Controllability and Fraction of Leaders in Infinite Network
Chinwendu Enyioha, Mohammad Amin Rahimian, George J. Pappas, Ali Jadbabaie
CControllability and Fraction of Leaders in Infinite Networks
C. Enyioha, M. A. Rahimian, G. J. Pappas and A. Jadbabaie † Abstract —In this paper, we study controllability of a networkof linear single-integrator agents when the network size goesto infinity. We first investigate the effect of increasing size byinjecting an input at every node and requiring that networkcontrollability Gramian remain well-conditioned with the in-creasing dimension. We provide theoretical justification to theintuition that high degree nodes pose a challenge to networkcontrollability. In particular, the controllability Gramian for thenetworks with bounded maximum degrees is shown to remainwell-conditioned even as the network size goes to infinity. In thecanonical cases of star, chain and ring networks, we also provideclosed-form expressions which bound the condition number ofthe controllability Gramian in terms of the network size. Wenext consider the effect of the choice and number of leadernodes by actuating only a subset of nodes and considering theleast eigenvalue of the Gramian as the network size increases.Accordingly, while a directed star topology can never be madecontrollable for all sizes by injecting an input just at a fraction f < of nodes; for path or cycle networks, the designer canactuate a non-zero fraction of nodes and spread them throughoutthe network in such way that the least eigenvalue of the Gramiansremain bounded away from zero with the increasing size. Theresults offer interesting insights on the challenges of control inlarge networks and with high-degree nodes. I. I
NTRODUCTION & B
ACKGROUND
The literature on the control of networks is vast and con-tinues to attract much attention amongst diverse communitiesranging from controls and theoretical physics to biology andapplied sciences. In [1] for instance, an interpretation of thecontrollability matrix is presented and applied to networks inbiology for monitoring protein concentrations; while in [2],controllability of Brain networks is investigated.In the control community as well, Pasqualetti et al. in[3] study the problem of controlling complex networks andquantify the difficulty of the control problem as a function ofthe minimum energy control. There, they also derived boundsto analyze the trade-off between control energy and numberof driver nodes. Whereas earlier works started by [4] and latercarried through by Mesbahi, Egerstedt and their collaborators[5], [6] have been focused on Laplacian dynamics, whereleader nodes update their state values based on exogenousinputs and non-leader nodes update their states according totheir relative states with their neighbors. Existing literature oncontrollability of networks has mostly focused on undirectednetworks.In this paper, we consider the problem of controllabilityfor a directed or undirected network of linear single-integrator † All authors are with the Department of Electrical and Systems Engineer-ing, University of Pennsylvania, Philadelphia, PA 19104-6228 USA (email: { cenyioha, mohar, pappasg jadbabai } @seas.upenn.edu ).This work was supported in part by TerraSwarm, one of six centersof STARnet, a Semiconductor Research Corporation program sponsored byMARCO and DARPA, and in part by AFOSR Complex Networks Program. agents and investigate the core challenges of control as net-work size increases. To begin, we assume that each agentis injected with an exogenous control signal and there ourprimary contribution is in bounding the condition number ofthe controllability Gramian in terms of the singular valuesof the network matrix, such that the Gramian remains nu-merically stable with the increasing dimension. In particular,we show that in structures with a bounded maximum degreethe controllability Gramian remains well-conditioned even asthe network size increases. Controllability of large networksand the interplay between structure and degree distribution hasbeen a focus of recent studies [7], [8]. Our results supplementthe existing literature by providing the Gramian conditionnumber as a metric to test controllability with the increasingnetwork size; hence, highlighting the challenges posed by thehigh degree nodes on the network controllability. We next shiftattention to the choice of leaders, i.e. exogenously actuatednodes, in the canonical cases of star, path and cycle networksand point out their