PHYSICAL REVIEW E 90, 042804 (2014)
Edge orientation for optimizing controllability of complex networks Yan-Dong Xiao, Song-Yang Lao, Lv-Lin Hou, and Liang Bai College of Information Systems and Management, National University of Defense Technology, Changsha 410073, China (Received 12 July 2014; published 9 October 2014) Recently, as the controllability of complex networks attracts much attention, how to design and optimize the controllability of networks has become a common and urgent problem in the field of controlling complex networks. Previous work focused on the structural perturbation and neglected the role of edge direction to optimize the network controllability. In a recent work [Phys. Rev. Lett. 103, 228702 (2009)], the authors proposed a simple method to enhance the synchronizability of networks by assignment of link direction while keeping network topology unchanged. However, the controllability is fundamentally different from synchronization. In this work, we systematically propose the definition of assigning direction to optimize controllability, which is called the edge orientation for optimal controllability problem (EOOC). To solve the EOOC problem, we construct a switching network and transfer the EOOC problem to find the maximum independent set of the switching network. We prove that the principle of our optimization method meets the sense of unambiguity and optimum simultaneously. Furthermore, the relationship between the degree-degree correlations and EOOC are investigated by experiments. The results show that the disassortativity pattern could weaken the orientation for optimal controllability, while the assortativity pattern has no correlation with EOOC. All the experimental results of this work verify that the network structure determines the network controllability and the optimization effects. DOI: 10.1103/PhysRevE.90.042804
PACS number(s): 89.75.Fb, 89.75.Hc, 02.30.Yy
I. INTRODUCTION
As the networked systems appear around us at an increasing rate, dynamics in complex networks has become one of the most popular research fields [1–4]. The ultimate proof of our understanding of such systems and their basic principle is reflected in our ability to control them [5]. Questions concerning how to manage and control complex networks are important in science, engineering, and biology. Recently, the controllability of complex networks has been a hot topic [5–8], the main point of which is applying the external signals to some nodes so that the system can achieve the desirable state in finite time. Controllability theory offers us a general, rigorous, and well-understood framework for the design and analysis of control systems. The seminal work by Lin [9] provides a graph-theoretical formulation which simplifies and generalizes this problem. A series of work [5,9] focuses on the identification of so-called driver nodes (ND ) using maximum matching, in which the nodes controlled directly by input signals are called driver nodes. Thus, the number of driver nodes is equal to the number of input signals. Controllability is characterized by the minimum number of driver nodes which could offer full control over the network. If a single driver node could fully control the whole network, i.e., ND = 1, we say that the network has perfect controllability. Therefore, the method of reducing ND or even approaching to the state of perfect controllability is of utmost significance in real-world networks. The most representative methods of optimizing of network controllability can be summarized as three methods: adding edges [10], rewiring edges [11], and assigning the direction of edges [12]. Wang et al. [10] describe a perturbation strategy based on adding edges for connecting the separated control path in a proper sequence, which arrives at perfect controllability by adding the least number of edges (only ND − 1 edges). According to the role of different edge categories (critical, ordinary, and redundant edge set) of network controllability, deleting a redundant edge cannot increase the number of driver 1539-3755/2014/90(4)/042804(8)
nodes, and adding an edge may enhance the controllability of the network but never weaken its controllability. Thus the way of rewiring redundant edges is also effective for optimizing controllability [11]. The approach of assigning edge direction to optimize the controllability has been studied by Hou et al. [12]. Compared with the above two methods of structural perturbations, assigning the direction of edges to optimize the network controllability is more practical and economic than that of reconstructing networks in practice. With the view of designing networks having better controllability, this method has the advantage of paying lower costs and being more practical than any other method because it neither changes the original structure of the network nor pays the additional cost for adding links. For example, the installation of new power lines between a pair of power plants sometimes would skyrocket costs and be impractical. However, the effect of edge direction on controllability has not been systematically considered. In the following section, we will give a more detailed analysis of previous work and its shortcomings when using edge orientation to optimize the network controllability. This work is introducing the problem of edge orientation for optimal controllability and the framework for solving this problem. Furthermore, we also investigate the correlations between the network structures and the edge orientation of optimal controllability. The paper is organized as follows. In Sec. II, we first introduce the previous work on this problem. In Sec. III, we systematically study the edge orientation for optimal controllability from the definitions and methodologies. Finally, the correlations between the network structures and the edge orientation of optimal controllability are presented in Sec. IV.
II. PREVIOUS WORKS
Reference [12] presented an efficient heuristic approximation algorithm for assigning edge direction to optimize the
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FIG. 1. (Color online) Illustration of the orientation of node residual degree (NRD). The original network (a) consists of eight nodes and 12 edges. The above control configuration has two driver nodes, while the under configuration has only one driver node. The yellow node is the start node in each orientation process and the red arcs are edge directions oriented by NRD. The gray dashed lines are not yet oriented. If there is more than one suitable choice, NRD randomly selects a node or edge from the network. As node 5 (black node) has two available choices to orient edge, it results in two different numbers of driver nodes, shown as blue nodes in (d). The orientation process of steps (i), (ii), and (iii) are repeated until all nodes and edges are exhausted.
