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Dominating communities for hierarchical control of complex networks
Information Sciences 414 (2017) 247–259
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Information Sciences journal homepage: www.elsevier.com/locate/ins
Dominating communities for hierarchical control of complex networks Peng Gang Sun a,b,∗, Xiaoke Ma a,∗ a b
School of Computer Science and Technology, Xidian University, P. O. Box 163, No. 2 South Taibai Road, Xi’an, 710071 China Center for Complex Data and Network Science, Xidian University, Xi’an, 710071, China
a r t i c l e
i n f o
Article history: Received 1 January 2017 Revised 21 April 2017 Accepted 31 May 2017 Available online 8 June 2017 Keywords: Hierarchical control Structure-driven minimum dominating set
a b s t r a c t Well-selected nodes driven by external signals in complex networks are of great importance for a dynamic system’s structural controllability, which can also be achieved by selecting nodes in the minimum dominating set (MDS) of the networks as driver nodes. However, hierarchical structures such as communities are widely observed in complex networks, which motivates us to develop the hierarchical control of complex networks by dominating communities, and the control over whole networks is transformed into that within communities, i.e., a non-driver node is effectively controlled by its driver node if they belong to the same community. For the hierarchical control, we propose the structure-driven minimum dominating set (SD-MDS), which assumes that a driver node can control a non-driver node if the link between them belongs to a cyclic structure of the networks. We further investigate the impact of community dynamics on our framework. The results show that it is easier to control the networks with stronger communities and more heterogeneous community sizes, and the SD-MDS achieves more effective control for non-driver nodes than the MDS. This work also indicates that the number of driver nodes for the SD-MDS is associated with the degree distribution of the network. © 2017 Elsevier Inc. All rights reserved.
1. Introduction The study of complex networks has been a hot topic and attracted a large amount of attention for many years [1,41]. Network structure analysis, such as community detection, as an important research area of complex networks, provides us with new insight into the hierarchical structures of complex networks, since communities described as densely connected subnetworks often correspond to important organizational/functional units in the networks [2,3,5–7,9–11,17,24–26,32,34– 38,43,44]. Recently, network control in complex networks has grown rapidly [4,12,20–23,27–31,33,39,45–51]. This area studies a dynamic system’s controllability. A dynamic system is controllable if suitable inputs of external signals can drive the system from any initial state to any desired final state in finite time [15,16,40]. Liu et al. [20] studied structural controllability on directed networks and transformed the problem into the identification of minimum driver nodes, which correspond to unmatched nodes in the maximum bipartite matching of the networks, and unmatched nodes as targets driven by external signals can achieve the structural controllability. Interestingly, Nacher and Akutsu [27,31] introduced the minimum dominating set (MDS) to study the structural controllability on undirected networks, which can be achieved by selecting nodes in ∗
Corresponding authors at: School of Computer Science and Technology, Xidian University, P. O. Box 163, No. 2 South Taibai Road, Xi’an, 710071 China. E-mail addresses: [email protected] (P.G. Sun), [email protected] (X. Ma).
http://dx.doi.org/10.1016/j.ins.2017.05.052 0020-0255/© 2017 Elsevier Inc. All rights reserved.
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the MDS of the network as driver nodes, since each link in the network is assumed to be bi-directional, and a driver node to control all its links separately, i.e., non-driver nodes are controllable if they are at least adjacent to a driver node. Therefore, driver nodes within the MDS can control the whole network. Remarkably, Nacher and Akutsu [27,31] also mentioned that the difference between the two control models is that the former assumes that external signals can only directly control driver nodes (unmatched nodes), which tend not to be high-degree nodes, while the latter assumes that each driver node can independently control its links, and driver nodes tend to be high-degree nodes, i.e., the MDS tends to identify important nodes in controlling the complex network [22,27,31]. Nevertheless, in both models, the number of driver nodes is mainly determined by the degree distribution of the network [20,22,27,31,45]. Hierarchical structures such as communities are widely observed in complex networks. To control a network-based system, we should fully dominate its structures since the system’s structures correspond to its functional units, and each unit has its realm of authority [2,3,5–7,9–11,17,24–26,32,34–37,43,44]. Therefore, we are motivated to develop a new framework, hierarchical control of complex networks by dominating communities, and the control over whole networks is transformed into that within communities, i.e., a non-driver node is effectively controlled by its driver node if they belong to the same community. For hierarchical control, we propose the structure-driven minimum dominating set (SD-MDS) model, in which a driver node can control a non-driver node if the link between them belongs to a cyclic structure of the networks, while for the MDS model, a non-driver node is controlled so long as it is adjacent to a driver node. We analyze the hierarchical control of random networks as well as real-world networks, and further investigate the impact of community dynamics on our framework. The rest of the paper is organized as follows. In Section 2, we introduce the SD-MDS model and illustrate the impact of community dynamics on our framework. In Section 3, we analyze the hierarchical control of random networks and realworld networks. The conclusion is provided in Section 4. 