main difference with respect to the spectralradius of the controllability Gramian inverse. In particular,while the star network can never be made controllable forall sizes just by selecting a fraction f < of nodes as leaders,in cases of the path and ring networks, one can select a non-zero fraction of nodes and spread them across the network tomaintain controllability with the increasing size.The rest of this paper is organized as follows. The model andproblem formulation are presented in Section II. In Section III,we present our main result on the numerical stability ofthe Gramian with the increasing dimension, and follow upwith illustrations on canonical networks. In section IV, weinvestigate the effect of the ratio and location of designatedleader nodes on the controllability properties of star, pathand cycle networks and with the increasing sizes. Concludingremarks are provided in Section V.II. P RELIMINARIES
A. Network Information Flow Graph
Throughout the paper, R is the set of all real numbers, N is the set of all natural numbers, N ∈ N denotes thenetwork size, and N = { , . . . , N } . Matrices are representedby capital letters, vectors are expressed by boldface lower-caseletters, and the superscript T indicates the matrix transpose.Moreover, for a matrix D , [ D ] ij indicates the element of D which is located at its i − th row and j − th column, and D issymmetric if D = D T . We denote as G = ( N , E ) a (directedor undirected) graph comprising N nodes labeled by N , and E ⊂ N × N the set of edges of G . Agents i and j arecalled neighbors if ( i, j ) ∈ E , graphs are used to capture thenetwork information flow structure and we say that ( i, j ) isan edge from i to j , and represent it by an arrow starting a r X i v : . [ c s . S Y ] O c t rom i and ending at j . Given G , we denote the network(weighted adjacency) matrix of the graph G by A ∈ R N × N ,where the entries of A are such that [ A ] ji = 0 if edge ( i, j ) (cid:54)∈ E . A is a symmetric matrix iff the graph is undirected(symmetric). The eigenvalues of the matrix A are denotedby λ ( A ) ≥ λ ( A ) ≥ . . . ≥ λ N ( A ) , and its the singularvalues are denoted by σ ( A ) ≥ σ ( A ) ≥ . . . ≥ σ N ( A ) andgiven as { σ i ( A ) , i ∈ N } = { λ i ( AA T ) , i ∈ N } . An infinitenetwork is a network G = ( N , E ) , in which N is countablyinfinite so that N ↔ N . A locally M -bounded network is anetwork G = ( N , E ) together with its associated matrix A ,satisfying ∀ j, (cid:80) i ∈N | a ji | < M , and ∀ i, (cid:80) j ∈N | a ij | < M ,where M < ∞ is a bounded constant. B. The Model
We consider a network of N single integrator agents, whichare labeled from to N and whose interaction structure isexpressed by the graph G . We assume discrete-time dynamicsin the interaction of the networked agents and let x i , i ∈ N represent the scalar state of agent i such that the temporalevolution of the agents after a fixed initial time t ∈ N isgiven by: x ( t + 1) = A x ( t ) + B u ( t ) , t > t , t ∈ N , (1)where A is the network (or adjacency) matrix de-scribing the interaction links between agents, x ( t ) =[ x ( t ) , x ( t ) , . . . , x N ( t )] T is the state vector of the nodes, B = I is the input matrix and u ( t ) ∈ R n is an exogenouscontrol input signal injected at each node in the network. Wemake the following assumptions in our modeling. Assumption 1.
The network matrix A is Schur stable; that is,all its eigenvalues are strictly inside the unit circle. Assumption 2.
The input matrix B is an N × N diagonalmatrix, whose diagonal entries consist only of and . Remark 1.
Notably, Assumption 2 is significant in that anysuch choice of matrix B indicates a particular selection ofleader nodes, which are those nodes to which a designer hasaccess and can feed them with control signals. The diagonalstructure of B further implies that the leader nodes are drivenindependently of each other. In particular setting B = I , toimply that the exogenous control input signals are injectedat each node in the network, allows us to investigate thecontrollability properties of the network as reflected throughthe solution of (2) and solely determined by the networkmatrix A . As we shall see, this feature plays a key role inhelping us characterize the influence of network size andmaximum degree on controllability, and is distinct from muchof the existing literature where the notion of driver nodes aretypically considered [3], [9], [10]. Assumption 3.
The network matrix A is locally M -bounded. Remark 2.