network controllability. This orientation method is based on the node residual degree (NRD) which originates from the optimization of synchronization of complex networks [13,14]. However, controllability is fundamentally different from synchronization. Structural controllability theorem states that directed loops play a significant role in controlling networks, while the original intention of improving synchronization avoids producing loops since the large number of loops is not beneficial to synchronization [13,14]. Therefore, the method of Ref. [12] can be recognized as the opposite of improving synchronization, which aims to generate the directed loops as much as possible. NRD can be described as below: (i) choose the node with the largest residual degree k˜ in the network as V1 ; (ii) choose the node with the largest k˜ in the neighborhood of V1 , denoted as V2 ; (iii) if koi (V1 ) koi (V2 ), then assign the direction from V1 to V2 and the residual degree of the selected nodes will be reduced by 1. Note that koi = kout − kin and randomly pick nodes if there are multiple such nodes. Obviously, this heuristic approximation algorithm is not the optimal orientation of controllability for any given topology of network because of the random feature of selecting nodes or edges. Here is a simple example to demonstrate this conclusion. From Fig. 1, an undirected network consists of eight nodes and 12 undirected edges. The red edges represent the orientation and the yellow node is the start node in each process of assigning edge direction. Both the edge and node could be selected randomly if there are more than one. In Fig. 1(b), as node 5 (black node) has two available choices to orient edge, it could result in a different number of driver nodes finally [blue nodes in Fig. 1(d)]. From Fig. 1(c), although the remaining four edges have different choices of edge direction, they do not influence the control configuration. Since sometimes the nodes and edges are randomly selected, this method may produce different orientation sets with a different
number of driver nodes. Therefore, it is necessary to propose an algorithm to find the optimal orientation set. III. EDGE ORIENTATION FOR OPTIMAL CONTROLLABILITY PROBLEM (EOOC)
According to the above arguments, we formulate the edge orientation for optimal controllability problem (EOOC) as follows. Definition 1. Let G = (V ,E) be an undirected and unweighted network. If there is an edge orientation set (G = (V ,E ),E = {vi → vj |vi ,vj ∈ V ,i,j = 1 · · · n}) for optimal controllability, it corresponds to the least number of driver nodes. According to the maximum matching theorem of Liu’s controllability framework, the optimal orientation set actually produces a maximum matching which covers the maximal number of nodes in the networks. So the EOOC can be interpreted to find a set of node-disjoint paths or cycles that contains the maximal nodes in the network. Noteworthy is that we only need to find a few edges to assign the edge directions rather than all the edges. If it can approach the state of perfect controllability, we only assign |V | edges in a proper sequence which consists of cycles containing all the nodes. In addition, this optimal orientation set must meet two conditions. (i) Unambiguity. The algorithm cannot exist with the sense of ambiguity, that is, the sizes of all the driver node sets are the same. The network maybe has some optimal orientation sets but each optimal orientation set produces the same number of driver nodes. (ii) Optimum. No other method can produce fewer numbers of driver nodes than the optimal algorithm. To solve this problem, we construct a corresponding network, called the switching network.
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FIG. 2. (Color online) Illustration of finding the maximum vertex independent set to solve the orientation problem of EOOC. (a) An undirected network G = (V ,E) with five nodes and five undirected edges. G is the symmetric directed network. (b) The switching network H with 2 · |E| nodes. Rule I avoids orienting the network with many two-cycle patterns between the pairs of nodes. Rule II prevents generating some in edges or out edges sharing the common tail or head node. (c) The maximum independent set of H corresponds to an orientation set of G which produces the least number of driver nodes. Even though there are a few different sets and they result in some different control configurations, all the control configurations have the same number of driver nodes. A. Introduction of switching network
Definition 2. Every network can be regarded as a symmetric directed graph no matter whether it is a directed or undirected network. For an undirected network, the symmetric directed graph has two symmetric edges in place of any single edge in the undirected representation. For a directed network, the symmetric directed graph is adding the edges with the opposite direction. Definition 3. According to the above definition and explanation, here we present a method of network transform for solving the EOOC problem. (i) Consider a network G = (V ,E) and its corresponding symmetric directed network G = (V ,E ) [see Fig. 2(a)]. (ii) Construct an undirected network H = (A,B) as follows. Each directed edge (vi ,vj ) in G corresponds to a node aij in H. (iii) Link rules of H . Rule I: two nodes in H corresponding to the symmetric directed edges of G need to be connected [shown as the green edges in Fig. 2(b)]. Rule II: the nodes in H corresponding to the edges sharing the tail or head node in G need to be connected [see the orange and purple edges in Fig. 2(b)]. The nodes connected by orange edges in H represent that they point from the common head node, while the nodes connected by purple edges indicate that they point to the common tail node.
B. Optimal edge orientation for EOOC 1. Case I: EOOC of undirected networks
Lemma 1. Consider an undirected network G; the maximum vertex independent set of H corresponds to the orientation set
for optimal controllability in G. The optimal controllability oriented by EOOC can be denoted as below, opt
ND = max{N − |XI (H )|,1}, opt
(1)
where ND is the number of driver nodes of optimal controllability and XI (H ) is the maximum independent set of H . Fox example, in Fig. 2, |XI (H )| of the left and right graphs opt are 4, so both the graphs’ ND is 1. However, the right graph approachesthe perfect controllability oriented by EOOC. Proof. According to the rule of constructing H , three types of edges need to be connected: (1) the pair of symmetric directed edges [vab and vba , green links in Figs. 2(b) and 2(e)], (2) the out edges sharing the same head tail [vab and vad , orange links in Figs. 2(b) and 2(e)], (3) the in edges sharing the same tail [vba and vda , purple links in Figs. 2(b) and 2(e)]. According to the definition of the vertex independent set, once a node appears in the independent set (IS), the rest of the connected nodes could never be present in the IS. For the first type, IS avoids producing the 2-symmetric cycle; for the second type, IS ensures that two or more out edges cannot point from the same node; for the third type, IS guarantees that two or more in edges cannot point to the same node. Finally, IS of H corresponds to a few directed paths or cycles, such as Figs. 4(c) and 4(f). (i) Optimum. If the independent set of H is the maximum, the directed paths or cycles comprising the independent set cover the maximum edges which can be recognized as the maximum edge-controlled set. Here the number of edges in the maximum edge-controlled set is |XI (H )|. Therefore, the maximum edge-controlled set produces the least number of opt driver nodes, that is, ND = N − |XI (H )|. (ii) Considering
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controllability than SF networks. Due to the properties of the SF model, the controllability of the SF model becomes better as γ increases. The results of Fig. 3 indicate the optimal orientation outperforms the other two methods. Compared with SF networks, the optimization effects of ER networks are better. In addition, the fraction of driver nodes decreases with the increase of k because the denser networks are easy to control. When k becomes larger, the network even approaches perfect controllability, i.e., nD = 1/N , because nD is determined by k [5]. So the optimization effect by our method is almost equal to the random orientation when k is larger. The computation of the maximum vertex independent set is a classical NP-hard problem [19]. In this work, we use an efficient algorithm based on the vertex support algorithm [20] to find the maximum independent set of a graph.