2. Dominating complex networks In this section, we first describe the MDS model and then introduce the SD-MDS model. Finally, we illustrate the impact of community dynamics on the SD-MDS. 2.1. Minimum dominating set In this paper, we use G(V, E) to denote an unweighted, undirected graph, where V and E correspond to the set of nodes and the set of edges, respectively. The minimum dominating set (MDS) of a graph is defined as an optimized subset of nodes such that each node of the network either belongs to the subset or is adjacent to an element of the subset [27–31]. Formally, Definition 1. (dominating set) V is a dominating set of G if V ⊆V, V = ∅, and ∀i ∈ V − V , ∃j ∈ V , (i, j) ∈ E, and i = j, where |V | = n and i, j ∈ {1, 2, · · ·, n}. Definition 2. (minimum dominating set) V is a minimum dominating set ofG if ∀V ∈ D, |V | ≤ |V |, where D is the set containing all the dominating sets of G. 2.2. MDS model For the MDS model, in which nodes belonging to the MDS are called driver nodes, the problem of dominating a network is transformed into the determination of its MDS, and we simply achieve full control of the network by dominating the driver nodes since each non-driver node is controllable so long as it is at least adjacent to a driver node [27–31]. The determination of the MDS can be solved by binary integer programming [27–31]. Formally,
min
i∈V
xi
(1)
subject to
AX ≥ B,
(2)
a11 . . . a1n . . .. . ), X = [x1 , x2 , · · ·, xn ]T and B = [b1 , b2 , · · ·, bn ]T . where A = ( .. . . an1 ··· ann Binary integer programming (also called “0–1 integer programming) is a mathematical optimization problem in which the variables in X are restricted to being either 0 or 1: xi = 1 indicates i ∈ MDS, and xi = 0 otherwise. Eqs. (1) and (2) correspond to the objective function and the constraint, respectively. A = (ai j )n×n is the adjacency matrix of G, where n denotes the number of nodes in the network, and ai j = 1 indicates (i, j) ∈ E and ai j = 0otherwise, where i = j.
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Fig. 1. Illustration of the MDS and the SD-MDS of a complex network. (a) A simple network is roughly classified into two communities separated by two dashed red lines. (b) The MDS of the network. (c) A problem in the MDS model caused by a link failure. (d) For the SD-MDS model, a driver node can control a non-driver node if the link between them belongs to a triangle. Non-driver nodes and driver nodes are blue and green colored, respectively. The failed link is denoted by a dashed blue line with a red cross, and the uncontrolled node is orange colored. The bold red lines pointing at non-driver nodes from driver nodes show that the links satisfy the cyclic structures for the SD-MDS model. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
1 . Here, A = ( .. an1
xi +
j∈V
... .. . ···
a1n . . ) and B = [1, 1, · · ·, 1]T . Therefore, Eq. (2) can be rewritten as . 1
ai j x j ≥ 1, i, j = 1, 2, · · ·, n
(3)
Fig. 1(a) and (b) correspond to a network and its MDS, respectively, where non-driver nodes and driver nodes are blue and green colored, respectively. From the figure, we can see that the MDS model assumes that each non-driver node is controllable so long as it is adjacent to a driver node. 2.3. SD-MDS model Hierarchical structures such as communities are widely observed in complex networks. To control a network-based system, we should fully dominate its structures since the system’s structures correspond to its functional units, and each unit has its realm of authority. To achieve effective control for non-driver nodes, it is better that a non-driver node and its driver node belong to the same unit. Whether a non-driver node can be fully driven by a driver node is highly associated with their neighbors, since Shang and Bouffanais [42] have found that in the consensus-reaching process, the behavior of a node is highly influenced by its neighbors. In Fig. 1, we illustrate the problems of the MDS model. The network in Fig. 1(a) is roughly classified into two communities separated by two dashed red lines. Nodes within same communities are densely connected compared with those of different communities. For the MDS model, a hub node tends to be a driver node, such as node 3 in Fig. 1(c), and a non-driver node (node 5) and its driver node (node 3) belong to different communities. If a link failure occurs, node 5 is unreachable from node 3 since the single link, (3, 5), as a bridge connecting different units, is more likely to be a target of attack. Fortunately, Radicchi et al. [36] observed that links connecting nodes of different communities are included in few or no triangles, while many triangles exist within communities. Therefore, we are motivated to develop a new framework, hierarchical control of complex networks by dominating communities, and the control over whole networks is transformed into that within communities, i.e., a non-driver node is effectively controlled by its driver node if they belong to the same community. For hierarchical control, we propose the structure-driven minimum dominating set (SD-MDS) model, which assumes that a driver node can control a non-driver node if the link between them belongs to a cyclic structure (e.g., triangle or quadrangle) of the network. In the SD-MDS model, node 2 in Fig. 1(d) can control node 1, since the link (2, 1) belongs to a triangle. Node 4 can only be a driver node, since the link (3, 4) does not belong to any triangles. In addition, non-driver nodes and their driver nodes belong to the same community, e.g., node 1 and node 3 can be driven by node 2 in the community on the left. The determination of the SD-MDS can also be solved by binary integer programming. Formally,
min
i∈V
xi
(4)
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Fig. 2. Illustration of the SD-MDSs based on different cyclic structures. (a)–(e) correspond to the SD-MDSs based on E(3) , E(3, 3) , E(4) , E(3, 4) and E(4, 4) , respectively. The SD-MDS based on E(3) illustrates that a non-driver node is controlled by a driver node so long as the link between them belongs to at least a triangle, which is similar to those of E(3, 3) , E(4) , E(3, 4) and E(4, 4) . The driver nodes are green colored, and the bold red lines pointing at non-driver nodes from driver nodes show that the links satisfy the cyclic structures. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
subject to
E(g,g ... ) X ≥ B, where E(g,g ... ) =
E(g,g ... ) =
(5) ... .. .
