Minimal Edge Addition for Network Controllability

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1 This article has been accepted for pblication in a ftre isse of this jornal, bt has not been flly edited. Content may change prior to final pblication. Citation information: DOI /TCNS , IEEE Minimal Edge Addition for Network Controllability Ximing Chen, Sérgio Peqito, George J. Pappas, and Victor M. Preciado Abstract We address the problem of optimally modifying the topology of a directed dynamical network to ensre strctral controllability. More precisely, given the strctre of a directed dynamical network (i.e., an existing networked infrastrctre), we propose a framework to find the minimm nmber of directed edges that need to be added to the network topology in order to render a strctrally controllable system. Or main contribtion is twofold: (i) we provide a fll characterization of all optimal network modifications, and (ii) we propose an algorithm able to find an optimal soltion in polynomial time. We illstrate the validity of or algorithm via nmerical simlations in random networked systems. I. INTRODUCTION Network control theory provides a plethora of tools to analyze the behavior of dynamical processes taking place in complex networked systems, sch as epidemic otbreaks in hman contact networks [1], information spreading in social networks [2], or synchronization in power systems [3]. The analysis and design of complex networks sing tools from graph theory have gained a growing interest in recent years [4]; in particlar, the classical control problem of steering the state of a dynamical network towards a desired state [5, 6]. However, in many practical scenarios, an exact qantitative description of the edges in the network may not be available de to measrement errors and/or modeling ncertainties [7]. In this scenario, it is still possible to analyze network control problems resorting to tools developed in the context of strctral systems theory [8 11]. Strctral controllability extends the classical controllability concept to the case of networks with ncertain edges. Loosely speaking, a network is strctrally controllable if it is controllable for almost all realizations of edge weights (see Section II for a formal description of this concept). In this context, given a strctrally ncontrollable system, one may be interested in enforcing strctral controllability by either (i) adding actation capabilities to the networked system, or (ii) modifying the topology of the dynamical network by, for example, adding new edges to the network topology. The former case is explored in [12 19]. Briefly, in [12, 18, 19], the athors proposed graph-theoretical algorithms to find the minimm nmber of driving nodes to ensre strctral controllability in complex networks. In [13, 14], Ximing Chen, George J. Pappas and Victor M. Preciado are with the Department of Electrical and Systems Engineering at the University of Pennsylvania, Philadelphia, PA {ximingch,pappasg,preciado}@seas.penn.ed. Sérgio Peqito is with the Department of Indstrial and Systems Engineering at the Rensselaer Polytechnic Institte, Troy, NY goncas@rpi.ed. This work is spported in part by the NSF award CAREER-ECCS the athors complement this work to obtain the minimm nmber of driven nodes in polynomial-time. Sbseqently, the minimm nmber of driven nodes reqired while acconting for actation costs was addressed in [15, 16]. Alternatively, if one seeks the minimm collection of inpts from an a priori defined collection of actation capabilities, then the problem is NP-hard [17]. Notwithstanding, there are several cases when adding actation capabilities to the network is either too expensive or not feasible. Therefore, whenever possible or cost-efficient, one can opt to modify the topology of the dynamical network. This case is the focs of the crrent paper, where we propose a polynomialtime algorithm to determine the minimm nmber of extra connections that mst be added to a given strctral system in order to ensre strctral controllability. In [20], Wang et al. proposed an approach to pertrb the strctre of an ndirected network to ensre strctrally controllability when only one driving node was considered. In [21], Ding et al. stdied a similar problem for directed networks. However, they assmed that all the nodes are already reachable from the driving nodes. Altogh they solved the problem sing a constrained integer program which is, in general, NP-hard [22], they did not discss the complexity of their algorithm. In contrast with previos works, we address the case of arbitrary directed network topologies with any nmber of driving nodes and show that the problem can be solved in polynomial time withot any assmption on reachability. These are the contribtions of this paper: (i) we characterize all possible soltions to the problem of determining the minimm nmber of additional edges reqired to ensre strctral controllability, and (ii) we provide a polynomial-time algorithm to find a soltion sitable for large complex networks. The rest of the paper is organized as follows. A formal description of the problem nder consideration are introdced in Section II. Preliminaries on graph theory and strctral system theory are introdced in Section III. The main reslts are provided in Section IV. In Section V, we illstrate or reslts in several complex network topologies. Finally, conclsions and discssion of ftre research are presented in Section VI. II. PROBLEM STATEMENT The dynamics of a linear networked dynamical system can be described as follows: ẋ(t) = Ax(t) B(t), (1) (c) 2018 IEEE. Personal se is permitted, bt repblication/redistribtion reqires IEEE permission. See for more information.

2 This article has been accepted for pblication in a ftre isse of this jornal, bt has not been flly edited. Content may change prior to final pblication. Citation information: DOI /TCNS , IEEE where x(t) R n is the state vector, (t) R m is the inpt vector, A R n n is the state transition matrix and B R n m is the inpt matrix. In the seqel, we refer to the system (1) by the matrix pair (A, B), and if the system is controllable, we say that the pair (A, B) is controllable. Frthermore, we define Ā {0, 1}n n to be the strctral pattern of A, i.e., Ā ij = 0 if [A] ij = 0, and Āij = 1 otherwise. Similarly, B {0, 1} n m encodes the sparsity pattern of B where B ij = 0 of [B] ij = 0, and B ij = 1 otherwise. We say that the strctral pattern (Ā, B) is strctally controllable if there exists a pair (Â, ˆB) with the same strctral pattern as (Ā, B) that is controllable [23]. Frthermore, if sch pair (Â, ˆB) exists, then almost all possible matrix pairs with the same strctral pattern as (Ā, B) are controllable [23]. In this paper, given a strctrally ncontrollable pair (Ā, B), we are interested in the problem of adding a minimm nmber of entries in Ā to obtain a strctrally controllable system. Intitively, if we add sfficient edges in the network sch that the reslting network is a complete graph, then the reslting system is strctrally controllable, provided that at least one node is actated, i.e., B 0. Nonetheless, adding new edges corresponds, in practice, to bilding new infrastrctre. Therefore, from a design and implementation perspective, one seeks to add the minimm nmber of edges to attain the design objective, which, in or case, consists in ensring strctral controllability. Formally, the problem is described as follows: Problem 1. Given the pair (Ā, B) with B 0, find à = arg min à 0 (2) à {0,1} n n s.t. (Ā Ã, B) is strctrally controllable, where à 0 denotes the nmber of non-zero entries in a matrix Ã, and the operator : {0, 1}n n {0, 1} n n {0, 1} n n is the element-wise exclsive-or for binary matrices. If (Ā Ã, B) is strctrally controllable, we refer to the matrix à as a feasible edge-addition matrix, and to à in (2) as the optimal edge-addition matrix. As part of the soltion proposed in this paper, we provide a characterization of all possible optimal edge-addition matrices by resorting to graph-theoretical tools. Frther, we provide a polynomialtime algorithm to obtain one sch soltion. III. NOTATION AND PRELIMINARIES In the rest of the paper, S denotes the cardinality of a set S. Let G = (V, E) denote a directed graph with vertex-set V = {1,..., n}, and edge-set E V V. Given an edge (i, j) E, we say that the tail vertex i is pointing towards the head vertex j, which we denote by i j. A path of length K in G is defined as an ordered seqence of distinct vertices (v 0, v 1,..., v K ) with v k V and (v k, v k1 ) E for all k = 0,..., K1. A cycle is either a path (v 0, v 1,..., v K ) with an additional edge (v K, v 0 ), or a vertex with an edge to itself (i.e. self-loop). A vertex v 2 V is reachable from v 1 V if there exists a path in G from v 1 to v 2. A directed graph G s = (V s, E s ) is a