It is worth highlighting that stability and control-lability properties of A differ in the sense that while stabilityof A is sensitive to perturbations in the network matrix A , controllability is not. Rather controllability is sensitive tostructural changes. As such, if the given network matrix A in (1) is not stable, it is possible to shift its eigenvalues tomake it Schur stable, by scaling its entries so that they liewithin the unit circle and without affecting its controllabilityproperty. We shall make use of this feature when consideringa family of networks with a particular structure but of varyingsizes, as we can ensure that all network matrices are Schurstable by uniformly scaling all members of the family by somelarge enough constant γ .C. Network Controllability Gramian, its Condition Number,and Relation to Minimum Energy Control The networked system in (1) is controllable if any state x ( t ) can be steered to the zero state = x ( t ) , for somefinite t > t and using an appropriate input signal u ( t ) , t ≤ t ≤ t . This controllability condition for a Schur stable matrix A is equivalent to requiring that the solution to the discreteLyapunov equation AG c A T − G c = − BB T (2)is invertible. The controllability Gramian is the symmetricpositive semi-definite matrix G c that uniquely satisfies (2) andis given by [11, Chapter 6], G c = lim t →∞ G ( t ) , where, G ( t ) = t (cid:88) τ =0 A τ BB T ( A T ) τ . The controllability condition is equivalent to positive-definiteness of G c . The difficulty of control can be quan-tified by the minimum amount of energy required to reacha state x ( t ) = x des from x (0) = , which is equal to x Tdes G − ( t ) x des and can be achieved through the least norminput u ( τ ) = u ∗ ( τ ) given by u ∗ ( τ ) = B T ( A T ) t − − τ G − ( t − x des for all τ ∈ [ t − . However, for A Schur stable perAssumption 1, G ( t ) converges to G c , exponentially fast andfor sufficiently large t , the two matrices can be made arbitrarilyclose. In particular, if G c is nearly singular, then large energyinputs are required to reach those states x des belonging to theeigenspace of its least eigenvalue λ N ( G c ) . This motivates theuse of the minimum eigenvalue of the controllability Gramianin [3], and we adopt the same measure of the worst casecontrol effort when investigating the role of the choice andfraction of leader nodes in Section IV.Moreover, when investigating the problem of network con-trollability with the increasing size, it becomes crucial for large N that computations of G − ( t ) for minimum energy controlremain numerically stable; that is, for G − ( t ) to be well-conditioned as the dimension N increases [12, Chapter III].To this end, we require that the Gramian condition number, κ ( G c ) (cid:44) σ max ( G c ) /σ min ( G c ) , with σ max ( G c ) and σ min ( G c ) being the maximal and minimal singular values of G c , remainbounded uniformly in N . This is especially important when weguarantee that σ min ( G c ) is bounded away from zero by taking B = I , as then even though the network is controllable for anyfinite N , for certain networks as N → ∞ the Gramin conditionnumber grows unbounded. Examples of such networks aretar and complete networks, as shown in Fig. 1. Clearly,
60 65 70 75 80 85 90 95 100020406080100120 Network size G r a m i an c ond i t i on nu m be r , k ( G ) Condition number of a complete graphCondition number of a Star graph
Fig. 1: The plot above depicts how the condition numbers κ ( G c ) of undirected star and complete networks grow un-bounded as the network size N → ∞ .the controllability Gramian for certain networks becomes ill-conditioned as the network size increases. The importanceof condition number for the controllability Gramian and thenetwork control problem is also highlighted in [13], [14]and our main results in the next section provide a sufficientcharacterization of the networks for which κ ( G c ) remainsbounded as N → ∞ .III. C ONTROLLABILITY & B
OUNDED D EGREES
As a key insight, in this section we characterize how theincreasing local degrees in a network hinders its controllabilityproperty. The main result of this section provides a theoreticaljustification to this intuition, resulting in a sufficient conditionfor having a well-conditioned Gramian as network size in-creases. First, we state a lemma bounding the singular valuesof the Controllability Gramian which we use in the sequel.
