Control Centrality and Hierarchical Structure in Complex Networks

We introduce the concept of control centrality to quantify the ability of a single node to control a directed weighted network. We calculate the distribution of control centrality for several real networks and find that it is mainly determined by the network’s degree distribution. We show that in a directed network without loops the control centrality of a node is uniquely determined by its layer index or topological position in the underlying hierarchical structure of the network. Inspired by the deep relation between control centrality and hierarchical structure in a general directed network, we design an efficient attack strategy against the controllability of malicious networks.


Introduction
Complex networks have been at the forefront of statistical mechanics for more than a decade [1][2][3][4]. Studies of them impact our understanding and control of a wide range of systems, from Internet and the power-grid to cellular and ecological networks. Despite the diversity of complex networks, several basic universal principles have been uncovered that govern their topology and evolution [3,4]. While these principles have significantly enriched our understanding of many networks that affect our lives, our ultimate goal is to develop the capability to control them [5][6][7][8][9][10][11][12][13][14][15][16][17].
According to control theory, a dynamical system is controllable if, with a suitable choice of inputs, it can be driven from any initial state to any desired final state in finite time [18][19][20]. By combining tools from control theory and network science, we proposed an efficient methodology to identify the minimum sets of driver nodes, whose time-dependent control can guide the whole network to any desired final state [12]. Yet, this minimum driver set (MDS) is usually not unique, but one can often achieve multiple potential control configurations with the same number of driver nodes. Given that some nodes may appear in some MDSs but not in other, a crucial question remains unanswered: what is the role of each individual node in controlling a complex system? Therefore the question that we address in this paper pertains to the importance of a given node in maintaining a system's controllability.
Historically, various types of centrality measures of a node in a network have been introduced to determine the relative importance of the node within the network in appropriate circumstances. For example, the degree centrality, closeness centrality [21], betweenness centrality [22], eigenvector centrality [23,24], PageRank [25], hub centrality and authority centrality [26], routing centrality [27], and so on. Here, we introduce control centrality to quantify the ability of a single node in controlling the whole network. Mathematically, control centrality of node i captures the dimension of the controllable subspace or the size of the controllable subsystem when we control node i only. This agrees well with our intuitive notion about the ''power'' of a node in controlling the whole network. We notice that control centrality is fundamentally different from the concept of control range, which quantifies the ''duty'' or ''responsibility'' of a node i in controlling a network together with other driver nodes [28].

Control Centrality
Consider a complex system described by a directed weighted network of N nodes whose time evolution follows the linear timeinvariant dynamics.
where B [ R N|M is an N|M input matrix (MƒN) identifying the nodes that are controlled by the time dependent input vector u(t)( u 1 (t), u 2 (t), Á Á Á , u M (t)) T [ R M with M independent signals imposed by an outside controller. The matrix element b ij [R represents the coupling strength between the input signal u j (t) and node i. The system (1), also denoted as (A,B), is controllable if and only if its controllability matrix C~(B, AB, Á Á Á , A N{1 B) [ R N|NM has full rank, a criteria often called Kalman's controllability rank condition [18]. The rank of the controllability matrix C, denoted by rank(C), provides the dimension of the controllable subspace of the system (A,B) [18,19]. When we control node i only, B reduces to the vector b (i) with a single non-zero entry, and we denote C with C (i) . We can therefore use rank(C (i) ) as a natural