1 .. .
(
(g,g ... ) en1 (ei(jg,g ...) )n×n
(g,g ... )
e1n
···
.. .
).
1 (g )
is a “0–1 matrix, and ei j = 1 indicates that the link (i, j) belongs to a cyclic structure of order g
(e.g., for a triangle, g = 3, and for a quadrangle, g = 4), and 0 otherwise. In particular, ei(j3,3 ) = 1 or ei(j4,4 ) = 1 indicates that the link (i, j) belongs to two different triangles or quadrangles. Similarly, ei(j3,4 ) =1 indicates that the link (i, j) belongs to a triangle as well as a quadrangle. Similarly, B = [1, 1, · · ·, 1]T . Therefore, Eq. (5) can be rewritten as
xi +
j∈V
ei(jg,g ... ) x j ≥ 1, i, j = 1, 2, · · ·, n.
(6)
In Fig. 2, we illustrate the SD-MDS based on different cyclic structures. Fig. 2(a) shows the SD-MDS based on E(3) ; e.g., node 4 can control nodes 1, 2 and 3, since the links (4, 1), (4, 2) and (4, 3) belong to a triangle, respectively. Fig. 2(b) shows the SD-MDS based on E(3, 3) ; e.g., node 4 can control node 2, since the link (4, 2) belongs to two different triangles. Similarly, Fig. 2(c)–(e) correspond to the SD-MDSs based on E(4) , E(3, 4) and E(4, 4) , respectively. 2.4. Implementation of SD-MDS The implementation of the SD-MDS consists of two steps:
(1) The transformation from the adjacency matrix to the matrix of cyclic structures, A ⇒ E(g,g ... ) (see Eqs. (7) and (8)), where Eq. (9) corresponds to the network of Fig. 2(b).
⎛
1 .. ⎝ A= . an1
ei(jg,g ... ) =
⎛
1 ⎜1 A=⎝ 0 1
⎞
... .. . ···
0 1 1 1
(i, j ) ∈ a
⎞
⎞
... ) e1(g,g n .. ⎟ . ⎠ 1
... .. . ···
n1
ai j = 1 , 0,
1 1 1 1
⎛
a1n 1 .. ⎠ ⇒ E(g,g ... ) = ⎜ .. ⎝ . . 1 e(g,g ... )
cyclicstructure o f order otherwise
⎛
1 1 1⎟ ⎜0 ( 3,3 ) ⇒ E =⎝ 1⎠ 0 1 0
0 1 0 1
(2) The determination of the SD-MDS based on E(g,g ... ) .
0 0 1 0
(7)
g, g ... respectively
(8)
⎞
0 1⎟ 0⎠ 1
(9)
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Fig. 3. Illustration of the response of the SD-MDS based on E(3) to the change of community structures in complex networks. (a)–(d) We fix the number of nodes as well as the number of links of the four networks. (a)–(c) correspond to the networks that can be roughly divided into one, two, and three communities, respectively. (d) corresponds to a network after we weaken the communities in the network of (b). The driver nodes are green colored, and we separate communities by dashed red lines. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
The most important step in the implementation of the SD-MDS is A ⇒ E(g,g ... ) . Here, we give pseudocodes for A ⇒ E(3 ) and A ⇒ E(4 ) . Pseudocode forA ⇒ E(3) Input:A Output:E(3) 1: for ai j = 1 (i = jandi, j ← 1, 2, · · ·, n) do 2: if |Neighbors(i ) ∩ Neighbors( j )| ≥ 1 3: then doei(j3) = 1
Pseudocode forA ⇒ E(4) Input:A Output:E(4) 1: for ai j = 1 (i = jandi, j ← 1, 2, · · ·, n) do 2: if ∃k ∈ Neighbors(i ),∃l ∈ Neighbors( j ) 3: andakl = 1(k = l = i = j;k, l ∈ {1, 2, · · ·, n})
4: 5:
4: 5: 6:
end if end for
Neighbors(i )denotes the set containing all the neighbors of node
then doei(j4) = 1 end if end for i , i.e., Neighbors (i ) = { j |ai j = 1, i = j }
For the second step, lpSolve of the R package is used to solve the linear programming (LP) problem [47]. The search strategies are fully discussed by Wuchty [47]. In the R package, lpSolve uses a “branch-and-bound” algorithm [19,47] based on LP to solve the problem. In the “branching”, by repeatedly adding constraints, the algorithm creates a binary search tree for the problem [47]. In this step, the algorithm generates two branches with xi = 0, 1 for the tree [47]. In general, the algorithm solves a series of LP-relaxation problems to find an optimal solution to the problem, and the variables in the binary-integer optimization are replaced by a weaker constraint, 0 ≤ x1 , x2 , · · ·, xn ≤ 1 [19,47]. Based on the discussion above, the procedure consists of three steps: (1) we search for a feasible solution; (2) so long as the search tree grows, we update the best feasible solution; (3) we check whether there is no better feasible solution by solving a series of LPrelaxation problems [19,47]. By looking at the C source code of the lpSolve package, we observe that it repeatedly adds constraints to restrict the optimal solution. In general, lpSolve can find the optimal solution if it exists. 2.5. Impact of community dynamics on SD-MDS For the MDS model, it is easier to control the networks with stronger communities and more heterogeneous community sizes [45]. Therefore, the SD-MDS model is more likely to respond sensitively to the dynamics of communities. Here, we use a simple example to illustrate the response of the SD-MDS to the dynamics, such as the sizes and the strengths of community structures, in Fig. 3. Although the networks in Fig. 3(a)–(d) maintain a fixed number of nodes