Lemma 1 ([15]) . Let the matrix A in (1) be asymptoticallystable such that the solution G c = G Tc (cid:31) to (2) exists.Furthermore, let α ≥ . . . ≥ α N be the eigenvalues of G c , β ≥ . . . ≥ β N be the eigenvalues of B , Re ( λ ) ≥ . . . ≥ Re ( λ N ) be the eigenvalues of A , and > σ ≥ . . . ≥ σ N be the eigenvalues of AA T . Then, the eigenvalues of G c areupper and lower bounded by β i + σ N β N − σ N ≤ α i ≤ β i + β σ − σ , ∀ i ∈ N .Proof: The proof is a direct consequence of the Ostroskyinequalities for the eigenvalue of a sum of symmetric matricesand for the eigenvalue of a matrix product. We refer readers to[15, Theorem 3.1] for proof of Lemma 1. Similar and relatedresults are presented in [16] and [17].Observe that since G c = G Tc , its singular values andeigenvalues coincide and Lemma 1 can be used to bound thecondition number of controllability Gramian for the networkmodel given by (1) under the Assumptions 1 to 3. Theorem 2.
Given B = I , together with Assumptions 1and 3 for the network model (1) , the condition number of the controllability Gramian κ ( G c ) is bounded in terms of thesingular values of the network matrix A , as follows: κ ( G c ) = σ ( G c ) σ N ( G c ) ≤ − σ N ( A )1 − σ ( A ) . (3) Proof:
First since B = I , β i = 1 , ∀ i and the spectralbounds of G c from Lemma 1 become − σ N ≤ α i ≤ − σ , ∀ i ∈ N . (4)We can now upper bound κ ( G c ) as in (3), noting that κ ( G c ) is trivially lower bounded by .The bounds in (3) are in terms of the singular values of thenetwork matrix A , and the following result attributed to Schur allows us to uniformly bound σ i ( A ) , ∀ i ∈ N of an adjacencymatrix A , provided that its maximum degree does not scalewith the network size N . Lemma 3. (Schur Bound [18]) Let A be an N × N locally M -bounded network matrix; then its largest singular valuesatisfies σ ( A ) ≤ M .Proof: For all i, j ∈ N , let R i = (cid:80) k ∈N | [ A ] ik | and C j = (cid:80) k ∈N | [ A ] kj | . It follows by the Schur Bound [18], [19], that σ ( A ) ≤ max i ∈N (cid:80) j ∈N | [ A ] ij | C j ≤ max i,j ∈N , [ A ] ij (cid:54) =0 R i C j ,and by locally M -boundedness we get that R i < M and C j Let A be the network matrix corresponding toa locally M -bounded network and γ > M constant. Thecondition number of the Gramian for a network, followingthe dynamics in (1) with network matrix γ A and the inputmatrix B = I , is bounded uniformly in N , whence the Con-trollability Gramian is guaranteed to remain well-conditionedas N → ∞ .Proof: It follows from Lemma 3 that the singular valuesof γ A are bounded above by Mγ < . Replacing the latterinequality in (3) and noting that γ A is Schur stable we get κ ( G c ) ≤ − σ N ( A )1 − σ ( A ) ≤ γ γ − M < ∞ . (5) Remark 3. The result of Theorem 4 is to a great extent anartifact of our methodology. In particular, by taking B = I theminimum eigenvalue α N of G c is lower-bounded by one andaway from zero per (4) . Indeed, setting B = I and allowingfor an input signal to be injected at every node of the networkfactors out the variety of structural and dynamical influencesthat affect the control behavior, whence singling out the effectof network size N . This in turn enables us to highlight the rolef maximum degree, or more generally local boundedness, incontrollability of large networks. Our result shows that thoughfor each finite N the network is controllable, as N goes toinfinity being locally-bounded is a sufficient condition for thecontrollability Gramian to remain well-conditioned. By considering the condition number of the controllabilityGramian, we are able to use bounds on κ ( G c ) to investigatethe effect of network size N , and the limiting behavior as N → ∞ . This idea is explored further in the next subsection,where we consider the cases of star, path and cycle networksand proffer closed form expressions for the upper-bound in(3). A. Bounds on Condition Numbers for Canonical Networks In this