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FIG. 3. (Color online) Characteristics of nD as a function of k for random orientation, NRD orientation and optimal orientation on (a) ER networks, (b) SF networks with γ = 2.5, (c) SF networks with γ = 3, and (d) SF networks with γ = 4. The results are averaged over 50 independent realizations.
the sense of Unambiguity, for EOOC, the number of driver nodes only has the correlation with the maximum vertex independent sets. Although the network maybe has some maximum independent sets, such as Figs. 2(c) and 2(f), every maximum independent set contains the same number of nodes. Considering a special case, N = |XI (H )|, which means all the nodes consist of one or several cycles, also called perfect controllability. We only modify an arbitrary bud to a stem and obtain a cactus with one input ND = 1. The detailed explanation can be found in the supplementary information of Ref. [5]. Therefore, the EOOC for an undirected network contains two main steps: constructing the switching network H , and then orienting the undirected edges of G based on the maximum vertex independent set of H . Thus, the EOOC algorithm works with the following steps. (i) Constructing the switching network H following link rules I and II of Definition 3. (ii) Finding the maximum independent set of H and assigning the edge direction based on the maximum independent set of H . (iii) Reorienting the rest of the unassigned edges from step (ii). Stop until all the edges are assigned. The optimal orientation represents the method of this paper. As the comparative experiments, we consider the method of assigning direction randomly (random orientation) and the method based on node residual degree (NRD orientation). Figure 3 shows the effectiveness of our method on the two basic network models, the Erd¨os-R´enyi (ER) model [15,16], and the scale-free (SF) model [17,18]. The SF networks are generated from the static model in Ref. [18]. Note that in case γ → ∞, this model is equivalent to the classical ER random network model. Therefore, ER networks have better
In directed networks, there are some “inappropriate” edges which can weaken the network controllability, such as in Fig. 4(a). For the control configuration of (vad ,vcb ), the edges vab and vcd (the blue edges) weaken the controllability because of their “inappropriate” direction. So the EOOC problem of directed networks is to detect the “inappropriate” edges and change their directions. Lemma 2. Consider a directed network G. We assume the symmetric directed graph G = (V ,E ,W ) is edge weighted. The weight of real edges existing in G is 1; otherwise it is 0 [see Fig. 4(b)]. So the nodes of H = (A,B,W ) correspond to weight 0 or 1. The maximum-weighted vertex independent set of H corresponds to the orientation set for optimal controllability of directed network G. Proof. This basic theory follows from Lemma 1. The principle of maximum weight guarantees the independent set contains the maximal nodes with weight 1. Therefore, the orientation set only needs to be modified by the least number of edge directions. For example, the maximum-weighted independent set of Fig. 4(c) shows the optimal edge orientation and the least number of edges to be modified. The major difference of EOOC for undirected and directed networks is the node weight of H . It has been proven it is a typical NP-hard problem in computational complexity theory [21,22], and it is not plausible that there is a polynomial time algorithm to find a maximum-weighted independent set (MWIS) for arbitrary graphs. Next we show how to compute the MWIS using an ILP formulation, in which the optimal solution is calculated using the IBM ILOG CPLEX Optimizer v.12.6. From H = (A,B,W ), we construct the following ILP instance: max Z = wi xi , s.t. xi + xj 1,∀(vi ,vj ) ∈ B,
(2)
xi = {0,1},vi ∈ A. Then, the set {v|xv = 1} clearly gives a maximum-weighted independent set. The optimization procedure of the directed network is similar to the undirected network, where the only difference is finding MWIS for the directed network. Figure 5 shows the optimization effects and costs of EOOC for directed networks. From the results of the edge-modified percent, the
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FIG. 4. (Color online) Illustration of edge orientation of optimal controllability for directed networks. (a) A directed network G = (V ,E) with four nodes and four directed edges. For directed networks, some edges with inappropriate direction affect the controllability of networks, such as the blue edges. If we change the direction of blue edges, it can improve its controllability. (b) G is the symmetric directed network. Here we assume that the weight of the real edge is 1; otherwise it is 0. Thus it results that the nodes of H have weight. (c) The switching network H . The red nodes correspond to the real edges in G. The node with weight 0 in the maximum-weighted independent set implies the inappropriate edge of G. Therefore, the maximum-weighted independent set means not only the optimal edge orientation but also the least number of edges to change direction. In addition, although there are several different maximum-weighted independent sets, the same number of driver nodes is guaranteed for the different control configurations.
percent of inappropriate edges occupies a very small part, almost below 15% for ER networks and 10% for SF networks. The edge-modified percent decreases with the increment of k. However, unlike the structural perturbation, EOOC cannot delete original edges or add new edges between nodes to optimize controllability. It only mitigates the negative effect of dilation rather than eliminating the effect. So EOOC cannot decrease driver nodes to approach perfect controllability.
C. Structure properties of switching network H
Here, we summarize some basic structure properties of switching network H . We label the original undirected network as G(V ,E), the symmetric directed network as G (V ,E ), and
the switching network as H (A,B). N is the total number of nodes in G. vi ∈ V is the node i in G. ki is the degree of node vi , and k is the average degree of G. vij = vj i ∈ E implies the undirected edge of G. vij ∈ E indicates the directed edge (vi → vj ). a(i,j ) ∈ A represents the node of H which corresponds to the directed edge vij of G . ka(i,j ) is the degree of a(i,j ) in H , and kH is the average degree of H . Lemma 3. ka(i,j ) = ka(j,i) = ki + kj − 1. Proof. For vij ∈ E , there are (ki − 1) edges sharing the head node vi and (kj − 1) edges sharing the tail node vj . All the (ki − 1) edges and (kj − 1) edges need to be connected according to rule II of definition 3. Based on rule I, the edge vij and vj i also need to link together. Therefore, ka(i,j ) = (ki − 1) + (kj − 1) + 1. ka(j,i) follows the same analysis.