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as well as a fixed number of links, they consist of communities of different sizes. Intuitively, we can roughly classify the networks of Fig. 3(a)–(c) into networks of one, two, and three communities, respectively. The results in Fig. 3(a)–(c) indicate that the SD-MDS is more likely to be associated with the number of communities in the network. Fig. 3(d) corresponds to the network Fig. 3(b) after we weaken the network’s communities, and we can see that more driver nodes are needed to control the networks with weaker communities. From the results above, we can see that like the MDS, the SD-MDS is also highly associated with the strengths and the sizes of communities when the number of nodes and the number of links of the networks are fixed. 3. Results and discussions In this section, we first analyze the hierarchical control of Erdos–Renyi (E-R) networks [8] and scale-free (S-F) networks and then investigate the impact of community dynamics on the framework. Finally, we analyze hierarchical control of some real-world networks such as the Zachary karate-club network [51], the Dolphin network [18] and the co-appearance network in the novel Les Miserables [14]. 3.1. Hierarchical control of Erdos–Renyi and scale-free networks A random network with n nodes and m links based on the E-R model is chosen uniformly at random from the collection of all networks that have n nodes and m links [8]. For instance, for a network with 3 nodes and 2 links, each of the three possible networks on three nodes and two links are included with probability 1/3 [8]. A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. Here, the E-R networks and the S-F networks are generated by igraph of the R package based on a static model [13], and on average, over 10 realizations (n=10 0 0) are obtained for each. 3.1.1. Impact of average node degree on SD-MDS In this subsection, we study the hierarchical control of the E-R networks and the S-F networks and then analyze the impact of the average node degree of the networks on f mds =
nd n
, which is defined as the fraction of nodes in the networks
that are driver nodes, where nd denotes the number of driver nodes. Fig. 4 shows the results of fmds as a function of k for the E-R networks and the S-F networks, where k denotes the average node degree of the networks. In Fig. 4(a), we can see that fmds decreases with the increase of k for the E-R networks. For k ≥ 2, the curve of the MDS declines sharply compared with that of the SD-MDS, which indicates that the MDS is mainly associated with k, while the SD-MDS is highly associated with k as well as the network structures. We also find that for k > 2, the curve of the SD-MDS based on E(4) declines quickly compared with that of E(3) , which probably indicates that nodes are more likely to be members of quadruples than of triangles in the E-R networks with the increase of k. For k = 4, the SD-MDS based on E(4, 4) need more driver nodes to control the E-R networks than those of E(3) ; the opposite is true for k > 4. The results indicate that the structure E(4, 4) is more easily formed than that of E(3) in dense E-R networks. For k ≥ 10, the SD-MDSs based on E(3) and E(3, 4) yield nearly the same results. For k ≥ 12, the MDSs and the SD-MDSs based on E(4) and E(4, 4) also yield nearly the same results. On the whole, the MDS and the SD-MDS achieve the same results in complete graphs, i.e., k → n − 1. Similar results can be seen in Fig. 4(b) for the S-F networks. From the discussions above, we can conclude that the SD-MDS can more sensitively capture the structure characteristics of the E-R networks and the S-F networks compared with the MDS. In Fig. 4(c), we compare fmds of the MDS and the SD-MDS in the E-R networks and the S-F networks. We find that for the MDS, it is slightly easier to control the S-F networks with degree exponent γ = 2.2 compared with the E-R networks for fixed k, and fmds decreases as γ increases (see Fig. 4(c) and (d)). However, for the SD-MDS, this finding is remarkable, and the more complicated the structure is, the more obvious the finding is, e.g., the results of the SD-MDS based on E(3, 3) are more obvious than those of the SD-MDS based on E(3) . 3.1.2. Impact of degree heterogeneity on SD-MDS To test whether fmds is affected by the degree heterogeneity of the network, we use a measure called the degree heterogeneity, which is defined as the relative mean difference of the degree distribution of a network, proposed by Liu et al. [20]:
H=
s
abs(s − t ) · P (s ) · P (t )/k,
(10)
t
where P denotes the degree distribution of the network, s and t correspond to two variables ranging over all possible degrees of the nodes, and abs(· ) denotes the absolute value. Fig. 5 shows the results of fmds as a function ofHfor the E-R networks and the S-F networks. In Fig. 5(a), we can see that fmds increases as H increases for both the SD-MDS and the MDS, and the SD-MDS maintains a sensitive response to the change in H for the E-R networks. In general, the more complicated the structure is, the more sensitive the response is, e.g., the curve of the SD-MDS based on E(3, 3) rises quickly compared with that of E(4) . Similarly, we obtain the same results in
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Fig. 4. The impact of network parameters on the minimum driver nodes for the E-R networks and the S-F networks based on the MDS and the SD-MDS. (a) and (b) correspond to fmds as a function of k for the E-R networks and the S-F networks, respectively. (c) We compare the results of the MDSs and the SD-MDSs based on E(3) and E(3, 3) for variable γ . (d) We show the results of the SD-MDSs based on E(3,4) , E(4) and E(4,4) for variable γ .