subsection, we illustrate our key result on somecanonical graphs. For the cases considered, we computebounds on κ ( G c ) and consider the limit as N → ∞ of κ ( G c ) . In each case, based on the premise of Theorem 4 andper Remark 2, we scale the − adjacency matrices by acommon constant γ to ensure the Schur stability of γ A forevery network in the range of sizes considered. Undirected star graphs on N nodes have eigenvaluesthat are given by λ i = 0 , ∀ i ∈ { , . . . , N − } , and λ N = −√ N − , λ = + √ N − [20]. Based on (5) inthe proof of Theorem 4, we can bound κ ( G c ) as follows: κ ( G c ) ≤ γ γ − ( N − . First, we note that star networks do notsatisfy the premise of Theorem 4, since its maximum degreeis not bounded as N → ∞ . As shown in Fig. 1, a star networkis a perfect archetype of networks that become uncontrollablesince its maximum degree is unbounded as N → ∞ , causingthe condition number of its associated Gramian, κ ( G c ) to growunbounded as N → ∞ . Undirected path graphs have a maximum degree of twothat is constant, hence bounded, as the network size N → ∞ .The eigenvalues of an undirected path network with N nodesare given by λ i = 2 cos (cid:16) iπN +1 (cid:17) , ∀ i ∈ N [20]. Hence, κ ( G c ) for a path network is upper bounded by κ ( G c ) ≤ − (cid:18) ( (cid:98) N/ (cid:99) πN +1 ) γ (cid:19) − (cid:18) ( πN +1 ) γ (cid:19) = γ − (cid:16) (cid:98) N/ (cid:99) πN +1 (cid:17) γ − (cid:16) πN +1 (cid:17) . As N → ∞ , we can see that the upper bound of κ ( G c ) inundirected path graphs, as shown in Fig. 3, is bounded, andapproaches its bound from below. Specifically, as N → ∞ , cos ( πN +1 ) → and cos (cid:16) (cid:98) N/ (cid:99) πN +1 (cid:17) → , so that κ ( G c ) for apath network is essentially upper bounded by γ γ − . Undirected ring graphs remain locally bounded as thenetwork size increases, similarly to undirected path graphs.The eigenvalues of a ring network of size N is given by λ i = 2 cos (cid:16) π ( i − N (cid:17) , i ∈ { , . . . , N } ; hence, the conditionnumber of the controllability Grammian for a ring network isupper bounded as κ ( G c ) ≤ γ − ( (cid:98) N/ (cid:99) πN ) γ − . The behavior of the upper bound on κ ( G c ) is similar to that of the Pathgraph. Shown in Fig. 3, the periodic spikes observed in theplot are due to the term (cid:16) (cid:98) N/ (cid:99) πN (cid:17) . In particular, forlow values of N , the differences in the values of (cid:98) N/ (cid:99) πN arehigher; and as N increases, the differences reduce, resultingin the evening out of the ‘saw-tooth’ observed for low valuesof N ; and as N → ∞ the term ( (cid:98) N/ (cid:99) πN ) approaches .Observe that for both path and ring graphs we get the sameasymptotic bound of γ γ − , also captured by Fig. 3; and indeed,it is to be expected that ring and path networks should behaveincreasingly similar to each other as N → ∞ . Undirected complete graphs do not satisfy the premise ofTheorem 4. In particular, the eigenvalues of an undirectedcomplete network are given by λ = N − , λ i = − , ∀ i ∈ { , . . . , N } . Hence, the condition number of the control-lability Gramian for a complete network is upper bounded by κ ( G c ) ≤ γ − γ − ( N − . Like star networks, complete networksare not locally bounded as N → ∞ . Hence, the sufficientconditions in Theorem 4 are not satisfied and as we observedin Fig. 1, κ ( G c ) for complete graphs grows unbounded withincreasing network size. Directed star networks have a constant condition numberon the controllability Gramian, even though the bound on κ ( G c ) increases unbounded with as N → ∞ . This observationis intuitive, since an application of control input at the centralnode affects other nodes to control the network, implying thatthe network can be controlled with low energy. The squaredsingular values of a directed star networks with edge orien-tation as shown in Fig. 2a are σ i = 0 , ∀ i ∈ { , . . . , N − } and σ N = N − . Substituting