FIG. 5. (Color online) Characteristics of nD and percent of edge modification as a function of k for initial networks and modified networks on (a) and (b) ER networks; (c) and (d) SF networks with γ = 2.5; (e) and (f) SF networks with γ = 3. The results are averaged over 50 independent realizations. 042804-5
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Lemma 4. kH = 2k 2 /k − 1. vij ∈E ka(i,j ) P roof. kH = |A| vij ∈E (ki + kj − 1) = kN vij ∈E (ki + kj ) − kN = kN N 2ki ki − kN = i kN N 2 2 i ki − kN = kN
nodes with many connections [27,28], i.e., a preference for high-degree nodes to attach to other high-degree nodes. On the contrary, some networks show disassortativity—high-degree nodes are more likely to attach to low-degree ones. 1. Case I: undirected network
For the practical purpose of evaluating r on an observed network, Newman [29] rewrites the degree assortativity as below, −1 i ji k i − L i ji i ki r = 2 , 2 2 2 −1 −1 j − L j k − L i i i i i i i ki (4)
2k 2 2k 2 − k = − 1. = k k According to the above structure properties of switching network H , we can estimate the optimal controllability for random networks with any given degree distribution. A rigorous bound [19] exists for XI (H ): ln c/c < xI (c) < 1 − xl (c), where xI (c) is the fraction of the maximum independent set of H and c is the average degree of H . Here xl (c) is the root of xl (c) ln xl (c) + [1 − xl (c)] ln[1 − xl (c)] + (c/2)[1 − xl (c)]2 = 0.
(3)
There are many equations to measure the fraction of the maximum independent set for the different scale of average degree [23,24]. Especially in Zhou et al. [25,26], they use the cavity method to analyze the maximum vertex cover problem, which is the dual problem of the independent set. Furthermore, Liu et al. [5] also analyze the controllability of networks by the maximum matching theorem from the view of cavity theory. Our work provides a perspective to analyze the optimal controllability of networks with any given degree distribution.
where ji and ki are the degree of two nodes that are connected by ith link, and L is the number of edges. The value of r ranges from −1 to 1. If an undirected network has perfect assortativity (r = 1), then all nodes connect only to nodes with the same degree. If the network has no assortativity (r = 0), then any node can randomly connect to any other node. If a network is perfectly disassortativity (r = −1), all nodes have to connect to nodes with different degrees. Figure 6 shows how the degree-degree correlation affects the optimal orientation for undirected networks. Here we generate the different asssortativity coefficient by edge swap, which keeps the degree of every node unchanged. The results clearly show the disassortativity pattern could weaken the optimization effects of orientation for optimal controllability, while the assortativity pattern has no correlation with EOOC. In fact, the uncontrolled nodes are caused by the dilations of network structure. However, the disassortativity pattern means a pair of two nodes sharing the same edge exists with huge differences of degree, which produces more dilations in the network. No matter how the orientation method can assign the edge direction, it cannot eliminate the negative effect of dilations because the prerequisites of this method do not change the network skeleton.
IV. RELATIONS BETWEEN DEGREE-DEGREE CORRELATION AND EOOC
2. Case II: directed network
Obviously, the structure and degree distribution of network determines its controllability. We further investigate the relationship between EOOC and degree-degree correlation. A network generally shows the feature of assortativity patterns if the nodes that have many connections tend to connect to other
However, since nodes in directed networks have both in degrees and out degrees, the directed networks show common patterns across the four directed degree-degree correlation measures: r(in,in),r(in,out),r(out,in),r(out,out) [30], where the first element labels the degree of the source node of the
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directed edge, and the second represents the degree of the target node. Let α,β ∈ {in,out} index the degree type, and jiα and kiα be the α− and β− degree of the source node and target node for link i. Then the directed assortativity using the Pearson correlation is defined as β L−1 i jiα − jˆα ki − kˆ β r(α,β) = , (5) σ ασ β where L is the number of links in the network, j α is the β degree of the source node, and k isthe degree of the target node; jˆα = L−1 j α , σ α = L−1 (j α − jˆα )2 ; kˆ β and σ β i i
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are similarly defined. The four directed assortativity types are shown in subplot of Fig. 7. Reference [31] investigates that the driver nodes have linear, quadratic or independent correlations on the four degree-degree correlation coefficients. Here we generate the different directed degree-degree correlations by edge swap, but this swap procedure keeps the in and out degree of every node unchanged. Figure 7 shows the change tendency of driver nodes with different r(α,β) before and after edge modification. Although the different directed degree-degree correlations have different effects on network controllability, the change of driver nodes has the same tendency as the EOOC of the undirected network after edge modification. The disassortativity patterns weaken the orientation for optimal controllability, while the assortativity patterns have no correlations with EOOC.