sbgraph of G if V s V and E s E. In particlar, if V s = V, then G s is said to span G. Given a vertex set S V, we define the S-indced sbgraph of G by S as G S = (S, E S ), where E S = E (S S). A graph is said to be strongly connected if there exists a path between any two vertices in the graph. A strongly connected component (SCC) is a maximal sbgraph G s that is strongly connected. A condensation of G is a directed acyclic graph (DAG) generated by representing each SCC in G as a virtal vertex in the condensation and a directed edge between two virtal vertices in the condensation exists, if and only if, there exists a directed edge connecting the corresponding SCCs in G [24]. An SCC is said to be linked if it has at least one incoming/otgoing edge from another SCC. In particlar, a sorce SCC has no incoming edges from another SCC and a sink SCC has no otgoing edges to another SCC. Given a directed graph G = (V, E) and two vertex sets S 1, S 2 V, we define the (ndirected) bipartite graph B(S 1, S 2, E S1,S 2 ) as a graph, whose vertex set is S 1 S 2 and edge set 1 E S1,S 2 = {{s 1, s 2 } E : s 1 S 1, s 2 S 2 }. Given B(S 1, S 2, E S1,S 2 ), a matching M is a set of edges in E S1,S 2 that do not share vertices, i.e., given edges e = {s 1, s 2 } and e = {s 1, s 2}, e, e M only if s 1 s 1 and s 2 s 2. The vertex v is said to be right-nmatched (resp., left-nmatched) with respect to a matching M associated with B(S 1, S 2, E S1,S 2 ) if v S 2 (resp., S 1 ), and v does not belong to an edge in the matching M. A matching is said to be maximm if it is a matching with the maximm nmber of edges among all possible matchings. Additionally, a matching is called a perfect matching if it does not contain right-nmatched vertices. Given a bipartite graph B(S 1, S 2, E S1,S 2 ), the maximm matching problem can be solved efficiently in O( S 1 S 2 E S1,S 2 ) time [24]. Given a strctral pair (Ā, B), we associate a directed graph G(Ā, B) = (X U, E X,X E U,X ), which we refer to as the system digraph, where X = {,..., x n } and U = { 1,..., m } denote the set of state vertices and inpt vertices, and E X,X = {(x i, x j ): [Ā] ji 0} and E U,X = {( j, x i ): [ B] ij 0} denote its edge sets. In the remaining of the paper, nless otherwise specified, a state vertex being reachable means that it is reachable from some inpt vertex. Similarly, a vertex set is reachable if every vertex in the set is reachable. Also, de to the graph representation of the pair (Ā, B), when (Ā, B) is strctral controllable, we interchangeably say that G(Ā, B) is strctrally controllable. In addition, we can associate an ndirected bipartite graph with G(Ā, B), called the system bipartite graph and denoted by B(Ā, B) = B(X U, X, E X,X E U,X ), in which {x i, x j } E X,X if (x i, x j ) E X,X, and { i, x j } E U,X if ( i, x j ) E U,X. Sbseqently, for ease of notation, we se a signalnotation mapping s: E X,X E U,X E X,X E U,X to 1 We denote ndirected edges sing crly brackets {v i, v j }, in contrast with directed edges, for which we se parenthesis (c) 2018 IEEE. Personal se is permitted, bt repblication/redistribtion reqires IEEE permission. See for more information.

3 This article has been accepted for pblication in a ftre isse of this jornal, bt has not been flly edited. Content may change prior to final pblication. Citation information: DOI /TCNS , IEEE map edges from the system digraph into edges of the system bipartite graph, as follows: s(( i, x j )) = { i, x j } and s((x i, x j )) = {x i, x j }. In addition, de to the bijectivity of the signal-notation mapping, we have that s 1 ({ i, x j }) = ( i, x j ) and s 1 ({x i, x j }) = (x i, x j ). The concepts introdced in this section can be sed to determine if a strctral system is strctrally controllable, as follows: Theorem 1 ([11, 13]). The pair (Ā, B) is strctrally controllable if and only if the following two conditions hold: (a) every state vertex x X in the system digraph G(Ā, B) = (X U, E X,X E U,X ) is reachable (from some inpt vertex U); (b) any maximm matching M of the system bipartite graph B(Ā, B) = B(X U, X, E X,X E U,X ) has no right-nmatched vertices. Notice that both conditions in Theorem 1 can be verified in polynomial time [11]. Hence, one cold naively try to ensre both conditions by adding edges iteratively, bt sch an approach is, in general, non-optimal and does not provide optimality garantees. IV. MINIMUM TOPOLOGICAL CHANGES TO ENSURE STRUCTURAL CONTROLLABILITY In this section, we provide the main reslts of the paper. First, in Section IV-A, we reformlate Problem 1 as a graph-theoretical problem. Next, in Section IV-B, we sharpen or intition by exploring two particlar network topologies. In Section IV-C, we show that iterative soltions are sb-optimal. Next, sing graph-theoretical tools, we characterize the set of feasible soltions to Problem 1 (Theorem 2). Sbseqently, we obtain a feasible soltion containing the minimm nmber of additional edges to ensre strctral controllability (Theorem 3). Finally, we provide a polynomial-time algorithm (Algorithm 3) to obtain an optimal soltion to Problem 1, whose correctness and comptational complexity are proved in Theorem 4. A. Graph-Theoretical Optimization Problem At a first glance, Problem 1 may seem a prely combinatorial problem. Naively, one may find a soltion by exhastively exploring the set of n n binary matrices. However, Theorem 1 can be leveraged to shrink the search domain of (2). This motivates s to recast (2) as the following graph-theoretical problem. Recall that the system digraph is given by G(Ā, B) = (X U, E X,X E U,X ). Therefore, given a feasible edgeaddition matrix Ã, we can associate a digraph with the pertrbed strctral system (ĀÃ, B), which we denote by G(ĀÃ, B) = (X U, E X,X E U,X Ẽ), where the edge set Ẽ X X is sch that (x i, x j ) Ẽ if and only if Ãji = 1. Sbseqently, since there is an one-to-one correspondence between Ẽ and the strctral matrix Ã, we can provide the following eqivalent formlation of Problem 1: Problem 2. Given the system digraph G(Ā, B) = (X U, E X,X E U,X ), find Ẽ = arg min Ẽ X X Ẽ s.t. G(Ā Ã, B) = (X U, E X,X E U,X Ẽ) is strctrally controllable. Additionally, we define a feasible edge-addition configration as a set of edges that is a feasible soltion of Problem 2. Also, an optimal edge-addition configration is defined as an optimal soltion of Problem 2. B. Special Cases Next, before showing that iterative strategies can be sboptimal, we discss two special cases to sharpen or intition. First, recall that according to Theorem 1, the pair (Ā, B) is strctrally controllable, if and only if, two conditions are satisfied. Therefore, we explore two special cases, where in each case only one of the conditions in Theorem 1 is satisfied; hence, only the remaining condition needs to be ensred to attain feasibility. Case I: Consider a strctred system (Ā, B) sch that only Condition (a) in Theorem 1 holds, while Condition (b) is not satisfied. In other words, all state vertices are reachable while there exists a maximm matching of the system bipartite graph with right-nmatched vertices. As a reslt, the cardinality of a maximm matching M with respect to B(Ā, B) is strictly less than n. Sbseqently, let s denote by U L = {vi l : i {1,..., n l}} and U R = {vi r : i {1,..., n r}} the left- and rightnmatched vertices associated with a maximm matching M, respectively. In particlar, notice that n l n r since X U X, and M = n n r. Therefore, to ensre that G(Ā Ã, B) is strctrally controllable, it is sfficient to add edges between U L and U R withot common end-points and sch that all right-nmatched vertices belong to one of sch edges. However, sch approach is not necessarily a soltion to Problem 2 since some of the newly considered edges may correspond to edges between inpt and state vertices, while we are only allowed to connect pairs of state vertices. Conseqently, let (withot loss of generality) UL X = {vl i : i {1,..., n r}} U L be the set of n r left-nmatched state vertices. Therefore, an optimal edge-addition configration can be obtained as E = {(vi l, vr i ): vl i U L X, vr i U R, i {1,..., n r }}. In other words, M E is a maximm matching with respect to the bipartite graph B(Ā Ã, B) withot right-nmatched vertices, which implies that Theorem 1-(b) holds. Ths, Ẽ = {s 1 ({vi l, vr i })): {vl i, vr i } E } is an optimal soltion to Problem 2. Since there may exist mltiple maximm matchings of the system bipartite graph, the optimal edgeaddition configration constrcted sing the above procedre may not be niqe. However, the nmber of right-nmatched vertices are the same for all maximm matchings de to maximality. As a reslt, in this case, all optimal edgeaddition configrations contain n r edges (c) 2018 IEEE. Personal se is permitted, bt repblication/redistribtion reqires IEEE permission. See for more information.