Fig. 5(b) for the S-F networks. In Fig. 5(c), we compare fmds of the MDS and the SD-MDS in the E-R networks and the S-F networks. Overall, fmds increases as H increases, and fmds decreases as γ increases for fixed H (see Fig. 5(c) and (d)), which indicates that it is easier to control the S-F networks as γ increases for fixed H. 3.2. Hierarchical control of complex networks with community dynamics As we discussed in Section 2.5, the SD-MDS is closely associated with the dynamics of the network structures, such as the strengths and sizes of communities. Therefore, we study the hierarchical control of random networks with built-in variable communities generated by the Lancichinetti, Fortunato and Radicchi (LFR) benchmark [17]. 3.2.1. Impact of community strength on SD-MDS We produce three groups of random networks: (1) n = 10 0 0,k= 15, maxk= 50, minc= 20 and maxc= 50; (2) n = 10 0 0,k= 15, maxk= 50, minc= 40 and maxc= 50; and (3) n = 10 0 0,k= 25, maxk= 50, minc= 20 and maxc= 50. maxk is the maximum degree of the nodes. The maximum and the minimum community sizes are denoted by maxc and minc, respectively [17]. In Fig. 6(a), we find that fmds increases as mu increases, i.e., the communities in the networks become increasingly weak as mu increases, where mu is a mixing parameter. For each node, a fraction 1-mu of its links are associated with other nodes in same communities, and mu ∈ [0, 1]. The results indicate that the networks with built-in weaker communities are harder to control than those with stronger communities. We can also see that the SD-MDS maintains a more sensitive response to the change in the strengths of communities than the MDS. 3.2.2. Impact of community sizes on SD-MDS Furthermore, we analyze the impact of community sizes on the SD-MDS. Fig. 6(b) shows fmds as a function of mu for fixed k and variable minc, and the results indicate that the networks with more heterogeneous community sizes are easier to control. Like the findings in the subsection above, the SD-MDS also has a more sensitive response to the change of community sizes than the MDS. In addition, fmds decreases as k increases for fixed minc in Fig. 6(c). Similarly, the same finding can be observed in Fig. 6(c) for the SD-MDS.
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Fig. 5. The impact of degree heterogeneity on the minimum driver nodes for the E-R networks and the S-F networks based on the MDS and the SD-MDS. (a) and (b) correspond to fmds as a function ofHfor the E-R networks and the S-F networks, respectively. (c) We compare the results of the MDSs and the SD-MDSs based on E(3) and E(3, 3) for variable γ . (d) We show the results of the SD-MDSs based on E(3,4) , E(4) and E(4,4) for variable γ .
Fig. 7(a) shows kmds as a function of mu for fixed k and minc based on the MDS and the SD-MDS, where kmds denotes the average degree of driver nodes. We find in Fig. 7(a) that on the whole, kmds decreases as mu increases. Fig. 7(b) shows kmds as a function of mu for fixed k and variable minc, and kmds decreases as minc increases. Of course, kmds increases as k increases in Fig. 7(c). We also find that kmds is far above k, which directly indicates that highdegree nodes are preferentially selected as driver nodes in both the MDS and the SD-MDS. Therefore, kmds decreases as fmds increases. 3.2.3. Results on randomized versions Here, we randomize the above LFR networks based on two randomization procedures [2]: (1) node-degree conservation (NDC) and (2) degree-distribution conservation (DDC). For the NDC, each node in the randomized version has the same degree as in the original network (edge randomization) [2]. For the DDC, the global distribution of node degrees remains the same as in the original networks, but the degrees of nodes do not need to be preserved [2]. In Fig. 6(d), we can see that for the MDS, the results based on the randomization procedure of the DDC are closer to those of the original networks. Similarly, the SD-MDS yields the same results as the MDS (see Fig. 6(e) and (f)). In other words, fmds is associated with the degree distribution of the network for both the MDS and the SD-MDS. 3.3. Robustness analysis of hierarchical control of complex networks In this subsection, we test the robustness of the MDS and the SD-MDS from two aspects: (1) whether a non-driver node and its driver node belong to the same community and (2) the control rate of nodes after link removal. To test whether a non-driver node can be driven by a driver node in the same community, we propose a measure gmds , which is defined as the fraction of non-driver nodes such that the non-driver node and its driver node belong to the same community. Fig. 7(d) shows gmds as a function of mu for fixed k and minc based on the MDS and the SD-MDS. In Fig. 7(d), we find that gmds decreases as mu increases, and the SD-MDS based on E(3, 3) yields the best results. Fig. 7(e) shows gmds as a function of mu for fixed k and variable minc, and gmds decreases as minc increases for mu> 0.6; the opposite is observed for mu<= 0.6. Of course, gmds decreases as k increases in Fig. 7(f). The above results indicate that the SD-MDS
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Fig. 6. The impact of the dynamics of communities on fmds . (a) fmds as a function of mu for fixed k and minc. (b) fmds as a function of mu for fixed k and variable minc. (c) fmds as a function of mu for variable k and fixed minc. (d) fmds of the randomized versions of the networks based on the NDC (node-degree conservation) and the DDC (degree-distribution conservation) for the MDS. (e) and (f) fmds of the randomized versions of the networks based on the NDC and the DDC for the SD-MDS.