However, when r(α,β) ∈ (−0.5,0), the driver nodes decrease more dramatically than others. Especially for the SF network, the disassortativity patterns weaken the controllability more than that of ER networks. We also display the cost of edge modification and find that the edge-modification percent increases with the increment of r(α,β) except r(out,in), which means although the assortativity pattern cannot affect the network controllability, it increases the cost in the optimization procedure. For r(out,in), unlike the other correlations, the best optimization effects occur in r(out,in)
Edge orientation for optimizing controllability of complex networks Yan-Dong Xiao, Song-Yang Lao, Lv-Lin Hou, and Liang Bai College of Information Systems and Management, National University of Defense Technology, Changsha 410073, China (Received 12 July 2014; published 9 October 2014) Recently, as the controllability of complex networks attracts much attention, how to design and optimize the controllability of networks has become a common and urgent problem in the field of controlling complex networks. Previous work focused on the structural perturbation and neglected the role of edge direction to optimize the network controllability. In a recent work [Phys. Rev. Lett. 103, 228702 (2009)], the authors proposed a simple method to enhance the synchronizability of networks by assignment of link direction while keeping network topology unchanged. However, the controllability is fundamentally different from synchronization. In this work, we systematically propose the definition of assigning direction to optimize controllability, which is called the edge orientation for optimal controllability problem (EOOC). To solve the EOOC problem, we construct a switching network and transfer the EOOC problem to find the maximum independent set of the switching network. We prove that the principle of our optimization method meets the sense of unambiguity and optimum simultaneously. Furthermore, the relationship between the degree-degree correlations and EOOC are investigated by experiments. The results show that the disassortativity pattern could weaken the orientation for optimal controllability, while the assortativity pattern has no correlation with EOOC. All the experimental results of this work verify that the network structure determines the network controllability and the optimization effects. DOI: 10.1103/PhysRevE.90.042804
PACS number(s): 89.75.Fb, 89.75.Hc, 02.30.Yy
I. INTRODUCTION
As the networked systems appear around us at an increasing rate, dynamics in complex networks has become one of the most popular research fields [1–4]. The ultimate proof of our understanding of such systems and their basic principle is reflected in our ability to control them [5]. Questions concerning how to manage and control complex networks are important in science, engineering, and biology. Recently, the controllability of complex networks has been a hot topic [5–8], the main point of which is applying the external signals to some nodes so that the system can achieve the desirable state in finite time. Controllability theory offers us a general, rigorous, and well-understood framework for the design and analysis of control systems. The seminal work by Lin [9] provides a graph-theoretical formulation which simplifies and generalizes this problem. A series of work [5,9] focuses on the identification of so-called driver nodes (ND ) using maximum matching, in which the nodes controlled directly by input signals are called driver nodes. Thus, the number of driver nodes is equal to the number of input signals. Controllability is characterized by the minimum number of driver nodes which could offer full control over the network. If a single driver node could fully control the whole network, i.e., ND = 1, we say that the network has perfect controllability. Therefore, the method of reducing ND or even approaching to the state of perfect controllability is of utmost significance in real-world networks. The most representative methods of optimizing of network controllability can be summarized as three methods: adding edges [10], rewiring edges [11], and assigning the direction of edges [12]. Wang et al. [10] describe a perturbation strategy based on adding edges for connecting the separated control path in a proper sequence, which arrives at perfect controllability by adding the least number of edges (only ND − 1 edges). According to the role of different edge categories (critical, ordinary, and redundant edge set) of network controllability, deleting a redundant edge cannot increase the number of driver 1539-3755/2014/90(4)/042804(8)
nodes, and adding an edge may enhance the controllability of the network but never weaken its controllability. Thus the way of rewiring redundant edges is also effective for optimizing controllability [11]. The approach of assigning edge direction to optimize the controllability has been studied by Hou et al. [12]. Compared with the above two methods of structural perturbations, assigning the direction of edges to optimize the network controllability is more practical and economic than that of reconstructing networks in practice. With the view of designing networks having better controllability, this method has the advantage of paying lower costs and being more practical than any other method because it neither changes the original structure of the network nor pays the additional cost for adding links. For example, the installation of new power lines between a pair of power plants sometimes would skyrocket costs and be impractical. However, the effect of edge direction on controllability has not been systematically considered. In the following section, we will give a more detailed analysis of previous work and its shortcomings when using edge orientation to optimize the network controllability. This work is introducing the problem of edge orientation for optimal controllability and the framework for solving this problem. Furthermore, we also investigate the correlations between the network structures and the edge orientation of optimal controllability. The paper is organized as follows. In Sec. II, we first introduce the previous work on this problem. In Sec. III, we systematically study the edge orientation for optimal controllability from the definitions and methodologies. Finally, the correlations between the network structures and the edge orientation of optimal controllability are presented in Sec. IV.
II. PREVIOUS WORKS
Reference [12] presented an efficient heuristic approximation algorithm for assigning edge direction to optimize the
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FIG. 1. (Color online) Illustration of the orientation of node residual degree (NRD). The original network (a) consists of eight nodes and 12 edges. The above control configuration has two driver nodes, while the under configuration has only one driver node. The yellow node is the start node in each orientation process and the red arcs are edge directions oriented by NRD. The gray dashed lines are not yet oriented. If there is more than one suitable choice, NRD randomly selects a node or edge from the network. As node 5 (black node) has two available choices to orient edge, it results in two different numbers of driver nodes, shown as blue nodes in (d). The orientation process of steps (i), (ii), and (iii) are repeated until all nodes and edges are exhausted.