measure of node i's ability to control the system: if rank(C (i) )~N, then node i alone can control the whole system, i.e. it can drive the system between any points in the Ndimensional state space in finite time. Any value of rank(C (i) ) less than N provides the dimension of the subspace i can control. In particular if rank(C (i) )~1, then node i can only control itself. The precise value of rank(C) is difficult to determine because in reality the system parameters, i.e. the elements of A and B, are often not known precisely except the zeros that mark the absence of connections between components of the system [29]. Hence A and B are often considered to be structured matrices, i.e. their elements are either fixed zeros or independent free parameters [29]. Apparently, rank(C) varies as a function of the free parameters of A and B. However, it achieves the maximal value for all but an exceptional set of values of the free parameters which forms a proper variety with Lebesgue measure zero in the parameter space [30,31]. This maximal value is called the generic rank of the controllability matrix C, denoted as rank g (C), which also represents the generic dimension of the controllable subspace. When rank g (C)~N, the system (A,B) is structurally controllable, i.e. controllable for almost all sets of values of the free parameters of A and B except an exceptional set of values with zero measure [29,30,32,33]. For a single node i, rank g (C (i) ) captures the ''power'' of i in controlling the whole network, allowing us to define the control centrality of node i as C c (i):rank g (C (i) ): ð2Þ The calculation of rank g (C) can be mapped into a combinatorial optimization problem on a directed graph G(A,B) constructed as follows [31]. Connect the M input nodes fu 1 , Á Á Á , u M g to the N state nodes fx 1 , Á Á Á , x N g in the original network according to the input matrix B, i.e. connect u j to x i if b ij =0, obtaining a directed graph G(A,B) with NzM nodes (see Fig. 1a and b). A state node j is called accessible if there is at least one directed path reaching from one of the input nodes to node j. In Fig. 1b, all state nodes fx 1 , Á Á Á , x 7 g are accessible from the input node u 1 . A stem is a directed path starting from an input node, so that no nodes appear more than once in it, e.g. Fig. 1b. Denote with G s the stem-cycle disjoint subgraph of G (A,B), such that G s consists of stems and cycles only, and the stems and cycles have no node in common (highlighted in Fig. 1b). According to Hosoe's theorem [31], the generic dimension of the controllable subspace is given by with G the set of all stem-cycle disjoint subgraphs of the accessible part of G(A,B) and DE(G s )D the number of edges in the subgraph G s . For example, the subgraph highlighted in Fig. 1b, denoted as G max s , contains the largest number of edges among all possible stem-cycle disjoint subgraphs. Thus, C c (1)~rank g (C (1) )~6, which is the number of red links in Fig. 1b. Note that rank g (C (1) )~6vN~7, the whole system is therefore not structurally controllable by controlling x 1 only. Yet, the nodes covered by the G max s highlighted in Fig. 1b, e.g. fx 1 , x 2 , x 3 , x 4 , x 5 , x 7 g, constitute a structurally controllable subsystem [33]. In other words, by controlling node x 1 with a time dependent signal u 1 (t) we can drive the subsystem fx 1 , x 2 , x 3 , x 4 , x 5 , x 7 g from any initial state to any final state in finite time, for almost all sets of values of the free parameters of A and B except an exceptional set of values with zero measure. In general G max s is not unique. For example, in Fig. 1b we can get the same cycle x 2 ?x 3 ?x 4 ?x 2 together with a different stem u 1 ?x 1 ?x 5 ?x 6 , which yield a different G max s and thus a different structurally controllable subsystem fx 1 , x 2 , x 3 , x 4 , x 5 , x 6 g. Both subsystems are of size six, which is exactly the generic dimension of the controllable subspace. Note that we can fully control each subsystem individually, yet we cannot fully control the whole system.
The advantage of Eq. (3) is that max Gs[G DE(G)D can be calculated via linear programming [34], providing us an efficient numerical tool to determine the control centrality and the structurally controllable subsystem of any node in an arbitrary complex network (see Fig. S1).