maintains effective control for non-driver nodes in the same community, and the more complicated the structure is, the more controllable the non-driver nodes are. To test the robustness under link failure, we use the control rate (r), which is defined as the fraction of nodes in the network that are controllable after the removal of links, i.e., r = that are controllable under link failure. Here, r ≥ fmds =
nd n
nc +nd n ,
where nc denotes the number of non-driver nodes
, i.e., lower bound r = fmds =
nd n
, since driver nodes are
always controllable in the process of link removal. To improve the resistance against link failure, Nacher and Akutsu [30] introduced robust MDS (RMDS), which assumes that each non-driver node is adjacent to two or more driver nodes, i.e., an MDS with cover C ≥ 2. Here, we show the control rates of the MDS, the RMDS and the SD-MDS after a fraction of the links are removed (denoted by l) in Fig. 8(a) and (b). In general, the RMDS is more robust than the MDS and the SD-MDS. In addition, since sparse networks contain a few triangles, which leads to a large r=0.68, the SD-MDS is more robust than the MDS against link removal (see Fig. 8(a)). However, as
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Fig. 7. kmds and gmds as functions of mu for the MDS and the SD-MDS. (a) kmds as a function of mu for fixed k and minc. (b) kmds as a function of mu for fixed k and variable minc. (c) kmds as a function of mu for variable k and fixed minc. (d) gmds as a function of mu for fixed k and minc. (e) gmds as a function of mu for fixed k and variable minc. (f) gmds as a function of mu for variable k and fixed minc.
k increases, an approximate value of r=0.1 is obtained for both the SD-MDS and the MDS (see Fig. 8(b)), which indicates that each link tends be a member of triangles that are destroyed easily as the fraction of removed links increases. Therefore, the SD-MDS is more fragile than the MDS under link attacks on dense networks. 3.4. Hierarchical control of real-world networks In this subsection, we further study the hierarchical control of three real-world networks with communities: the Zachary karate-club network [52], the Dolphin network [18] and the co-appearance network in the novel Les Miserables [14]. In Fig. 8(c), we can see that by limiting driver nodes’ realms of authority within communities, the SD-MDS often needs more driver nodes to control the real-world networks than the MDS. In Fig. 8(d), we analyze the randomized versions of the real-world networks based on the NDC and the DDC, and the results indicate that the number of driver nodes based on the SD-MDS is also associated with the degree distribution of the network. Fig. 9 illustrates the driver nodes of the Zachary karate-club network based on the MDS and the SD-MDS. Driver nodes of the Zachary karate-club network based on the MDS can be seen in Fig. 9(a), where the driver nodes are green colored, and
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Fig. 8. Robustness analysis of random networks and the control of real-world networks. (a) and (b) correspond to the robustness analyses after link removal for the MDS, the RMDS and the SD-MDS on sparse and dense networks, respectively. (c) fmds of real-world networks based on the MDS and the SD-MDS. (d) fmds of the randomized versions of the networks based on the NDC and the DDC for E(3) , E(3, 3) , E(3, 4) , E(4) and E(4, 4) of the SD-MDS.
the results show that the MDS tends to choose high-degree nodes preferentially. Fig. 9(b)-(e) correspond to driver nodes of the Zachary karate-club network based on E(3) , E(3, 3) , E(4) , and E(4, 4) of the SD-MDS, respectively, where E(3, 4) shares the same results with E(3) . Compared with the MDS, the SD-MDS is highly associated with the network’s structures, e.g., the leaf node (node 12) in Fig. 9(b)–(e) can only be a driver node, since the link (1, 12) does not belong to any cyclic structures. From the results of both the random networks and the real-world networks, we can see that compared with the MDS, the SD-MDS effectively captures the structures of complex networks, maintains a sensitive response to the dynamics of aspects of network structure, such as the strengths and the sizes of communities, and achieves effective control for nondriver nodes. The results also indicate that like the MDS, the number of driver nodes of the SD-MDS is associated with the degree distribution of the network. The work in this paper provides us with a deep understanding of the control and the structures of complex networks.