network controllability. This orientation method is based on the node residual degree (NRD) which originates from the optimization of synchronization of complex networks [13,14]. However, controllability is fundamentally different from synchronization. Structural controllability theorem states that directed loops play a significant role in controlling networks, while the original intention of improving synchronization avoids producing loops since the large number of loops is not beneficial to synchronization [13,14]. Therefore, the method of Ref. [12] can be recognized as the opposite of improving synchronization, which aims to generate the directed loops as much as possible. NRD can be described as below: (i) choose the node with the largest residual degree k˜ in the network as V1 ; (ii) choose the node with the largest k˜ in the neighborhood of V1 , denoted as V2 ; (iii) if koi (V1 ) koi (V2 ), then assign the direction from V1 to V2 and the residual degree of the selected nodes will be reduced by 1. Note that koi = kout − kin and randomly pick nodes if there are multiple such nodes. Obviously, this heuristic approximation algorithm is not the optimal orientation of controllability for any given topology of network because of the random feature of selecting nodes or edges. Here is a simple example to demonstrate this conclusion. From Fig. 1, an undirected network consists of eight nodes and 12 undirected edges. The red edges represent the orientation and the yellow node is the start node in each process of assigning edge direction. Both the edge and node could be selected randomly if there are more than one. In Fig. 1(b), as node 5 (black node) has two available choices to orient edge, it could result in a different number of driver nodes finally [blue nodes in Fig. 1(d)]. From Fig. 1(c), although the remaining four edges have different choices of edge direction, they do not influence the control configuration. Since sometimes the nodes and edges are randomly selected, this method may produce different orientation sets with a different
number of driver nodes. Therefore, it is necessary to propose an algorithm to find the optimal orientation set. III. EDGE ORIENTATION FOR OPTIMAL CONTROLLABILITY PROBLEM (EOOC)
According to the above arguments, we formulate the edge orientation for optimal controllability problem (EOOC) as follows. Definition 1. Let G = (V ,E) be an undirected and unweighted network. If there is an edge orientation set (G = (V ,E ),E = {vi → vj |vi ,vj ∈ V ,i,j = 1 · · · n}) for optimal controllability, it corresponds to the least number of driver nodes. According to the maximum matching theorem of Liu’s controllability framework, the optimal orientation set actually produces a maximum matching which covers the maximal number of nodes in the networks. So the EOOC can be interpreted to find a set of node-disjoint paths or cycles that contains the maximal nodes in the network. Noteworthy is that we only need to find a few edges to assign the edge directions rather than all the edges. If it can approach the state of perfect controllability, we only assign |V | edges in a proper sequence which consists of cycles containing all the nodes. In addition, this optimal orientation set must meet two conditions. (i) Unambiguity. The algorithm cannot exist with the sense of ambiguity, that is, the sizes of all the driver node sets are the same. The network maybe has some optimal orientation sets but each optimal orientation set produces the same number of driver nodes. (ii) Optimum. No other method can produce fewer numbers of driver nodes than the optimal algorithm. To solve this problem, we construct a corresponding network, called the switching network.
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FIG. 2. (Color online) Illustration of finding the maximum vertex independent set to solve the orientation problem of EOOC. (a) An undirected network G = (V ,E) with five nodes and five undirected edges. G is the symmetric directed network. (b) The switching network H with 2 · |E| nodes. Rule I avoids orienting the network with many two-cycle patterns between the pairs of nodes. Rule II prevents generating some in edges or out edges sharing the common tail or head node. (c) The maximum independent set of H corresponds to an orientation set of G which produces the least number of driver nodes. Even though there are a few different sets and they result in some different control configurations, all the control configurations have the same number of driver nodes. A. Introduction of switching network
Definition 2. Every network can be regarded as a symmetric directed graph no matter whether it is a directed or undirected network. For an undirected network, the symmetric directed graph has two symmetric edges in place of any single edge in the undirected representation. For a directed network, the symmetric directed graph is adding the edges with the opposite direction. Definition 3. According to the above definition and explanation, here we present a method of network transform for solving the EOOC problem. (i) Consider a network G = (V ,E) and its corresponding symmetric directed network G = (V ,E ) [see Fig. 2(a)]. (ii) Construct an undirected network H = (A,B) as follows. Each directed edge (vi ,vj ) in G corresponds to a node aij in H. (iii) Link rules of H . Rule I: two nodes in H corresponding to the symmetric directed edges of G need to be connected [shown as the green edges in Fig. 2(b)]. Rule II: the nodes in H corresponding to the edges sharing the tail or head node in G need to be connected [see the orange and purple edges in Fig. 2(b)]. The nodes connected by orange edges in H represent that they point from the common head node, while the nodes connected by purple edges indicate that they point to the common tail node.
B. Optimal edge orientation for EOOC 1. Case I: EOOC of undirected networks
Lemma 1. Consider an undirected network G; the maximum vertex independent set of H corresponds to the orientation set
for optimal controllability in G. The optimal controllability oriented by EOOC can be denoted as below, opt
ND = max{N − |XI (H )|,1}, opt
(1)
where ND is the number of driver nodes of optimal controllability and XI (H ) is the maximum independent set of H . Fox example, in Fig. 2, |XI (H )| of the left and right graphs opt are 4, so both the graphs’ ND is 1. However, the right graph approachesthe perfect controllability oriented by EOOC. Proof. According to the rule of constructing H , three types of edges need to be connected: (1) the pair of symmetric directed edges [vab and vba , green links in Figs. 2(b) and 2(e)], (2) the out edges sharing the same head tail [vab and vad , orange links in Figs. 2(b) and 2(e)], (3) the in edges sharing the same tail [vba and vda , purple links in Figs. 2(b) and 2(e)]. According to the definition of the vertex independent set, once a node appears in the independent set (IS), the rest of the connected nodes could never be present in the IS. For the first type, IS avoids producing the 2-symmetric cycle; for the second type, IS ensures that two or more out edges cannot point from the same node; for the third type, IS guarantees that two or more in edges cannot point to the same node. Finally, IS of H corresponds to a few directed paths or cycles, such as Figs. 4(c) and 4(f). (i) Optimum. If the independent set of H is the maximum, the directed paths or cycles comprising the independent set cover the maximum edges which can be recognized as the maximum edge-controlled set. Here the number of edges in the maximum edge-controlled set is |XI (H )|. Therefore, the maximum edge-controlled set produces the least number of opt driver nodes, that is, ND = N − |XI (H )|. (ii) Considering
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controllability than SF networks. Due to the properties of the SF model, the controllability of the SF model becomes better as γ increases. The results of Fig. 3 indicate the optimal orientation outperforms the other two methods. Compared with SF networks, the optimization effects of ER networks are better. In addition, the fraction of driver nodes decreases with the increase of k because the denser networks are easy to control. When k becomes larger, the network even approaches perfect controllability, i.e., nD = 1/N , because nD is determined by k [5]. So the optimization effect by our method is almost equal to the random orientation when k is larger. The computation of the maximum vertex independent set is a classical NP-hard problem [19]. In this work, we use an efficient algorithm based on the vertex support algorithm [20] to find the maximum independent set of a graph.