4 This article has been accepted for pblication in a ftre isse of this jornal, bt has not been flly edited. Content may change prior to final pblication. Citation information: DOI /TCNS , IEEE Remark 1. Under the assmption that all state vertices in the system digraph are reachable, Problem 1 can be also solved via an integer program, as proposed in [21]. Case II: Sppose that a network (Ā, B) is sch that Condition (b) in Theorem 1 holds, while Condition (a) does not, i.e., some state vertex might be nreachable in G(Ā, B). Since at least one state vertex is assmed to be actated (i.e., B 0), the set of reachable state vertices is nonempty. Therefore, we propose to partition the state vertices of the system digraph into two disjoint sets according to their reachability. Let R 1 and N be the sets containing all the reachable and nreachable state vertices, respectively. Then, we define G r (respectively, G ) as the R 1 -indced (respectively, N -indced) sbgraph. Now, notice that if an edge is added to ensre the reachability of any vertex v in some sorce SCC in G (i.e., the tail of the edge is a reachable state vertex), then all state vertices reachable from this particlar sorce SCC become reachable as well. Conseqently, to ensre reachability of all state vertices, it is sfficient to add edges to ensre reachability of one vertex per each nreachable sorce SCCs. Additionally, it is also necessary to have an edge pointing towards each sorce SCC in G, since otherwise the vertices belonging to it remain nreachable. Therefore, we first need to identify the sorce SCCs in the DAG associated with the nreachable sbgraph G (these sorce SCCs can be efficiently fond sing, for example, [24]). Also, withot loss of generality, assme there are r of these sorce SCCs, whose vertex sets are denoted by S j N, j = 1,..., r. Sbseqently, to ensre the reachability of all state vertices in N, we need to add r edges which tails are in a reachable vertex and each head points towards one of the vertices in one of the r sorce SCCs. Ths, the set Ẽ = {(v r, v j ): v r R 1, v j S j, j {1,..., r}} is an optimal edge-addition configration. Notwithstanding, notice that Ẽ does not characterize all possible optimal edge-addition configrations, since when an edge is added from a reachable vertex towards an nreachable sorce SCC, all state vertices reachable from this particlar sorce SCC become reachable; ths, the tail of an edge in an optimal edge-addition configration shold be in R 1 and its head shold be in an nreachable sorce SCC. From Case I, we notice that selecting new edges for the edge-addition configration do not increase the nmber of right-nmatched vertex associated with the system bipartite graph. Similarly, adding more edges never decreases the nmber of reachable state vertices in the system digraph. As a conseqence, one may select edges to ensre both conditions in Theorem 1 are satisfied iteratively. Nonetheless, sch a selection scheme often leads to sb-optimal soltions, as we show next. C. Iterative Soltions are Sb-Optimal In order to motivate the need for an algorithm that solves a general instance of the problem proposed in Problem 2, we describe below a naive iterative approach leading to sboptimal soltions. The steps in this iterative algorithm are (c) 2018 IEEE. Personal se is permitted, bt repblication/redistribtion reqires IEEE permission. See for more information. (a) (b) (c) (d) Figre 1: In (a), we illstrate a system digraph G({,, }, {(, )}) with three vertices and one edge depicted in black. The goal is to find the smallest sbset of state edges (depicted by red edges) to ensre strctral controllability. Let s consider the iterative strategy described in Sbsection IV-C. In (b), we depict a possible soltion to the first step described in Case I, i.e., the edge (, ) sffices to satisfy Theorem 1-(b). In (c), we depict a possible soltion to the second step described in Case II when the system digraph considered is the one depicted in (b). In contrast, the edge (, ) sffices to satisfy Theorem 1-(a), reslting in the system digraph in (d). (a) (b) (c) (d) Figre 2: In (a), we illstrate a system digraph G({,,, }, {(, )}) in black. The goal is to find the smallest sbset of state edges (depicted by red) to ensre strctral controllability. Let s consider the iterative strategy described in Sbsection IV-C. In (b), we depict a possible soltion to the first step described in Case II. In (c), we depict a possible soltion to the second step, which was compted by performing the soltion described in Case I when the system digraph considered is the one depicted in (b). In contrast, the edge (, ) sffices to satisfy Theorem 1-(b), reslting in the system digraph in (d). based on the cases described in Section IV-B. Specifically, each iteration consists of a two-stage process. In the first stage, we find the minimm nmber of edges reqired to satisfy Theorem 1-(b) sing the methodology described in Case I. The second stage in each iteration is described in Case II, whose aim is to satisfy Condition (a) in Theorem 1. To show how this iterative approach can lead to sboptimal soltions, we show in Figre 1 an instance where we initially se the method proposed in Case I to ensre that Theorem 1-(b) holds, followed by the method proposed in Case II is applied to ensre Theorem 1-(a). As we explain in the caption of Figre 1, the naive strategy reqires two edges, whereas the digraph depicted in Figre 1-(d) is also feasible and reqires only one edge. Alternatively, in Figre 2, we provide an instance where the strategy adopted aims first to ensre Theorem 1-(a), followed by Theorem 1- (b), sing the soltions in Case II and Case I, respectively. Again, in this case, the naive strategy reqires three edges, whereas the digraph depicted in Figre 2-(d) is also feasible and reqires only two edges. In smmary, naive strategies are (in general) sb-optimal. D. General Case Hereafter, we characterize the soltions to Problem 2 when no assmptions are made on the topology of the

5 This article has been accepted for pblication in a ftre isse of this jornal, bt has not been flly edited. Content may change prior to final pblication. Citation information: DOI /TCNS , IEEE network. First, we introdce a definition reqired to characterize the smallest collection of edges needed to attain reachability, i.e., satisfy Condition (a) in Theorem 1. In order to introdce this definition, we need to define the following notation. Let G(Ā, B) = (X U, E X,X E U,X ) be the system digraph, and partition the set of state vertices X into two sets based on their reachability (from an inpt), namely, X = R 1 N, where R 1 is the set of reachable vertices and N is the set of nreachable vertices. Additionally, withot loss of generality, let s assme there are r sorce SCCs that are nreachable, which vertex sets are denoted by N 1,..., N r N. Also, let (N h ) denote the set of vertices that are reachable in G(Ā, B) from the vertices in N h, for h = 1,..., r. Definition 1. A set S B is called a set of bridging edges if it can be generated by the following recrsive algorithm: Algorithm 1 Set of bridging edges Inpt: Sets R 1 and N 1,..., N r ; 1: Initialize K = {1,..., r}, t 1 as any vale in K, and the set S B to contain a single edge (i, j) where i is any vertex in R 1 and j is any vertex in N t1 ; 2: for k = 2 : r do 3: R k R k1 (N tk1 ); 4: Assign t k to any vale in K \ k1 h=1 {t h}; 5: S B S B {(i, j)} for any i R k and any j N tk ; 6: end for Algorithm 1 is illstrated in Figre 3. In particlar, notice that at the end of this algorithm N = r h=1 (N t k ), which implies that all nreachable states become reachable. Frthermore, notice that the set of bridging edges contains the minimm nmber edges reqired to ensre that all state vertices are reachable. In fact, it readily follows that the soltions to Case II in Section IV-B can be characterized by the possible sets of bridging edges. Frthermore, the set of bridging edges only ensre Condition (a) in Theorem 1, which is not sfficient to ensre strctral controllability in general. More specifically, to ensre strctral controllability and, sbseqently, to obtain a feasible edge-addition configration, two types of edges are reqired: (i) a set of bridging edges, and (ii) edges that connect left-nmatched state vertices to right-nmatched vertices in some maximm matching associated with the system bipartite graph (recall Case I in Section IV-B). In what follows, we state necessary and sfficient conditions to obtain a feasible edge-addition configration: Theorem 2. Let G(Ā, B) be a system digraph and B(Ā, B) be its bipartite representation. Frthermore, let M be a maximm matching associated with B(Ā, B) and U L (M) = {vi l : i {1,..., n l}} and U R (M) = {vi r : i {1,..., n r}} be the left- and right-nmatched vertices of M. Withot loss of generality, let UL X (M) = {vl i : i {1,..., n