Distribution of Control Centrality
We first consider the distribution of control centrality. Shown in Fig. 2 is the distribution of the normalized control centrality (c c (i):C c (i)=N) for several real networks. We find that for the intra-organization network, P(c c ) has a sharp peak at c c~1 , suggesting that a high fraction of nodes can individually exert full control over the whole system (Fig. 2a). In contrast, for company-ownership network, P(c c ) follows an approximately exponential distribution or a very short power-law distribution ( Fig. 2d), indicating that most nodes display low control centrality. Even the most powerful node, with c c *0:01, can control only one percent of the total dimension of the system's full state space. For other networks P(c c ) displays a mixed behavior, indicating the coexistence of a few powerful nodes with a large number of nodes that have little control over the system's dynamics (Fig. 2b,c). Note that under full randomization, turning a network into a directed Erdös-Rényi (ER) random network [35,36] with number of nodes (N) and number of edges (L) unchanged, the c c distribution changes dramatically. In contrast, under degree-preserving randomization [37,38], which keeps the in-degree (k in ) and out-degree (k out ) of each node unchanged, the c c distribution does not change significantly. This result suggests that P(c c ) is mainly determined by the underlying network's degree distribution P(k in , k out ). (Note that similar results were also observed for the minimum number of driver nodes [12] and the distribution of control range [28].) This result is very useful in the following sense: P(k in , k out ) is easy to calculate for any complex network, while the calculation of P(c c ) requires much more computational efforts (both CPU time and memory space). Studying P(c c ) for model networks of prescribed P(k in , k out ) will give us qualitative understanding of how P(c c ) changes as we vary network parameters, e.g. mean degree SkT. See Fig. S7 for more details.

Control Centrality and Topological Features
To understand which topological features determine the control centrality itself, we compared the control centrality for each node in the real networks and their randomized counterparts (denoted as rand-ER and rand-Degree). The lack of correlations indicates that both randomization procedures eliminate the topological feature that determines the control centrality of a given node (see Fig. S2). Since accessibility plays an important role in maintaining structural controllability [29], we conjecture that the control centrality of node i is correlated with the number of nodes N r (i) that can be reached from it. To test this conjecture, we calculated N r (i) and C c (i) for the real networks shown in Fig. 2, observing only a weak correlation between the two quantities (see Fig. S3). This lack of correlation between N r (i) and C c (i) is obvious in a directed star, in which a central hub (x 1 ) points to N{1 leaf nodes (x 2 , Á Á Á , x N ) (Fig. 1c). As the central hub can reach all nodes, N r (1)~N, suggesting that it should have high control centrality. Yet, one can easily check that the central hub has control centrality C c (1)~2 for any N §2 and there are N{1 structurally controllable subsystems, i.e. fx 1 , x 2 g, Á Á Á ,fx 1 , x N{1 g. In other words, by controlling the central hub we can fully control each leaf node individually, but we cannot control them collectively.
Note that in a directed star each node can be labeled with a unique layer index: the leaf nodes are in the first layer (bottom layer) and the central hub is in the second layer (top layer). In this case the control centrality of the central hub equals its layer index (see Fig. 1c). This is not by coincidence: we can prove that for a directed network containing no cycles, often called a directed acyclic graph (DAG), the control centrality of any node equals its layer index.
Indeed, lacking cycles, a DAG has a unique hierarchical structure, which means that each node can be labeled with a unique layer index (l i ), calculated using a recursive labeling algorithm [39]: (1) Nodes that have no outgoing links (k out~0 ) are labeled with layer index 1 (bottom layer). (2) Remove all nodes in layer 1. For the remaining graph identify again all nodes with k out~0 and label them with layer index 2. (3) Repeat step (2) until all nodes are labeled. As the DAG lacks cycles, each subgraph in the set G of the directed graph G(A,b (i) ) consists of a stem only, which starts from the input node pointing to the state node i and ends at a state node in the bottom layer, e.g. u 1 ?x 1 ?x 2 ?x 4 in Fig. 1d. The number of edges in this stem is equal to the layer index of node i, so rank g (C (i) )~C c (i)~l i . Therefore in DAG the higher a node is in the hierarchy, the higher is its ability to control the system. Though this result agrees with our intuition to some extent, it is surprising at the first glance because it indicates that in a DAG the control centrality of node i is only determined by its topological position in the hierarchical structure, rather than any other importance measures, e.g. degree or betweenness centrality. This result also partially explains why driver nodes tend to avoid hubs [12]. (Note that similar phenomena about have been observed in other problems, e.g. networked transportation [40], synchronization [41] and epidemic spreading [42]).