these into (3), we have that κ ( G c ) ≤ γ γ − ( N − , where for the range of values that N takes, the sale factor γ is such that it dominates thelargest N , thence for the directed star the bound increases as N increases. Numerical experiments indicate that the actualcondition number of the Gramian associated with the directedstar network is bounded, pointing out that locally-boundednessin Theorem 4 is a sufficient but not necessary condition. Directed path networks have maximum degree that isbounded as N → ∞ . For directed path graphs with edgeorientation shown in Fig. 2b, the squared singular values are σ i = 1 , for i = 1 , . . . , N − and σ N = 0 , which yield anupper bound of κ ( G c ) ≤ γ γ − , applying (3). Observe that thebound is constant; in fact, κ ( G c ) = γ γ − , ∀ N and as N → ∞ ,in directed path networks. Directed ring networks with edge orientation as shown inFig. 2c have squared singular values given as σ i = 1 , ∀ i ∈ N .From (3) we can get the bound κ ( G c ) ≤ , which is constantand in fact binding: κ ( G c ) = 1 , ∀ N and as N → ∞ .IV. C ONTROLLABILITY & F RACTION OF L EADERS IN D IRECTED C ANONICAL S TRUCTURES Thus far, in analyzing the effect of increasing size onnetwork controllability, we have assumed that all nodes areinjected with an input, so that B = I . In this section, westudy how the fraction and spread of leaders in the three 10 20 30 40 50 60 70 80 90 100050100150200250 Network size k ( G ) and i t s uppe r bound k(G) for a Path graphupperbound for Path graphk(G) for a Ring graphupperbound for Ring graph Fig. 3: Gramian condition numbers for ring and path graphs.directed structures (depicted in Fig. 2), affect their control-lability properties. Thence, rather than set the input matrix B = I , we inject the inputs only into a subset of nodes,dubbed leaders. While condition number has been effective ininvestigating the effects of increasing size in Section III, herewe adopt minimum required energy in the worst case capturedby /λ N ( G c ) as the measure of interest for investigating therole of leader nodes.To begin, consider the cases of the star and path network inFigs. 2a and 2b with their respective N × N adjacencies A s and A p given by A s = · · · 01 0 · · · ... ... . . . ... · · · , A p = · · · 01 0 . . . ... . . . . . . ... . . . . . . . . . . We can now replace the scaled adjacencies γ A s and γ A p in(2) and with B given per Assumption 2, we can solve for thecorresponding Gramians G sc and G pc as follows. Directed star networks have a controllability Gramian G sc whose entries are given by [ G sc ] = [ B ] , [ G sc ] ii = γ − [ G sc ] + [ B ] ii , ∀ i > , [ G sc ] j = [ G sc ] j = [ B ] j = 0 , ∀ j > , and [ G sc ] ij = γ − [ G sc ] , ∀ j > , j (cid:54) = i . In particular,all entries on the first row of the Gramian are zero except forthe , entry which is equal to [ B ] . Hence, in order for a star topology to be controllable the designer should always selectthe first (central) node as a leader. Further calculation of theeigenvalues indicate that we alway need to select all but oneperipheral node of the star network to order to have a full rankGramian or a controllable network, λ N ( G sc ) > . Therefore, there is no fraction f < of nodes that can be chosen toensure controllability of a star network, as N → ∞ . Thisbehavior is in sharp contrast with the directed path and cycletopologies analyzed next. In the latter cases, although no finitecollection of leaders can ensure controllability as N → ∞ ,the designer can still select an asymptotically non-vanishingfraction of nodes as leaders and obtain a controllable ring orpath network for all N and as N → ∞ . Directed path networks have a diagonal controllabilityGramian G pc whose diagonal entries are give by [ G pc ] ii = (cid:80) ik =1 γ k − i ) [ B ] kk , ∀ i . The eigenvalues of G pc are the sameas its diagonal entries, and the designer would again need toselect the