r}} denotes the set of n r left-nmatched state vertices of M. A set Ẽ is a feasible edge-addition configration if and only if it contains the nion of the following two sets: R 1 e 1 (a) N 1 N 2 e 2 e 1 R 1 (b) N 1 N 2 Figre 3: This figre provides an illstration of Algorithm 1. All vertices (ble or black), together with all black edges, form the initial system digraph G(Ā, B). The black vertices, except the inpt vertex, constitte the set of reachable state vertices R 1 (enclosed by the black dashed ellipsoid). Ble vertices constitte the set of nreachable state vertices N. The nreachable state sorce SCCs, N 1 and N 2, are contained in red dashed sqares. In Figre (a), we depict one possible reslt for Algorithm 1. In the initialization step, or algorithm initializes S B as the set containing edge e 1 only. Sbseqently, after e 1 is added to S B, all the states reachable from N 1 become reachable (we encircle these reachable states by a ble dashed ellipsoid in Figre (a)). Afterwards, in the FOR loop, edge e 2 in Figre (a) is added to S B (in Step 5 of Algorithm 1), reslting in a digraph in which all vertices are reachable from the inpt node. An alternative otpt of Algorithm 1 is plotted in Figre (b). Notice that both in Figres (a) and (b), all vertices are reachable after adding two red edges. Therefore, S B = {e 1, e 2 } and S B = {e 1, e 2 } are two possible sets of bridging edges. (a) S B is the set of bridging edges; and (b) S M = {s 1 ({v l i, vr i }): vl i U X L (M), vr i U R (M), and i = {1,..., n r }}, for some maximm matching M associated with the system bipartite graph. From Theorem 2, we can readily obtain a lower-bond on the nmber of edges in a feasible edge-addition configration. Corollary 1. The cardinality of an optimal edge-addition configration Ẽ satisfies Ẽ max{n r, r}, where n r is the nmber of right-nmatched vertices of any given maximm matching M associated with the system bipartite graph B(Ā, B), and r is the nmber of nreachable state sorce SCCs in the DAG associated with the system digraph G(Ā, B). In particlar, it is easy to verify that the eqality in Corollary 1 is ensred when both special cases addressed in Section IV-B are considered. Althogh Theorem 2 characterizes feasible edge-addition configrations, we seek to find a feasible edge-addition configration of minimm cardinality. To achieve this goal, we notice that it is preferable to obtain a maximm matching whose set of right-nmatched vertices are spread across different nreachable sorce SCCs. This is becase the edges connecting left- to right-nmatched vertices in this particlar maximm matching are sefl to simltaneosly satisfy both Conditions (a) and (b) in Theorem 2. To formalize this reasoning, we introdce the following concept. Definition 2. Let G(Ā, B) be the system digraph and M be a maximm matching associated with its bipartite representation B(Ā, B). Frthermore, denote by U R (M) the set of right-nmatched vertices of M. An nreachable state sorce SCC of the DAG associated with the system digraph e (c) 2018 IEEE. Personal se is permitted, bt repblication/redistribtion reqires IEEE permission. See for more information.

6 This article has been accepted for pblication in a ftre isse of this jornal, bt has not been flly edited. Content may change prior to final pblication. Citation information: DOI /TCNS , IEEE G(Ā, B) is said to be nreachable-assignable if it contains at least one right-nmatched vertex in U R (M). Whether an nreachable state sorce SCC S is nreachable-assignable depends on the specific maximm matching M. In other words, given two sets U R (M 1 ) and U R (M 2 ) of right-nmatched vertices associated with two different maximm matchings M 1 and M 2, it is possible that U R (M 1 ) contains a vertex from S while U R (M 2 ) does not. We introdce the following definition to characterize the maximm nmber of possible nreachable-assignable state sorce SCCs. Definition 3. The nreachable sorce assignability nmber (USAN) of the system digraph G(Ā, B) is defined as the maximm nmber of nreachable-assignable state sorce SCCs among all the maximm matchings associated with the system bipartite graph B(Ā, B). Remark 2. According to Definition 3, for every system digraph G(Ā, B), the USAN mst be less or eqal to the nmber of right-nmatched vertices associated with any maximm matching of the B(Ā, B) and the total nmber of nreachable state sorce SCCs in G(Ā, B). To find a maximm matching associated with the system bipartite graph that attains the USAN, one can naively enmerate all possible maximm matchings associated with B(Ā, B), bt this approach incrs into a problem that is comptationally P -complete 2 [25]. Instead of sing an exhastive search, it is possible to determine in polynomialtime a maximm matching attaining the USAN sing the following algorithm. Algorithm 2 Maximm matching attaining the USAN Inpt: A system digraph G(Ā, B); Otpt: A maximm matching M attaining the USAN; 1: Partition the set of state vertices in the system digraph G(Ā, B) based on their reachability. Obtain the set containing all the nreachable vertices of G(Ā, B), denoted as N, and its N -indced sbgraph, denoted as G. 2: Obtain the sorce SCCs of G and denote their vertex sets as N 1,..., N r, where r is the total nmber of sorce SCCs in G ; 3: Define a vertex set I = {γ 1,..., γ r } comprising r slack vertices. Constrct a weighted bipartite graph B w = B(X U I, X, E X,X E U,X E I), where E I = r i=1 {{γ i, x j }: x j N i }. The weights in B w are as follows: every edge in E X,X E U,X is assigned to have nit weight, whereas every edge in E I has weight two; 4: Let M be the minimm-weighted maximm matching of B w ; 5: Retrn M = M \ E I. 2 The class of P -complete problems is a class of comptationally eqivalent conting problems that are at least as difficlt as the NP-complete problems. (a) γ 1 (b) γ 1 (c) Figre 4: This figre presents an example illstrating Algorithm 2. The black vertices and edges in (a) form the initial system digraph G(Ā, B). In this case, N = {,, } is the set of nreachable state vertices. Moreover, there is only one nreachable sorce SCC, whose vertex set is N 1 = {, }. The black vertices and edges in (b) constitte the original system bipartite graph B(Ā, B), while the ble vertex γ 1 represents a slack variable associated with N 1. In addition, the ble dashed edges {γ 1, } and {γ 1, } together constitte E I. The minimm-weighted maximm matching M of B w is depicted sing red edges in (c). By removing {γ 1, x 2 } E I, we have that M = {{, x 1 }, {x 2, }, {, }} is a maximm matching of B(Ā, B). In (d), we depict in red the edges from the system digraph G(Ā, B) associated with those in the maximm matching M. Notice that is a right-nmatched vertex of M and it is in N 1 ; hence, M is a maximm matching attaining the USAN of G(Ā, B). Remark 3. The proof of correctness of the algorithm described above is very similar to the proof of Theorem 11 in Section VI of [13]. Essentially, in order to find a maximm matching attaining the USAN, we associate a slack vertex γ i with each nreachable sorce SCC N i. We create additional edges from each slack vertex to every state vertex of its corresponding SCC. In other words, we let E I = r i=1 {{γ i, x j }: x j N i }. Next, we set the weights of edges E I higher than the weights of edges in B(Ā, B). With this particlar selection of weights, the minimm-weighted maximm matching M prefers selecting edges in B(Ā, B) to edges in E I. In particlar, edges are selected from E I if it helps to increase the matching. As a conseqence, the vertices that are matched sing edges in E I mst correspond to rightnmatched vertices in the matching M \ E I. Frthermore, these right-nmatched vertices are spread across different nreachable sorce SCCs. Finally, de to maximality of matching, we can ensre that M achieves the USAN. To frther illstrate the algorithm, we present an example in Figre 4. Remark 4. De to maximality, the USAN is niqe for every system digraph G(Ā, B). Nonetheless, there may exist mltiple maximm matchings that attains this vale. Algorithm 2 obtains one particlar soltion. Althogh the maximm matching that achieves the USAN can be efficiently obtained as described in Algorithm 2, this is not sfficient to obtain an optimal feasible edge-addition configration. To illstrate this claim, let s consider the example depicted in Figre 5. In this case, the optimal feasible edge-addition configration depends on the maximm matching achieving the USAN. Specifically, if all the leftnmatched vertices are nreachable state vertices, then, after flfilling Condition (b) in Theorem 2, we shold add extra edges to form a set of bridging edges to ensre Condition (a) in Theorem 2. This wold reslt in a sb-optimal soltion. (d) γ (c) 2018 IEEE. Personal se is permitted, bt repblication/redistribtion reqires IEEE permission. See for more information.