Despite the simplicity of Eq. (4), we cannot apply it directly to real networks, because most of them are not DAGs. Yet, we note that any directed network has a underlying DAG structure based on the strongly connected component (SCC) decomposition (see Fig. S4). A subgraph of a directed network is strongly connected if there is a directed path from each node in the subgraph to every other node. The SCCs of a directed network G are its maximal strongly connected subgraphs. If we contract each SCC to a single supernode, the resulting graph , called the condensation of G, is a DAG [43]. Since a DAG has a unique hierarchical structure, a directed network can then be assigned an underlying hierarchical structure. The layer index of node i can be defined to be the layer index of the corresponding supernode (i.e. the SCC that node i belongs to) in . With this definition of l i , it is easy to show that C c (i) §l i for general directed networks (see Fig. S6 for more details). Furthermore, for an edge (i?j) in a general directed  [49]. (b) Hyperlinks between weblogs on US politics [50]. (c) Email network in a university [51]. (d) Ownership network of US corporations [52]. In-and out-degree distributions for each network are shown in the insets. See Table 1  network, if node i is topologically ''higher'' than node j (i.e. l i wl j ), then C c (i)wC c (j). Since C c (i) has to be calculated via linear programming which is computationally more challenging than the calculation of l i , the above results suggest an efficient way to calculate the lower bound of C c (i) and to compare the control centralities of two neighboring nodes. Note that if l i wl j and there is no directed edge (i?j) in the network, then in general one cannot conclude that C c (i)wC c (j) (see Fig. S5 for more details).

Attack Strategy
Our finding on the relation between control centrality and hierarchical structure inspires us to design an efficient attack strategy against malicious networks, aiming to affect their controllability. The most efficient way to damage the controllability of a network is to remove all input nodes fu 1 , u 2 , Á Á Á , u M g, rendering the system completely uncontrollable. But this requires a detailed knowledge of the control configuration, i.e. the wiring diagram of G (A,B), which we often lack. If the network structure (A) is known, one can attempt a targeted attack, i.e. rank the nodes according to some centrality measure, like degree or control centrality, and remove the nodes with highest centralities [44,45]. Though we still lack systematic studies on the effect of a targeted attack on a network's controllability, one naively expects that this should be the most efficient strategy. But we often lack the knowledge of the network structure, which makes this approach unfeasible anyway. In this case a simple strategy would be random attack, i.e. remove a randomly chosen P fraction of nodes, which naturally serves as a benchmark for any other strategy. Here we propose instead a random upstream attack strategy: randomly choose a P fraction of nodes, and for each node remove one of its incoming or upstream neighbors if it has one, otherwise remove the node itself. A random downstream attack can be defined similarly, removing the node to which the chosen node points to. In undirected networks, a similar strategy has been proposed for efficient immunization [45] and the early detection of contagious outbreaks [46], relying on the statistical trend that randomly selected neighbors have more links than the node itself [47,48]. In directed networks we can prove that randomly selected upstream (or downstream) neighbors have more outgoing (or incoming) links than the node itself. Thus a random upstream (or downstream) attack will remove more hubs and more links than the random attack does. But the real reason why we expect a random upstream attack to be efficient in a directed network is because C c (i) §C c (j) for most edges (i?j), i.e. the control centrality of the starting node is usually no less than the ending node of a directed edge (see Fig.  S8). In DAGs, for any edge (i?j), we have strictly C c (i)wC c (j). Thus, the upstream neighbor of a node is expected to play a more important or equal role in control than the node itself, a result deeply rooted in the nature of the control problem, rather than the hub status of the upstream nodes.
To show the efficiency of the random upstream attack we compare its impact on fully controlled networks with several other strategies. We start from a network that is fully controlled (rank g (C)~N) via a minimum set of N D driver nodes. After the attack a P faction of nodes are removed, denoting with rank g (C 0 ) the dimension of the controllable subspace of the damaged network. We calculate rank g (C 0 ) as a function of P, with P tuned from 0 up to 1. Since the random attack serves as a natural benchmark, we calculate the difference of rank g (C 0 ) between a given strategy and the random attack, denoted as d~½rank Strategy{j g (C 0 ){rank Random g (C 0 )=N. Obviously, the more negative is d, the more efficient is the strategy compared to a fully random attack. We find that for most networks random upstream attack results in dv0 for 0vPv1, i.e. it causes more damage to the network's controllability than random attack (see Fig. 3b,c,d). Moreover, random upstream attack typically is more efficient than random downstream attack, even though in both cases we remove more hubs and more links than in the random attack. This is due to the fact that the upstream (or downstream) neighbors are usually more (or less) ''powerful'' than the node itself.