first (root) node if the system is to be controllable.However, with just the root node as the leader λ N ( G pc ) = γ − N ) → as N → ∞ so that injecting the input just atthe first node cannot ensure the controllability of an infiniteintegrator chain with the increasing length. Indeed, with anyfinite collection of leaders it follows that the distance to theleader nodes grows for the nodes further through the chain andthe minimum eigenvalue of the Gramian would approach zerogeometrically fast as N → ∞ . On the other hand, by selectinga non-zero fraction f of nodes as leaders and spreading themuniformly throughout the chain one can ensure a distanceof at most /f to the closest leader for every node in thechain and the above calculation of the Gramian would thenimply a lower bound of λ N ( G pc ) ≥ γ − /f , which holds evenas N → ∞ . By selecting a non-zero fraction of nodes andspreading them uniformly throughout the network, the designercan ensure the controllability of an infinite integrator chain. The numerical experiments in what follows indicate that thisobservation applies also to the case of networks with directedring topology. Directed ring networks demonstrate an asymptotic behaviorthat resembles that of the path networks as N → ∞ . Here,we investigate the effect of the fraction of leaders on the leasteigenvalue of the Gramian in a ring network of nodes. Tothis end, we first divide the nodes into consecutive blocks of afixed length and with varying number of leader nodes at eachblock. We next consider the effect of varying the block lengthby fixing only one leader at each block and increasing the (a) A directed − node star network (b) A directed − node path network (c) A directed − node ring network Fig. 2: Edge orientations of Directed Graphs consideredlock length. The two experiments in Fig. 4 indicate althoughthe worst case least control effort decreases with the increasingfraction of leader nodes, when a single leader is fixed at eachblock better control can be achieved with a smaller fraction ofleaders, since the leaders are better spread throughout the thenetwork. Indeed, in the extreme case where all the leaders areclustered together then no fraction f < of leaders can ensurecontrollability as N increases. This can be attributed to the factthat even though the number of leader nodes increases withthe network size, when all the leaders are clustered togetherand not spread through the network there will always be somenodes in the network that get arbitrarily far from all the leaderas the network size increases. −1 λ N ( G c ) Fraction of Leaders (f) Fig. 4: Effect of the fraction and spread of leaders for an -node directed cycle network: putting a single leader in eachblock and varying the block lengths over the first divisorsof for the green curve; and fixing block length at andincreasing the number of nodes at each block for blue curve.V. C ONCLUSIONS In this paper, we investigated the controllability of a linearsingle integrator network as the number of nodes increases.We first injected input signals at every node and requiredthe controllability Gramian to remain well-conditioned evenas the network size increases. Accordingly, with a propernormalization that is uniform in the size of the network,the Gramian condition number for graphs with a boundedmaximum degrees was shown to remain bounded, uniformlyin the size. The results provide theoretical insights on thechallenges of controllability for large networks in general,and highlights the role of bounded degrees in particular.Furthermore, we proffered bounds on the condition numberof the controllability Gramian, which in the cases of cycle,path or star topologies were expressible in terms of thenetwork size and could guarantee numerical stability withthe increasing dimension. We next shifted our attention tothe question of choice and number of leader nodes for largenetworks, and showed that while a star