7 This article has been accepted for pblication in a ftre isse of this jornal, bt has not been flly edited. Content may change prior to final pblication. Citation information: DOI /TCNS , IEEE (a) (b) (c) (d) (e) Figre 5: This figre presents two examples where different maximm matchings lead to sets of feasible edge-addition configrations with different cardinalities. The black vertices and edges in (a) form the initial system digraph G(Ā, B). The red edges in (c) and (e) constitte two different maximm matchings associated with B(Ā, B). The red edges in (b) and (d) are direct graph representations of the edges determined by the maximm matchings in (b) and (d), respectively. The edge-set Ẽ 2 = {(, )} (depicted by ble dashed arrows in (d)) is a feasible edge-addition configration, since the addition of (, ) ensres both conditions in Theorem 1. In contrast, in Fig. (b) we also need to add edge (, ) (in addition to (, )) to ensre that Theorem 2-(b) holds, which leads to a feasible edge-addition configration given by Ẽ1 = {(, ), (, )}. Ths, Ẽ 2 is an optimal edge-addition configration with cardinality 1 while Ẽ1 is not. Since B 0 0, one can find a path rooted at an inpt vertex U whose end vertex is some state vertex x X. Ths, x is a left-nmatched vertex in the maximm matching containing the path. Conseqently, it is always possible to obtain a maximm matching associated with B(Ā, B) with at least one reachable left-nmatched state vertex see Proof of Theorem 3 in Appendix A for more details. Moreover, when an edge is added from the reachable left-nmatched vertex to a right-nmatched state vertex in an nreachable sorce SCC, the set of reachable state vertices can be extended. We will se this fact to circmvent the sb-optimality isse mentioned above. In or next reslt, we characterize the relationship between the USAN and the optimal vale to Problem 2: Theorem 3. Given the system digraph G(Ā, B) and its bipartite representation B(Ā, B), if B 0 > 0, then the cardinality of an optimal edge-addition configration p = Ẽ satisfies p = n r r q, (3) where n r is the nmber of right-nmatched vertices in any maximm matching associated with B(Ā, B), r is the nmber of nreachable sorce state SCCs in the DAG associated with G(Ā, B), and q is the USAN. In fact, based on the constrctive proof of Theorem 3 in Appendix A, we propose a procedre (described in Algorithm 3) to find an optimal edge-addition configration in polynomial-time. Briefly, Algorithm 3 consists of the following for main steps: (Step 1) Decompose the system digraph based on the reachability of state vertices. (Step 2) Determine a maximm matching that achieves the USAN; if the obtained maximm matching admits no reachable left-nmatched vertex, then we alter the matching by finding a path rooted at certain inpt vertex. (Step 3) Based on the obtained maximm matching, in order to ensre both conditions in Theorem 2, select the edges from reachable left-nmatched vertices to right-nmatched vertices in nreachable sorce SCCs iteratively. (Step 4) If the system is still not strctrally controllable, then add the smallest collection of edges ensring that both conditions in Theorem 2 hold independently. The correctness and comptational complexity of this procedre are described in the following reslt. Theorem 4. Given the system digraph G(Ā, B) = (X U, E X,X E U,X ), Algorithm 3 provides an optimal soltion to Problem 2. Frthermore, the comptational complexity of Algorithm 3 is O( X U 3 ). Remark 5. The comptational complexity incrred by Algorithm 3 is comparable to that incrred by the algorithms reqired to solve the special cases described in Section IV-B. Specifically, the soltion to Case I can be determined throgh the comptation of a maximm matching, whose comptational complexity is given by O( X U E X,X E U,X ) [24]. Alternatively, the soltion to Case II can be obtained by determining the strongly connected components of the system digraph, which can be obtained by rnning a depth-first search algorithm twice [24] and incrring in O( X U 2 ) comptational complexity. A MATLAB implementation of Algorithm 3 can be fond in [26]. V. SIMULATIONS In this section, we illstrate the se of the main reslts of this paper. In particlar, given a strctrally ncontrollable system, we determine the minimm nmber of additional edges reqired for ensring strctral controllability in a some artificial network models. First, in Section V-A, we provide a pedagogical example captring the otcome of the different steps of Algorithm 1. In Section V-B, we evalate the minimm nmber of edges reqired in the context of large-scale randomly generated networks. A. Illstrative Example Consider the pair (Ā, B), whose system digraph is depicted in Figre 6. Notice that the system is not strctrally controllable since both conditions in Theorem 1 fail to hold. Therefore, additional edges are reqired to ensre strctral controllability. Towards this goal, we invoke Algorithm 3 to obtain an optimal edge-addition configration that solves Problem 1 given (Ā, B). In this algorithm, we need to decompose the system digraph G(Ā, B) according to the reachability of its state vertices. In particlar, the set of reachable state vertices is given by R 1 = {,..., }, while the set of nreachable state vertices is N = {x 5,..., 0 }. Sbseqently, we find the nreachable sorce SCCs, whose vertex sets are denoted by N 1, N 2, and N 3 in Figre 6; hence, the set of states in nreachable sorce SCCs is {x 5, x 7, x 8, 0 }. Step 2 of Algorithm 3 comptes a maximm matching M sing Algorithm 2. In Figre 7- (a), we present in red sch maximm matching, whose set of left-nmatched state vertices and right-nmatched vertices are U X L ( M) = {, x 9 } and U R ( M) = {x 5, 0 }, respectively. Notice that x 5 and 0 belong to two different nreachable sorce SCCs; hence, the nreachable sorce assignability nmber (USAN) eqals two, i.e., q = 2. As (c) 2018 IEEE. Personal se is permitted, bt repblication/redistribtion reqires IEEE permission. See for more information.

8 This article has been accepted for pblication in a ftre isse of this jornal, bt has not been flly edited. Content may change prior to final pblication. Citation information: DOI /TCNS , IEEE Algorithm 3 Compting an optimal edge-addition configration Ẽ to Problem 2 Inpt: The system digraph G(Ā, B); Otpt: An optimal edge-addition configration Ẽ ; Step 1: System digraph decomposition 1: Obtain the set of all reachable (resp. nreachable) state vertices R 1 (resp. N in G(Ā, B). Step 2: Maximm matching attaining the USAN 2: Obtain a maximm matching M associated with B(Ā, B) attaining the USAN q sing Algorithm 2; 3: if UL X ( M) R 1 = then 4: Find v sch that v R 1 and (, v) E U,X \ M; 5: Find ˆv sch that {ˆv, v } M; 6: M ( M \ {{ˆv, v }} ) {{, v }}; 7: else 8: Set M eqal to M; 9: end if Step 3: Add edges to satisfy (a) and (b) in Theorem 2 10: Obtain the niqe set of disjoint paths P = q i=1 P i in the matching M, where the starting vertex of each P i is in some nreachable sorce SCC and the end vertex is a left-nmatched state vertex; We remark that the niqeness of P is a direct conseqence of M being a matching. 11: Constrct two sets of vertices S = {s 1,..., s q } and T = {t 1,..., t q } sch that s i and t i are the starting and ending vertices of each path P i, respectively; 12: Let Ẽ and k 1; 13: if T R 1 = then 14: Select a t 0 sch that t 0 U L X (M) and t 0 R 1 ; 15: for k q do 16: Ẽ Ẽ {(t k1, s k )}; k k 1; 17: end for 18: UL X (M) U L X (M) \ {t 0,..., t q1 }; 19: else 20: Find and apply a permtation of the i indexes associated to the paths P i sch that t 1 R 1 (accordingly, permte the elements in S and T ); 21: for k < q do 22: Ẽ Ẽ {(t k, s k1 )}; k k 1; 23: end for 24: Ẽ Ẽ {(t q, s 1 )}; UL X (M) U L X (M) \ T ; 25: end if 26: U R (M) U R (M) \ S; Step 4: Add extra edges to satisfy Theorem 2 27: for v l UL X (M) do % to satisfy Theorem 2-(b) 28: if U R (M) then 29: Ẽ Ẽ {(v l, v r )}, for some vr U R (M); 30: UL X (M) U L X (M)\v l ; U R(M) U R (M)\vr ; 31: end if 32: end for 33: Constrct a graph G ag = (X U, E X,X E U,X Ẽ ). Let C i, i = 1,..., β, be the vertex-sets of β nreachable sorce SCCs in the DAG of G ag. Additionally, let R ag be the set of all reachable vertices in G ag ; 34: for i = 1 : β do % to satisfy Theorem 2-(a) 35: Ẽ Ẽ {(v i, z i )}, for some v i R ag, z i C i. 36: end for a reslt, by invoking Theorem 3, it follows that an optimal edge-addition configration consists of p = 3 edges. Figre 6: System digraph G(Ā, B) containing a single inpt vertex and ten state vertices {,..., 0 } (depicted in black dots). Black arrows correspond to the edges of G(Ā, B). The dashed ble ellipsoid contains all the reachable state vertices, i.e., R 1 = {,..., }, whereas each red dashed sqare contains an nreachable sorce SCC, whose vertex sets are N 1 = {x 5 }, N 2 = {0 }, and N 3 = {x 7, x 8 }, respectively. Now, notice