4. Conclusions A dynamic system’s structural controllability can be achieved by selecting nodes in the minimum dominating set (MDS) of the networks as driver nodes. However, hierarchical structures such as communities are widely observed in complex networks, which motivates us to develop a new framework, called the hierarchical control of complex networks by dominating communities, and the control over whole networks is transformed into that within communities, i.e., a non-driver node is effectively controlled by its driver node if they belong to the same community. For the hierarchical control, we propose the structure-driven minimum dominating set (SD-MDS) model, which assumes that a driver node can control a non-driver node if the link between them belongs to a cyclic structure of the network, while for the MDS model, a nondriver node is controlled so long as it is adjacent to a driver node. The results show that compared with the MDS model, the SD-MDS model effectively captures hierarchical structures of complex networks, maintains a sensitive response to the dynamics of aspects of network structure, such as the strengths and the sizes of communities, and achieves effective control for non-driver nodes. This work also indicates that like the MDS model, the number of driver nodes of the SD-MDS model is also associated with the network degree distribution. In future work, we will focus on the model and study the control of biological networks.
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Fig. 9. Illustration of the control of the Zachary karate-club network based on the MDS and the SD-MDS. (a) Driver nodes of the Zachary karate-club network based on the MDS. (b)–(e) Driver nodes of the Zachary karate-club network based on E(3) , E(3, 3) , E(4) and E(4, 4) of the SD-MDS, respectively, where E(3, 4) shares the same results with E(3) . The driver nodes are green colored. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Acknowledgments This work is supported by China Scholarship Council (Grant No. 201306965011) for Studying Abroad, the National Natural Science Foundation of China (Grant No. 61202175, 61502363), the Fundamental Research Funds for the Central Universities (Grant No. BDY181417, No. JB160303, No. JB160306), and Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2016JQ6044), and Natural Science Basic Research Plan in Ningbo City (Grant No. 2016A610034). References [1] R. Albert, A.-L. Barabási, Statistical mechanics of complex networks, Rev. Mod. Phys. 74 (2002) 47–97. [2] S. Brohée, K. Faust, G. Lima-Mendez, G. Vanderstocken, J. van Helden, Network analysis tools: from biological networks to clusters and pathways, Nat. Protocols 3 (10) (2008) 1616–1629. [3] L. Chen, Q. Yu, B. Chen, Anti-modularity and anti-community detecting in complex networks, Inf. Sci. 275 (2014) 293–313. [4] S.P. Cornelius, W.L. Kath, A.E. Motter, Realistic control of network dynamics, Nat. Commun. 4 (2013) 1942.
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Contents lists available at ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins
Dominating communities for hierarchical control of complex networks Peng Gang Sun a,b,∗, Xiaoke Ma a,∗ a b
School of Computer Science and Technology, Xidian University, P. O. Box 163, No. 2 South Taibai Road, Xi’an, 710071 China Center for Complex Data and Network Science, Xidian University, Xi’an, 710071, China
a r t i c l e
i n f o
Article history: Received 1 January 2017 Revised 21 April 2017 Accepted 31 May 2017 Available online 8 June 2017 Keywords: Hierarchical control Structure-driven minimum dominating set
a b s t r a c t Well-selected nodes driven by external signals in complex networks are of great importance for a dynamic system’s structural controllability, which can also be achieved by selecting nodes in the minimum dominating set (MDS) of the networks as driver nodes. However, hierarchical structures such as communities are widely observed in complex networks, which motivates us to develop the hierarchical control of complex networks by dominating communities, and the control over whole networks is transformed into that within communities, i.e., a non-driver node is effectively controlled by its driver node if they belong to the same community. For the hierarchical control, we propose the structure-driven minimum dominating set (SD-MDS), which assumes that a driver node can control a non-driver node if the link between them belongs to a cyclic structure of the networks. We further investigate the impact of community dynamics on our framework. The results show that it is easier to control the networks with stronger communities and more heterogeneous community sizes, and the SD-MDS achieves more effective control for non-driver nodes than the MDS. This work also indicates that the number of driver nodes for the SD-MDS is associated with the degree distribution of the network. © 2017 Elsevier Inc. All rights reserved.
1. Introduction The study of complex networks has been a hot topic and attracted a large amount of attention for many years [1,41]. Network structure analysis, such as community detection, as an important research area of complex networks, provides us with new insight into the hierarchical structures of complex networks, since communities described as densely connected subnetworks often correspond to important organizational/functional units in the networks [2,3,5–7,9–11,17,24–26,32,34– 38,43,44]. Recently, network control in complex networks has grown rapidly [4,12,20–23,27–31,33,39,45–51]. This area studies a dynamic system’s controllability. A dynamic system is controllable if suitable inputs of external signals can drive the system from any initial state to any desired final state in finite time [15,16,40]. Liu et al. [20] studied structural controllability on directed networks and transformed the problem into the identification of minimum driver nodes, which correspond to unmatched nodes in the maximum bipartite matching of the networks, and unmatched nodes as targets driven by external signals can achieve the structural controllability. Interestingly, Nacher and Akutsu [27,31] introduced the minimum dominating set (MDS) to study the structural controllability on undirected networks, which can be achieved by selecting nodes in ∗
Corresponding authors at: School of Computer Science and Technology, Xidian University, P. O. Box 163, No. 2 South Taibai Road, Xi’an, 710071 China. E-mail addresses: [email protected] (P.G. Sun), [email protected] (X. Ma).
http://dx.doi.org/10.1016/j.ins.2017.05.052 0020-0255/© 2017 Elsevier Inc. All rights reserved.