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the sense of Unambiguity, for EOOC, the number of driver nodes only has the correlation with the maximum vertex independent sets. Although the network maybe has some maximum independent sets, such as Figs. 2(c) and 2(f), every maximum independent set contains the same number of nodes. Considering a special case, N = |XI (H )|, which means all the nodes consist of one or several cycles, also called perfect controllability. We only modify an arbitrary bud to a stem and obtain a cactus with one input ND = 1. The detailed explanation can be found in the supplementary information of Ref. [5]. Therefore, the EOOC for an undirected network contains two main steps: constructing the switching network H , and then orienting the undirected edges of G based on the maximum vertex independent set of H . Thus, the EOOC algorithm works with the following steps. (i) Constructing the switching network H following link rules I and II of Definition 3. (ii) Finding the maximum independent set of H and assigning the edge direction based on the maximum independent set of H . (iii) Reorienting the rest of the unassigned edges from step (ii). Stop until all the edges are assigned. The optimal orientation represents the method of this paper. As the comparative experiments, we consider the method of assigning direction randomly (random orientation) and the method based on node residual degree (NRD orientation). Figure 3 shows the effectiveness of our method on the two basic network models, the Erd¨os-R´enyi (ER) model [15,16], and the scale-free (SF) model [17,18]. The SF networks are generated from the static model in Ref. [18]. Note that in case γ → ∞, this model is equivalent to the classical ER random network model. Therefore, ER networks have better
In directed networks, there are some “inappropriate” edges which can weaken the network controllability, such as in Fig. 4(a). For the control configuration of (vad ,vcb ), the edges vab and vcd (the blue edges) weaken the controllability because of their “inappropriate” direction. So the EOOC problem of directed networks is to detect the “inappropriate” edges and change their directions. Lemma 2. Consider a directed network G. We assume the symmetric directed graph G = (V ,E ,W ) is edge weighted. The weight of real edges existing in G is 1; otherwise it is 0 [see Fig. 4(b)]. So the nodes of H = (A,B,W ) correspond to weight 0 or 1. The maximum-weighted vertex independent set of H corresponds to the orientation set for optimal controllability of directed network G. Proof. This basic theory follows from Lemma 1. The principle of maximum weight guarantees the independent set contains the maximal nodes with weight 1. Therefore, the orientation set only needs to be modified by the least number of edge directions. For example, the maximum-weighted independent set of Fig. 4(c) shows the optimal edge orientation and the least number of edges to be modified. The major difference of EOOC for undirected and directed networks is the node weight of H . It has been proven it is a typical NP-hard problem in computational complexity theory [21,22], and it is not plausible that there is a polynomial time algorithm to find a maximum-weighted independent set (MWIS) for arbitrary graphs. Next we show how to compute the MWIS using an ILP formulation, in which the optimal solution is calculated using the IBM ILOG CPLEX Optimizer v.12.6. From H = (A,B,W ), we construct the following ILP instance: max Z = wi xi , s.t. xi + xj 1,∀(vi ,vj ) ∈ B,
(2)
xi = {0,1},vi ∈ A. Then, the set {v|xv = 1} clearly gives a maximum-weighted independent set. The optimization procedure of the directed network is similar to the undirected network, where the only difference is finding MWIS for the directed network. Figure 5 shows the optimization effects and costs of EOOC for directed networks. From the results of the edge-modified percent, the
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FIG. 4. (Color online) Illustration of edge orientation of optimal controllability for directed networks. (a) A directed network G = (V ,E) with four nodes and four directed edges. For directed networks, some edges with inappropriate direction affect the controllability of networks, such as the blue edges. If we change the direction of blue edges, it can improve its controllability. (b) G is the symmetric directed network. Here we assume that the weight of the real edge is 1; otherwise it is 0. Thus it results that the nodes of H have weight. (c) The switching network H . The red nodes correspond to the real edges in G. The node with weight 0 in the maximum-weighted independent set implies the inappropriate edge of G. Therefore, the maximum-weighted independent set means not only the optimal edge orientation but also the least number of edges to change direction. In addition, although there are several different maximum-weighted independent sets, the same number of driver nodes is guaranteed for the different control configurations.
percent of inappropriate edges occupies a very small part, almost below 15% for ER networks and 10% for SF networks. The edge-modified percent decreases with the increment of k. However, unlike the structural perturbation, EOOC cannot delete original edges or add new edges between nodes to optimize controllability. It only mitigates the negative effect of dilation rather than eliminating the effect. So EOOC cannot decrease driver nodes to approach perfect controllability.
C. Structure properties of switching network H
Here, we summarize some basic structure properties of switching network H . We label the original undirected network as G(V ,E), the symmetric directed network as G (V ,E ), and
the switching network as H (A,B). N is the total number of nodes in G. vi ∈ V is the node i in G. ki is the degree of node vi , and k is the average degree of G. vij = vj i ∈ E implies the undirected edge of G. vij ∈ E indicates the directed edge (vi → vj ). a(i,j ) ∈ A represents the node of H which corresponds to the directed edge vij of G . ka(i,j ) is the degree of a(i,j ) in H , and kH is the average degree of H . Lemma 3. ka(i,j ) = ka(j,i) = ki + kj − 1. Proof. For vij ∈ E , there are (ki − 1) edges sharing the head node vi and (kj − 1) edges sharing the tail node vj . All the (ki − 1) edges and (kj − 1) edges need to be connected according to rule II of definition 3. Based on rule I, the edge vij and vj i also need to link together. Therefore, ka(i,j ) = (ki − 1) + (kj − 1) + 1. ka(j,i) follows the same analysis.