that is a reachable left-nmatched vertex, i.e., UL X ( M) R 1. Ths, Step 2 of Algorithm 3 sets M eqal to M. To obtain an optimal edge-addition configration Ẽ, we shold add an edge with tail in and head in some right-nmatched nreachable state vertex. According to M, we obtain P = P 1 P 2, where P 1 = {x 5, x 6,, } and P 2 = {0, x 9 }. From P, the set S = {s 1 = x 5, s 2 = 0 } and T = {t 1 =, t 2 = x 9 } are constrcted accordingly. As a reslt, Step 3 in Algorithm 3 adds the edge (, 0 ) to the edgeaddition configration Ẽ. By selecting this edge, all vertices reachable from 0 become reachable. Sbseqently, the algorithm adds (x 9, x 5 ) to Ẽ, after which Condition (b) in Theorem 2 is satisfied, since M Ẽ B is a maximm matching of G(ĀÃ, B) withot right-nmatched vertices, where Ẽ B = {(, x 10 ), (x 9, x 5 )} represents the bipartite representation of the edges in Ẽ in G(Ā Ã, B). Finally, it remains to ensre that every state vertex is reachable, i.e., that Condition (a) in Theorem 2 is satisfied by G(Ā Ã, B). Towards this end, notice that the only remaining nreachable state sorce SCC is given by N 3 = {x 7, x 8 }. Conseqently, it sffices to add (, x 7 ) into Ẽ to ensre their reachability. However, there are mltiple choices of edges to ensre the reachability of N 3. More specifically, instead of adding (, x 7 ) into Ẽ, one can add any edge (x i, x j ) with i {1,..., 6, 10} and j {7, 8} as an alternative. In smmary, an optimal edge-addition configration, i.e., a soltion to Problem 2, is given by Ẽ = {(, 0 ), (x 9, x 5 ), (, x 7 )}, which contains p = 3 edges, as prescribed by Theorem 3. B. Random Networks In this section, we explore the minimm nmber of edges p contained in an optimal edge-addition configration Ẽ reqired to ensre strctral controllability of random networks. We assme that the strctre of Ā is generated sing an Erdős-Renyi model, i.e., [Ā] ij = 1 with probability 0 < p a < 1 for all i, j; 0 otherwise. In or simlations, the size of Ā is assmed to be n = We let c {0.1, 0.3, 0.5, 0.7, 0.9, 1.5, 2, 3, 4} and define p a = c n for every c accordingly. Ths, c represents the average sm (c) 2018 IEEE. Personal se is permitted, bt repblication/redistribtion reqires IEEE permission. See for more information.

9 This article has been accepted for pblication in a ftre isse of this jornal, bt has not been flly edited. Content may change prior to final pblication. Citation information: DOI /TCNS , IEEE 0 x 9 x 8 x 7 x 6 x c = 0. 1 c = 1. 5 c = p b = 0. 1 p b = 0. 5 p b = x 9 x 8 x 7 x 6 x 5 (a) 600 p p (b) Figre 7: This figre shows a maximm matching M obtained sing Step 2 in Algorithm 3. In (a), we depict the system bipartite graph associated with the pair (Ā, B), whose edges are depicted in black and red (edges in red are those in the maximm matching M). In (b), we depict in red the edges from the system digraph G(Ā, B) associated with those in the maximm matching M. of in-degree and ot-degree of each vertex in the graph represented by A. Moreover, we assme B to be a random diagonal matrix with p b n entries eqal to 1, and 0 otherwise, where p b (0, 1) represents the fraction of vertices to be set eqal to 1. With this particlar setp, we examine the vale of p as we vary c and p b, independently. In Figre 8, we plot the empirical average of p (over 10 random realizations). Notice that p decreases as c or p b increase. Intitively, a larger vale of c reslts in a denser state digraph. Ths, both conditions in Theorem 1 are more likely to be satisfied. In other words, the nmber of right-nmatched vertices associated with the maximm matching of the system bipartite graph and the nmber of nreachable state vertices are smaller as c increases. Frthermore, when p b becomes close to one, almost every state vertex is actated by an individal inpt. Ths, (a) in Theorem 1 holds with high probability. Since p = n r rq, it follows that p decreases as c or p b increase. To emphasize the effect of varying p b (respectively, c) on the minimm nmber of additional edges to ensre strctral controllability, we plot in Figre 8-(a) (respectively, Figre 8-(b)) the evoltion of p when c is fixed (respectively, p b is fixed). In Figre 8-(a), we observe that for a reasonably small vale of c (e.g., c = 3), the impact of p b in the size of the optimal edge-addition configration is almost negligible. Intitively, as c increases towards log(n), the nmber of isolated vertices in the random sbgraph indced by state vertices decreases. In particlar, if c log(n), then the state digraph presents a niqe giant strongly connected component [27]. Sbseqently, p is small even when there is only one state being actated by an inpt. Indeed, in or experiment, p = 1.1 when c = 7 and p b = In Figre 8-(b), we observe an almost exponential decrease of p with respect to c. x 5 x 6 x 7 x 9 VI. CONCLUSIONS We have addressed the problem of designing the topology of a networked dynamical system in order to achieve strc- x p b (a) c Figre 8: In this figre, we plot the evoltion of the average vale of p as c and p b vary. In (a), we fix the vale of c and show the evoltion of p verss p b, when p b ranges from 0.1 to 0.8 with step size 0.1. The red, ble, and black lines correspond to c = 0.1, c = 1.5, and c = 3, respectively. In (b), we plot the evoltion of p when c varies in the interval c {0.1, 0.3, 0.5, 0.7, 0.9, 1.5, 2, 3, 4}, while fixing p b. The red, ble, and black lines show the vale of p when p b = 0.1, p b = 0.5, and p b = 0.8, respectively. In both figres, the error bars represent the standard deviation of p. tral controllability. In particlar, given a system digraph, we have developed an efficient methodology to find the minimm nmber of edges that mst be added to the digraph to render a strctrally controllable system. As part of or analysis, we have characterized the set of all possible soltions to this problem, and provided a polynomial-time algorithm to obtain an optimal soltion. Additionally, we have presented scalable algorithms to solve or problem nder additional assmptions that are commonly fond in engineering applications. Finally, we have nmerically illstrated or reslts in the context of random networked systems. In ftre research, we will extend these reslts to the case when the cost of adding a particlar edge is not a fixed vale. Frthermore, since strctral controllability can be achieved by either (1) adding edges to the system or (2) actating more state vertices, we will explore the trade-offs between these two alternative strategies. In certain scenarios, it is of definite practical and theoretical interest to find efficient sb-optimal algorithms with qality garantees. Finally, it wold be interesting to solve the optimal design problem nder consideration when only a sbset of variables is reqired to be nder control. REFERENCES [1] C. Nowzari, V. M. Preciado, and G. J. Pappas, Analysis and control of epidemics: A srvey of spreading processes on complex networks, IEEE Control Systems, vol. 36, no. 1, pp , [2] S. P. Borgatti, A. Mehra, D. J. Brass, and G. Labianca, Network analysis in the social sciences, Science, vol. 323, no. 5916, pp , [3] F. Dörfler, M. Chertkov, and F. Bllo, Synchronization in complex oscillator networks and smart grids, Proceedings of the National Academy of Sciences, vol. 110, no. 6, pp , [4] Y. Li and A. L. Barabási, Control principles of complex systems, Rev. Mod. Phys., vol. 88, p , Sep [5] R. E. Kalman, Mathematical description of linear dynamical systems, Jornal of the Society for Indstrial and Applied Mathematics, Series A: Control, vol. 1, no. 2, pp , [6] F. Pasqaletti, S. Zampieri, and F. 