topology can neverbe made controllable for all N by selecting any fixed fraction f < of nodes as leaders; in the cases of path and ring networks, by selecting a non-zero fraction of nodes as leadersand having them spread across the network such that no nodesgets arbitrarily far from all leaders, the designer can ensurethat the minimum eigenvalue of the Gramian is bounded awayfrom zero even as the network size increases. This distinctionbetween the star topology and path or rings with respect tothe required asymptotic fraction of leaders for controllabilitywith the increasing size, further highlights the challengesimposed by the high-degree nodes on the controllability oflarge networks. R EFERENCES[1] A. Lombardi and M. H¨ornquist, “Controllability analysis of networks,” Physical Review E , vol. 75, no. 5, p. 056110, 2007.[2] S. Gu, F. Pasqualetti, M. Cieslak, S. T. Grafton, and D. S. Bassett,“Controllability of brain networks,” arXiv preprint arXiv:1406.5197 ,2014.[3] F. Pasqualetti, S. Zampieri, and F. Bullo, “Controllability metrics,limitations and algorithms for complex networks,” Control of NetworkSystems, IEEE Transactions on , vol. 1, no. 1, pp. 40–52, March 2014.[4] H. Tanner, “On the controllability of nearest neighbor interconnections,”in Proceedings of the 43rd IEEE Conference on Decision and Control ,2004, pp. 2467–2472.[5] A. Rahmani, M. Ji, M. Mesbahi, and M. Egerstedt, “Controllability ofmulti-agent systems from a graph-theoretic perspective,” SIAM Journalon Control and Optimization , vol. 48, no. 1, pp. 162–186, 2009.[6] S. Martini, M. Egerstedt, and A. Bicchi, “Controllability analysis ofmulti-agent systems using relaxed equitable partitions,” InternationalJournal of Systems, Control and Communications , vol. 2, no. 1, pp.100–121, 2010.[7] Q. Miao, Z. Rong, Y. Tang, and J. Fang, “Effects of degree correlationon the controllability of networks,” Physica A: Statistical Mechanics andits Applications , vol. 387, no. 24, pp. 6225 – 6230, 2008.[8] M. P´osfai, Y.-Y. Liu, J.-J. E. Slotine, and A.-L. Barab´asi, “Effect ofcorrelations on network controllability,” Scientific Reports , vol. 3, Jan.2013.[9] Y.-Y. Liu, J.-J. Slotine, and A.-L. Barab´asi, “Controllability of complexnetworks,” Nature , vol. 473, no. 7346, pp. 167–173, 2011.[10] M. A. Rahimian and A. G. Aghdam, “Structural controllability of multi-agent networks: Robustness against simultaneous failures,” Automatica ,vol. 49, no. 11, pp. 3149–3157, 2013.[11] C.-T. Chen, Linear System Theory and Design , 3rd ed. New York, NY,USA: Oxford University Press, Inc., 1998.[12] L. N. Trefethen and D. Bau III, Numerical linear algebra . Siam, 1997,vol. 50.[13] N. J. Cowan, E. J. Chastain, D. A. Vilhena, J. S. Freudenberg, andC. T. Bergstrom, “Nodal dynamics, not degree distributions, determinethe structural controllability of complex networks,” PLoS ONE , vol. 7,no. 6, p. e38398, 2012.[14] J. Sun and A. E. Motter, “Controllability transition and nonlocality innetwork control,” Physical Review Letters , vol. 110, p. 208701, May2013.[15] Z. Gaji´c and M. T. J. Qureshi, Lyapunov matrix equation in systemstability and control . DoverPublications. com, 2008.[16] V. Karanam, “Lower bounds on the solution of lyapunov matrix andalgebraic riccati equations,” Automatic Control, IEEE Transactions on ,vol. 26, no. 6, pp. 1288–1290, 1981.[17] T. Mori, N. Fukuma, and M. Kuwahara, “Explicit solution and eigen-value bounds in the lyapunov matrix equation,” Automatic Control, IEEETransactions on , vol. 31, no. 7, pp. 656–658, 1986.[18] I. Schur, “Bemerkungen zur theorie der beschrankten bilinearformenmit unendlich vielen veranderlischen,” Journal fur Reine und Angew,Mathematik , vol. 140, pp. 1–28, 1911.[19] G. H. Golub and C. F. Van Loan, Matrix Computations (3rd Ed.) .Baltimore, MD, USA: Johns Hopkins University Press, 1996.[20] Z. Yuan, C. Zhao, Z. Di, W.-X. Wang, and Y.-C. Lai, “Exact controlla-bility of complex networks,”