lcis the easiest to understand: for small Ækæ, all links are essential forcontrol (lc< 1). As Ækæ increases, the network’s redundancy increases,decreasing lc. The increasing redundancy suggests that the density ofredundant links, lr, should always increase with Ækæ, but it does not: itreaches amaximum at a critical value of Ækæ, Ækæc, after which it decays.This non-monotonic behaviour results from the competition of twotopologically distinct regions of a network, the core and leaves43.The core represents a compact cluster of nodes left in the networkafter applying a greedy leaf removal procedure48, and leaves are nodeswith kin5 1 or kout5 1 before or during leaf removal. The coreemerges through a percolation transition (Fig. 5b, d): for k, Ækæc,ncore5Ncore/N5 0, so the system consists of leaves only (Fig. 5e).At Ækæ5 Ækæc, a small core emerges, decreasing the number of leaves.For Erdos–Renyi networks, we predict that Ækæc5 2e< 5.436564 inagreement with the numerical result (Fig. 5a, b), a value that coincideswith Ækæ where lr reaches its maximum. Indeed, lr starts decayingat Ækæc because for Ækæ. Ækæc the number of distinct maximum
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Figure 3 | The impact of network structure onthe number of driver nodes. a–c, Characteristicsof the exploredmodel networks. A random-regulardigraph (a), shown here for Ækæ5 4, is the mostdegree-homogeneous network as kin5 kout5 Ækæ/2for all nodes. Erdos–Renyi networks (b) havePoisson degree distributions and their degreeheterogeneities are determined by Ækæ. Scale-freenetworks (c) have power-law degree distributions,yielding large degree heterogeneities. d, Drivernode density, nD, as a function of Ækæ for Erdos–Renyi (ER) and scale-free (SF) networks withdifferent values of c. Both the Erdos–Renyi and thescale-free networks are generated from the staticmodel38 with N5 105. Lines are analytical resultscalculated by the cavity method using the expecteddegree distribution in theNR‘ limit. Symbols arecalculated for the constructed discrete network:open circles indicate exact results calculated fromthe maximum matching algorithm, and plussymbols indicate the analytical results of the cavitymethod using the exact degree sequence of theconstructed network. For large Ækæ, nD approachesits lower bound, N21, that is, a single driver node(ND5 1) in a network of sizeN. e, nD as a functionof c for scale-free networks with fixed Ækæ. Forinfinite scale-free networks, nDR 1 as cR cc5 2,that is, it is necessary to control almost all nodes tocontrol the network fully. For finite scale-freenetworks, nD reaches itsmaximum as c approachescc (Supplementary Information). f, nD as a functionof degree heterogeneity, H, for Erdos–Renyi andscale-free networks with fixed c and variable Ækæ.g, nD as a function ofH for Erdos–Renyi and scale-free networks for fixed Ækæ and variable c. As cincreases, the curves converge to the Erdos–Renyiresult (black) at the corresponding Ækæ value.
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matchings increases exponentially (Supplementary Information, sec-tion IV.C) and, as a result, the chance that a link does not participatein any control configuration decreases. For scale-free networks, weobserve the same behaviour, with the caveat that Ækæc decreases with c(Fig. 5c, d).
Discussion and conclusionsControl is a central issue in most complex systems, but because ageneral theory to explore it in a quantitative fashion has been lacking,little is known about how we can control a weighted, directed net-work—the configuration most often encountered in real systems.Indeed, applying Kalman’s controllability rank condition (equation(3)) to large networks is computationally prohibitive, limiting pre-vious work to a few dozen nodes at most17–19. Here we have developedthe tools to address controllability for arbitrary network topologiesand sizes. Our key finding, thatND is determinedmainly by the degree
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Figure 4 | Link categories for robust control. The fractions of critical (red, lc),redundant (green, lr) and ordinary (grey, lo) links for the real networks namedin Table 1. To make controllability robust to link failures, it is sufficient todouble only the critical links, formally making each of these links redundantand therefore ensuring that there are no critical links in the system.
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Figure 5 | Control robustness. a, Dependence on Ækæ of the fraction of critical(red, lc), redundant (green, lr) and ordinary (grey, lo) links for an Erdos–Renyinetwork: lr peaks at Ækæ5 Ækæc5 2e and the derivative of lc is discontinuous atÆkæ5 Ækæc.b, Core percolation for Erdos–Renyi network occurs at k5 Ækæc5 2e,which explains the lr peak. c, d, Same as in a and b but for scale-free networks.The Erdos–Renyi and scale-free networks38 have N5 104 and the results are
averaged over ten realizations with error bars defined as s.e.m. Dotted lines areonly a guide to the eye. e, The core (red) and leaves (green) for small Erdos–Renyi networks (N5 30) at different Ækæ values (Ækæ5 4, 5, 7). Node sizes areproportional to node degrees. f, The critical (red), redundant (green) andordinary (grey) links for the above Erdos–Renyi networks at the correspondingÆkæ values.
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distribution, allows us to use the tools of statistical physics to predictND from P(kin, kout) analytically, offering a general formalism withwhich to explore the impact of network topology on controllability.The framework presented here raises a number of questions,
answers to which could further deepen our understanding of controlin complex environments. For example, although our analytical workfocused on uncorrelated networks, the algorithmic method wedeveloped can identify ND for arbitrary networks, providing a frame-work in which to address the role of correlations systematically40,49,50.Taken together, our results indicate thatmanyaspects of controllabilitycan be explored exactly and analytically for arbitrary networks if wecombine the tools of network science and control theory, opening newavenues to deepening our understanding of complex systems.
Received 18 November 2010; accepted 16 March 2011.
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Supplementary Information is linked to the online version of the paper atwww.nature.com/nature.
AcknowledgementsWe thank C. Song, G. Bianconi, H. Zhou, L. Vepstas, N. Gulbahce,H. Jeong, Y.-Y. Ahn, B. Barzel, N. Blumm, D. Wang, Z. Qu and Y. Li for discussions. Thiswork was supported by the Network Science Collaborative Technology Alliancesponsored by the US Army Research Laboratory under Agreement NumberW911NF-09-2-0053; the Office of Naval Research under Agreement NumberN000141010968; the Defense Threat Reduction Agency awards WMDBRBAA07-J-2-0035 and BRBAA08-Per4-C-2-0033; and the James S. McDonnellFoundation 21st Century Initiative in Studying Complex Systems.
Author Contributions All authors designed and did the research. Y.-Y.L. analysed theempirical data and did the analytical and numerical calculations. A.-L.B. was the leadwriter of the manuscript.
Author Information Reprints and permissions information is available atwww.nature.com/reprints. The authors declare no competing financial interests.Readers are welcome to comment on the online version of this article atwww.nature.com/nature. Correspondence and requests for materials should beaddressed to A.-L.B. ([email protected]).
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