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Rcinski, Random graphs. John Wiley & Sons, 2011, vol. 45. show that S M S B satisfies both conditions in Theorem 1 when the graph G ag (X U, E X,X E U,X S M S B ) is considered. Hereafter, we denote the bipartite representation of G ag by B ag B(X U, X, E X,X E U,X S± M S± B ), where S± M = {s(e): e S M } and S ± B = {s(e): e S B}. To verify Condition (a) of Theorem 1, we decompose the set of state vertices X, into R 1 and N based on their reachability as in Definition 1. Specifically, R 1 contains all the reachable state vertices and N contains all the nreachable state vertices. Since N = r h=1 (N t h ), every state vertex v N mst be contained in some (N th ) for some iteration step h. By the recrsive constrction of the bridging set S B as described in Definition 1, N th is reachable provided that N th1 is also reachable. Ths, we conclde that all v N become reachable in G ag. To verify Condition (b) of Theorem 1, let M be a maximm matching associated with the system bipartite graph. Next, we propose to consider a bipartite graph B M B(X U, E X,X E U,X S ± M ), which is a sbgraph of the bipartite graph B ag. By the constrction of S M, M S M is a matching in B M. Frthermore, it is a maximm matching since it has no right-nmatched vertices in B M. Since B ag has the same set of vertices as B M, it follows that M S M is also a maximm matching associated with B ag. Sbseqently, M S M satisfies Condition (b) in Theorem 1 for the system bipartite graph B ag. Therefore, if S M S B is added to the system digraph, the reslting system is strctrally controllable, which implies that S M S B is a feasible edge-addition configration. Next, we show that if Ẽ is a feasible edge-addition configration, then it mst contain the nion of the two sets as described in the theorem. Assme, by contradiction, that there is no sch S B in Ẽ, then there is a sorce SCC containing only state vertices that is nreachable. This implies that none of its states are reachable, which precldes the Condition (a) in Theorem 1 to hold; hence, a contradiction is attained. On the other hand, assme that for any maximm matchings M associated with B(Ā, B), we have S M \ Ẽ, then there exists at least one rightnmatched vertex corresponding to the head of an edge in S ± M \ M, which precldes Condition (b) in Theorem 1 to hold; hence, a contradiction is attained. Ths, a set Ẽ is a feasible edge-addition configration if and only if it contains S M and S B as sbsets. Proof of Corollary 1. From Theorem 2, any feasible edgeaddition configration contains S M, for some maximm matching M associated with the system bipartite graph, and S B, the bridging edges as sbsets, i.e., Ẽ S M S B. Conseqently, an optimal edge-addition configration shold satisfy Ẽ S M = n r and Ẽ S B = r. APPENDIX A Proof of Theorem 2. First, we show that if the set of edges Ẽ contains S M and S B as sbsets, then it mst be a feasible edge-addition configration. We notice that, given the system digraph G(Ā, B) = (X U, E X,X E U,X ), it sffices to Proof of Theorem 3. Briefly, the proof reqires the following steps. First, we show that an optimal edge-addition configration Ẽ mst satisfy Ẽ n r r q. Then, we constrct a feasible edge-addition configration sch that its cardinality achieves n r r q (c) 2018 IEEE. Personal se is permitted, bt repblication/redistribtion reqires IEEE permission. See for more information.

11 This article has been accepted for pblication in a ftre isse of this jornal, bt has not been flly edited. Content may change prior to final pblication. Citation information: DOI /TCNS , IEEE From Theorem 2, a feasible edge-addition configration mst satisfy Ẽ S M S B. As a reslt, the cardinality of a feasible edge-addition configration shold satisfy Ẽ S M S B, which implies that Ẽ S M S B S M S B. Notice that, S M = n r and S B = r, then Ẽ n r r S M S B. Ths, an optimal edgeaddition configration, which we denote as Ẽ, mst satisfy Ẽ n r r max M,SB S M S B, where the maximm is taken over all possible maximm matchings M of the system bipartite graph and possible bridging sets S B for the system digraph. To obtain the vale of max M,SB S M S B, we recall that maximizing the intersection between S M and S B gives the maximm nmber of right-nmatched vertices across all possible maximm matchings associated with B(Ā, B) in the nreachable sorce SCCs, i.e., the nreachable sorce assignability nmber q, from Definition 3. Therefore, we have that max M,SB S M S B = q, which implies that Ẽ n r r q. Next, we show that there exists a feasible edge-addition configration that achieves p = n r r q, which we approach by constrction. Given the system digraph G(Ā, B), we partition its state vertices based on reachability. Specifically, we denote R 1 as the set of all reachable state vertices and N as the set of all nreachable state vertices. Moreover, we se N 1,..., N r N to denote the vertex sets of r sorce SCCs that are nreachable, as in Definition 1. Frthermore, let G r be the R 1 -indced sbgraph of G(Ā, B). Next, we obtain a maximm matching M that attains the USAN sing Algorithm 2. Withot loss of generality, we assme there are q nreachable-assignable sorce SCCs whose vertex sets are denoted as N 1,..., N q with q r. Let UL X ( M) and U R ( M) be the set of left-nmatched and right-nmatched state vertices associated with M, respectively. We can obtain a digraph G(V(s 1 ( M)), E(s 1 ( M))) from M, where E(s 1 ( M))) = {s 1 (e) : e M} and V(s 1 ( M)), the vertices sed by the edges belonging to E(s 1 ( M)). In particlar, the set of edges E(s 1 ( M)) is spanned by a disjoint nion of paths {P i } i I and cycles {C j } j J, where I and J denote their indices. Frthermore, to constrct an optimal edge-addition configration, we define the following sets according to the correspondence between the maximm matching attaining the USAN q and the path and cycle decomposition captred by G(V(s 1 ( M)), E(s 1 ( M))). Let V L be the set of ending vertices of paths in {P i } i I whose starting vertex is in U. Let S be the set containing q starting vertices corresponding to disjoint paths in {P i } i I and belonging to different nreachable sorce SCCs. Lastly, let S ± = {x i : x i V L }, which by constrction is a sbset of left-nmatched vertices associated with M. Ths, either UL X ( M) S ± or UL X ( M) S ± = holds. We now begin to constrct a feasible edge-addition configration that achieves p nder the assmption that UL X ( M) S ± holds. We first initialize Ẽ to be an empty set. Then, at the initialization (k = 1), we add an edge (v 1, z 1 ) into Ẽ, where v 1 U L X ( M) S ± and z1 is a rightnmatched vertex associated with M in some nreachable sorce SCCs, i.e., z 1 N l for some l {1,..., q}. Since v 1 S±, it follows that v 1 R 1. Conseqently, if we add the edge (v 1, z 1 ) to the system digraph, then the vertex z 1 becomes reachable, which implies that all the state vertices in (N l ) become reachable as well. On the other hand, if z1 U R( M), then there mst exist a path in G(V(s 1 ( M)), E(s 1 ( M))) departing from z 1. In addition, the end of this path is a left-nmatched state vertex v 2 U L X ( M) with v 2 v 1. In particlar, v 2 (N l ) since it is reachable from z 1. Then, we can add another edge departing from v 2 to another right-nmatched vertex z2 in a different nreachable sorce SCC, i.e., to add the edge (v 2, z 2 ) to Ẽ. We iterate this procedre for another q1 steps, i.e., k = 2,..., q, ntil all q nreachable-assignable SCCs become reachable by adding edges into Ẽ. Now, withot loss of generality, let Ẽ = {(v k, z k ): k = 1,..., q}, where v k U L X ( M) and z k U R( M) for all k = 1,..., q, respectively. Nonetheless, there are r q remaining nreachable sorce SCCs, i.e., N q1,..., N r. To ensre reachability of all state vertices, it sffices to add edges from the set of reachable state vertices to each one of the remaining nreachable sorce SCCs. Conseqently, the complementary set of edges to accont in Ẽ is a set of bridging edges containing r edges by Definition 1. However, as implied by Theorem 2, to constrct a feasible edge-addition configration, we still need to inclde S M as a sbset. Towards this service, we notice that q rightnmatched vertices, i.e., those in the nreachable-assignable SCCs, have been matched dring the iterative procedre. Conseqently, it sffices to add n r q edges to ensre that all the remaining right-nmatched state vertices are matched, i.e., those in U R ( M) \ {z1,..., z q }. As sch, we have constrcted a set of edges considered to be added, i.e., Ẽ, that contains a set of bridging edges and S M for the maximm matching M. As a reslt, Ẽ is a feasible edgeaddition configration by Theorem 2. In addition, it contains n r r q edges, which implies that it is an optimal edgeaddition configration the constrction considered in this paragraph leads to Step 4 of Algorithm 3. Next, we discss the case when UL X ( M) S ± =. First, we define G r ± = {x i : x i R 1 } as the set of leftnmatched state vertices in G r. As a conseqence, two particlar cases may happen: either UL X ( M) G ± r = or UL X ( M) G ± r holds. Consider the first case, where UL X ( M) G ± r =, since UL X ( M) S ± =, then the sbgraph of G(V(s 1 ( M)), E(s 1 ( M))) constrained to the vertices in G r consists only of cycles. Therefore, and withot loss of generality, we let c r be the nmber of those cycles, whose set of vertices are denoted as C i, i = 1,..., c r. According to the assmption B 0 0, there exists an edge (, v) E U,X, with U and v V. Additionally, v belongs to the vertex set of some cycle, i.e., v C j for some j c r, which we represent by the ordered seqence (v, v 1,..., v k, v). If we replace the cycle (v, v 1,..., v k, v) by the path (, v, v 1,..., v k ), then the new digraph will correspond to another maximm matching ˆM associated with B(Ā, B) with a reachable left-nmatched state vertex v k. Additionally, UL X ( ˆM) S ±, and, as a reslt, we may redce the case with assmptions UL X ( M) S ± = (c) 2018 IEEE. Personal se is permitted, bt repblication/redistribtion reqires IEEE permission. See for more information.