The efficiency of the random upstream attack is even comparable to targeted attacks (see Fig. 3). Since the former requires only the knowledge of the network's local structure rather than any knowledge of the nodes' centrality measures or any other global information (i.e. the structure of the A matrix) while the latter rely heavily on them, this finding indicates the advantage of the random upstream attack. The fact that those targeted attacks do not always show significant superiority over the random attacks is intriguing and would be explored in future work. Notice that for the intra-organization network all attack strategies fail in the sense that d is either positive or very close to zero (Fig. 3a). This is due to the fact this network is so dense (with mean degree SkT&58) that we have C c (i)~C c (j)~N for almost all the edges (i?j). Consequently, both random upstream and downstream attacks are not efficient and the C c -targeted attack shows almost the same impact as the random attack. This result suggests that when the network becomes very dense its controllability becomes extremely robust against all kinds of attacks, consistent with our previous result on the core percolation and the control robustness against link removal [12]. We also tested those attack strategies on model networks (see Fig. S9, S10 and S11). The results are qualitatively consistent with what we observed in real networks.

Discussion
In sum, we study the control centrality of single node in complex networks and find that it is related to the underlying hierarchical structure of networks. The presented results help us better understand the controllability of complex networks and design an efficient attack strategy against network control. Due to the duality of controllability and observability [18,19], a similar centrality measure can be defined to quantify the ability of a single node in observing the whole system, i.e. inferring the state of the whole system. The modified digraph G'(A,B) used in solving the linear programming. Dotted and solid lines are assigned with weight w ij~0 and 1, respectively. The maximum-weight cycle partition is shown in red, which has weight 3, corresponding to the generic dimension of controllable subspace by controlling node x 1 or equivalently the control centrality of node x 1 . (TIF) Figure S2 Control centrality of nodes in several realworld networks and their randomized counterparts: rand-ER (red), rand-Degree (green). There are seven SCCs highlighted in different colors. The nodes are colored according to their control centrality. The edge (i?j) is colored in green, red, or blue if C c (i) is larger than, smaller than, or equal to C c (j), respectively. For all edges with l i wl j , we have C c (i)wC c (j). But this is not true for general node pairs fi, jg. (b) The condensation of the network in (a) is a DAG with three layers. Each node in the DAG represents a SCC in the original network. (TIF) Figure S5 Even if a lower node is accessible from a higher node, it is still possible that the control centrality of the higher node is smaller than or equal to the lower one.  Figure S9 Impact of different attack strategies on network controllability. rank Strategy{j g (C 0 ) represents the generic dimension of controllable subspace after removing a P fraction of nodes using strategy-j. The nodes are removed according to 10 different strategies (see text). Both ER and SF networks are generated from the Chung-Lu model with N~10 3 and the results are averaged over 10 random choices of P fraction of nodes with error bars defined as s.e.m. Lines are only a guide to the eye. (TIF) Figure S10 Impact of different attack strategies on network controllability with respect to random attack. d:½rank Strategy{j g (C 0 ){rank Strategy{0 g (C 0 )=N denotes the generic dimension difference of the controllable subspace after removing a P fraction of nodes using strategy-j and random attack. The more negative is d, the more efficient is the strategy compared to a random attack. Symbols are the same as used in Fig. S9. (TIF) Figure S11 Impact of different attack strategies on network connectivity. n lc represents the normalized size of the largest connected component of the network after removing a P fraction of nodes. The nodes are removed according to 10 different strategies (see text). Both ER and SF networks are generated from the Chung-Lu model with N~10 3 and the results are averaged over 10 random choices of P fraction of nodes with error bars defined as s.e.m. Lines are only a guide to the eye. (TIF)