FIG. 5. (Color online) Characteristics of nD and percent of edge modification as a function of k for initial networks and modified networks on (a) and (b) ER networks; (c) and (d) SF networks with γ = 2.5; (e) and (f) SF networks with γ = 3. The results are averaged over 50 independent realizations. 042804-5
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Lemma 4. kH = 2k 2 /k − 1. vij ∈E ka(i,j ) P roof. kH = |A| vij ∈E (ki + kj − 1) = kN vij ∈E (ki + kj ) − kN = kN N 2ki ki − kN = i kN N 2 2 i ki − kN = kN
nodes with many connections [27,28], i.e., a preference for high-degree nodes to attach to other high-degree nodes. On the contrary, some networks show disassortativity—high-degree nodes are more likely to attach to low-degree ones. 1. Case I: undirected network
For the practical purpose of evaluating r on an observed network, Newman [29] rewrites the degree assortativity as below, −1 i ji k i − L i ji i ki r = 2 , 2 2 2 −1 −1 j − L j k − L i i i i i i i ki (4)
2k 2 2k 2 − k = − 1. = k k According to the above structure properties of switching network H , we can estimate the optimal controllability for random networks with any given degree distribution. A rigorous bound [19] exists for XI (H ): ln c/c < xI (c) < 1 − xl (c), where xI (c) is the fraction of the maximum independent set of H and c is the average degree of H . Here xl (c) is the root of xl (c) ln xl (c) + [1 − xl (c)] ln[1 − xl (c)] + (c/2)[1 − xl (c)]2 = 0.
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There are many equations to measure the fraction of the maximum independent set for the different scale of average degree [23,24]. Especially in Zhou et al. [25,26], they use the cavity method to analyze the maximum vertex cover problem, which is the dual problem of the independent set. Furthermore, Liu et al. [5] also analyze the controllability of networks by the maximum matching theorem from the view of cavity theory. Our work provides a perspective to analyze the optimal controllability of networks with any given degree distribution.
where ji and ki are the degree of two nodes that are connected by ith link, and L is the number of edges. The value of r ranges from −1 to 1. If an undirected network has perfect assortativity (r = 1), then all nodes connect only to nodes with the same degree. If the network has no assortativity (r = 0), then any node can randomly connect to any other node. If a network is perfectly disassortativity (r = −1), all nodes have to connect to nodes with different degrees. Figure 6 shows how the degree-degree correlation affects the optimal orientation for undirected networks. Here we generate the different asssortativity coefficient by edge swap, which keeps the degree of every node unchanged. The results clearly show the disassortativity pattern could weaken the optimization effects of orientation for optimal controllability, while the assortativity pattern has no correlation with EOOC. In fact, the uncontrolled nodes are caused by the dilations of network structure. However, the disassortativity pattern means a pair of two nodes sharing the same edge exists with huge differences of degree, which produces more dilations in the network. No matter how the orientation method can assign the edge direction, it cannot eliminate the negative effect of dilations because the prerequisites of this method do not change the network skeleton.
IV. RELATIONS BETWEEN DEGREE-DEGREE CORRELATION AND EOOC
2. Case II: directed network
Obviously, the structure and degree distribution of network determines its controllability. We further investigate the relationship between EOOC and degree-degree correlation. A network generally shows the feature of assortativity patterns if the nodes that have many connections tend to connect to other
However, since nodes in directed networks have both in degrees and out degrees, the directed networks show common patterns across the four directed degree-degree correlation measures: r(in,in),r(in,out),r(out,in),r(out,out) [30], where the first element labels the degree of the source node of the
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0.12 (in,out) 0.08
0.04
0
1
0.6 (out,in)
0.08
0.04
1
nD 0.4
0.08
(in,in)
0.12
0.12
0.3
0.15
ER =3
nD 0.2
0.5
0.2
0.4 0.3
t (out, out)
0.04 4 -1
1 edge-modified percent
0 r(in, out) t
0.2 -1 0.6
0.05 5 -1
1
0.5
0 -1
0 r(in, in)
0.15
0.4
0 -1
0.3
0.05 5 -1
0.5
0 -1
nD 0.4
0.1
edge-modified percent
0.3
3 SF =3
0.5
edge-modified percent
nD
(in,in)
edge-modified percent
edge-modified percent
0.15
0.4
edge-modified percent
0.6
0.5
-1
0.12
(out,in)
0.08
0.04
-1
FIG. 7. (Color online) Plot of nD and percent of edge modification as a function of four different degree-degree correlations r(α,β),α,β ∈ {in,out} for initial networks and modified networks on ER networks. The subplots show the four directed degree-degree correlations of directed networks. The fuzzy directed links represent that they do not enter into the specific correlation. The results are averaged over 50 independent realizations.
directed edge, and the second represents the degree of the target node. Let α,β ∈ {in,out} index the degree type, and jiα and kiα be the α− and β− degree of the source node and target node for link i. Then the directed assortativity using the Pearson correlation is defined as β L−1 i jiα − jˆα ki − kˆ β r(α,β) = , (5) σ ασ β where L is the number of links in the network, j α is the β degree of the source node, and k isthe degree of the target node; jˆα = L−1 j α , σ α = L−1 (j α − jˆα )2 ; kˆ β and σ β i i
i
i
are similarly defined. The four directed assortativity types are shown in subplot of Fig. 7. Reference [31] investigates that the driver nodes have linear, quadratic or independent correlations on the four degree-degree correlation coefficients. Here we generate the different directed degree-degree correlations by edge swap, but this swap procedure keeps the in and out degree of every node unchanged. Figure 7 shows the change tendency of driver nodes with different r(α,β) before and after edge modification. Although the different directed degree-degree correlations have different effects on network controllability, the change of driver nodes has the same tendency as the EOOC of the undirected network after edge modification. The disassortativity patterns weaken the orientation for optimal controllability, while the assortativity patterns have no correlations with EOOC.
However, when r(α,β) ∈ (−0.5,0), the driver nodes decrease more dramatically than others. Especially for the SF network, the disassortativity patterns weaken the controllability more than that of ER networks. We also display the cost of edge modification and find that the edge-modification percent increases with the increment of r(α,β) except r(out,in), which means although the assortativity pattern cannot affect the network controllability, it increases the cost in the optimization procedure. For r(out,in), unlike the other correlations, the best optimization effects occur in r(out,in)