12 This article has been accepted for pblication in a ftre isse of this jornal, bt has not been flly edited. Content may change prior to final pblication. Citation information: DOI /TCNS , IEEE and ULX (M ) G± r = to the case previosly discssed by constrcting a new maximm matching M this procedre corresponds to steps 3 9 in Algorithm 3. Now, we sppose that ULX (M ) S ± = and ULX (M ) G± r 6= hold simltaneosly. Then, there exists v1 ULX (M ) G± r and vr UR (M ) sch that (vr,..., v1 ) is a path whose edges are associated with those in M throgh a signal-notation mapping. In particlar, vr / U. If vr is not a vertex in some nreachable sorce SCCs, then we may apply the procedre introdced in the case when ULX (M ) S ± 6= to constrct a feasible edgeaddition configration containing p edges. Nonetheless, if vr is a vertex in some nreachable sorce SCCs, then a modification of the iterative constrction mst be adopted. Specifically, recall that previosly, at the basis step of iteration, we add (v1, z1 ) into E, in which z1 Nl is arbitrarily chosen. Now, if z1 is chosen to be eqal to vr, then (v1, vr ) is added into E and follow-p iteration steps cannot be performed since the end of the path starting at z1 is v1. Conseqently, we mst adopt the following modification: if q = 1, then we mst add an edge (v1, vr ) into E ; otherwise, we add an edge (v1, z1 ) into E with z1 UR (M ) being a vertex in some nreachable sorce SCCs and z1 6= vr at the basis step. In other words, when constrcting the first q steps of a feasible edge-addition configration, we force zi UR (M ), zi Nl and zi 6= vr for all i = 1,..., q 1 and zq = vr, whereas the rest of the constrction readily follows as previosly discssed. As sch, we can obtain a feasible edge-addition configration achieving p if ULX (M ) S ± = and ULX (M ) G± r 6= simltaneosly this constrction procedre is smmarized in steps in Algorithm 3. Therefore, we conclde that if kb k0 > 0, we can constrct a feasible edge-addition configration achieving p = nr r q. Proof of Theorem 4. The correctness of the algorithm follows from the proof of Theorem 3. To determine the comptational complexity of the algorithm, we consider the comptational complexity incrred by each one of the major steps in the algorithm. Specifically, Step 1 reqires the comptation of strongly connected components, which can be achieved by applying the depth-first search algorithm twice with complexity O( X U EX,X EU,X ) [24]. Finding a minimm-weighted maximm matching in Step 2 incrs in O( X U 3 ), and can be achieved as described in Algorithm 2, and we can garantee that exists at least one left-nmatched vertex of M that is reachable in O( X ). In Step 3, we iteratively constrct an optimal edge-addition configration as described in the proof of Theorem 3, which can be attained in O( X U ), since it searches over the compted maximm matching and the sorce SCCs in the system digraph. Finally, in Step 4, we add the remaining edges to ensre conditions in Theorem 2, which incrs in O( X ). In smmary, the comptational complexity of Algorithm 3 is dominated by the second step, which implies an overall comptational complexity in O( X U 3 ). Ximing Chen is crrently a Ph.D. stdent in the Electrical and Systems Engineering program at the University of Pennsylvania. He obtained his bachelor s degree in Electronic Engineering with honors research option from the Hong Kong University of Science and Technology in His research interests inclde control and optimization of strctral systems and dynamic processes in large-scale complex networks, with applications in social networks, epidemic spreading processes and transportation systems. Sérgio Peqito (S 09, M 14) is an assistant professor at the Department of Indstrial and Systems Engineering at the Rensselaer Polytechnic Institte. From 2014 to 2017, he was a postdoctoral researcher in the GRASP lab at University of Pennsylvania. He obtained his Ph.D. in Electrical and Compter Engineering from Carnegie Mellon University and Institto Sperior Técnico in 2014, and B.Sc. and M.Sc. in Applied Mathematics from the latter instittion in 2007 and 2009, respectively. Peqito s research consists of nderstanding the global qalitative behavior of large-scale systems from their strctral or parametric descriptions and provide a rigoros framework for the design, analysis, optimization and control of large-scale (realworld) systems. Peqito was awarded the best stdent paper finalist in the 48th IEEE Conference on Decision and Control (2009). Peqito received the ECE Otstanding Teaching Assistant Award from the Electrical and Compter Engineering Department at Carnegie Mellon University, and the Carnegie Mellon Gradate Teaching Award (University- wide) honorable mention, both in Also, received the 2016 O. Hgo Schck Award in the Theory Category. George J. Pappas (F 09) received the Ph.D. degree in electrical engineering and compter sciences from the University of California, Berkeley, CA, USA, in He is crrently the Joseph Moore Professor and Chair of the Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA, USA. He also holds a secondary appointment with the Department of Compter and Information Sciences and the Department of Mechanical Engineering and Applied Mechanics. He is a Member of the GRASP Lab and the PRECISE Center. He had previosly served as the Depty Dean for Research with the School of Engineering and Applied Science. His research interests inclde control theory and, in particlar, hybrid systems, embedded systems, cyberphysical systems, and hierarchical and distribted control systems, with applications to nmanned aerial vehicles, distribted robotics, green bildings, and biomoleclar networks. Dr. Pappas has received varios awards, sch as the Antonio Rberti Yong Researcher Prize, the George S. Axelby Award, the Hgo Schck Best Paper Award, the George H. Heilmeier Award, the National Science Fondation PECASE award and nmeros best stdent papers awards. Victor M. Preciado received his Ph.D. degree in Electrical Engineering and Compter Science from the Massachsetts Institte of Technology. He is crrently the Raj and Neera Singh Assistant Professor of Electrical and Systems Engineering at the University of Pennsylvania. He is a member of the Networked & Social Systems Engineering (NETS) program, the Warren Center for Network & Data Sciences, and the Applied Math and Comptational Science (AMCS) program. He is a recipient of the 2017 National Science Fondation Faclty Early Career Development (CAREER) Award. His main research interests lie at the intersection of Data and Network Sciences; in particlar, in sing innovative mathematical and comptational approaches to captre the essence of complex, high-dimensional dynamical systems. Relevant applications of this line of research can be fond in the context of socio-technical networks, brain dynamical networks, healthcare operations, biological systems, and critical technological infrastrctre (c) 2018 IEEE. Personal se is permitted, bt repblication/redistribtion reqires IEEE permission. See for more information.