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the MDS of the network as driver nodes, since each link in the network is assumed to be bi-directional, and a driver node to control all its links separately, i.e., non-driver nodes are controllable if they are at least adjacent to a driver node. Therefore, driver nodes within the MDS can control the whole network. Remarkably, Nacher and Akutsu [27,31] also mentioned that the difference between the two control models is that the former assumes that external signals can only directly control driver nodes (unmatched nodes), which tend not to be high-degree nodes, while the latter assumes that each driver node can independently control its links, and driver nodes tend to be high-degree nodes, i.e., the MDS tends to identify important nodes in controlling the complex network [22,27,31]. Nevertheless, in both models, the number of driver nodes is mainly determined by the degree distribution of the network [20,22,27,31,45]. Hierarchical structures such as communities are widely observed in complex networks. To control a network-based system, we should fully dominate its structures since the system’s structures correspond to its functional units, and each unit has its realm of authority [2,3,5–7,9–11,17,24–26,32,34–37,43,44]. Therefore, we are motivated to develop a new framework, hierarchical control of complex networks by dominating communities, and the control over whole networks is transformed into that within communities, i.e., a non-driver node is effectively controlled by its driver node if they belong to the same community. For hierarchical control, we propose the structure-driven minimum dominating set (SD-MDS) model, in which a driver node can control a non-driver node if the link between them belongs to a cyclic structure of the networks, while for the MDS model, a non-driver node is controlled so long as it is adjacent to a driver node. We analyze the hierarchical control of random networks as well as real-world networks, and further investigate the impact of community dynamics on our framework. The rest of the paper is organized as follows. In Section 2, we introduce the SD-MDS model and illustrate the impact of community dynamics on our framework. In Section 3, we analyze the hierarchical control of random networks and realworld networks. The conclusion is provided in Section 4. 2. Dominating complex networks In this section, we first describe the MDS model and then introduce the SD-MDS model. Finally, we illustrate the impact of community dynamics on the SD-MDS. 2.1. Minimum dominating set In this paper, we use G(V, E) to denote an unweighted, undirected graph, where V and E correspond to the set of nodes and the set of edges, respectively. The minimum dominating set (MDS) of a graph is defined as an optimized subset of nodes such that each node of the network either belongs to the subset or is adjacent to an element of the subset [27–31]. Formally, Definition 1. (dominating set) V is a dominating set of G if V ⊆V, V = ∅, and ∀i ∈ V − V , ∃j ∈ V , (i, j) ∈ E, and i = j, where |V | = n and i, j ∈ {1, 2, · · ·, n}. Definition 2. (minimum dominating set) V is a minimum dominating set ofG if ∀V ∈ D, |V | ≤ |V |, where D is the set containing all the dominating sets of G. 2.2. MDS model For the MDS model, in which nodes belonging to the MDS are called driver nodes, the problem of dominating a network is transformed into the determination of its MDS, and we simply achieve full control of the network by dominating the driver nodes since each non-driver node is controllable so long as it is at least adjacent to a driver node [27–31]. The determination of the MDS can be solved by binary integer programming [27–31]. Formally,
min
i∈V
xi
(1)
subject to
AX ≥ B,
(2)
a11 . . . a1n . . .. . ), X = [x1 , x2 , · · ·, xn ]T and B = [b1 , b2 , · · ·, bn ]T . where A = ( .. . . an1 ··· ann Binary integer programming (also called “0–1 integer programming) is a mathematical optimization problem in which the variables in X are restricted to being either 0 or 1: xi = 1 indicates i ∈ MDS, and xi = 0 otherwise. Eqs. (1) and (2) correspond to the objective function and the constraint, respectively. A = (ai j )n×n is the adjacency matrix of G, where n denotes the number of nodes in the network, and ai j = 1 indicates (i, j) ∈ E and ai j = 0otherwise, where i = j.
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Fig. 1. Illustration of the MDS and the SD-MDS of a complex network. (a) A simple network is roughly classified into two communities separated by two dashed red lines. (b) The MDS of the network. (c) A problem in the MDS model caused by a link failure. (d) For the SD-MDS model, a driver node can control a non-driver node if the link between them belongs to a triangle. Non-driver nodes and driver nodes are blue and green colored, respectively. The failed link is denoted by a dashed blue line with a red cross, and the uncontrolled node is orange colored. The bold red lines pointing at non-driver nodes from driver nodes show that the links satisfy the cyclic structures for the SD-MDS model. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
1 . Here, A = ( .. an1
xi +
j∈V
... .. . ···
a1n . . ) and B = [1, 1, · · ·, 1]T . Therefore, Eq. (2) can be rewritten as . 1
ai j x j ≥ 1, i, j = 1, 2, · · ·, n
(3)
Fig. 1(a) and (b) correspond to a network and its MDS, respectively, where non-driver nodes and driver nodes are blue and green colored, respectively. From the figure, we can see that the MDS model assumes that each non-driver node is controllable so long as it is adjacent to a driver node. 2.3. SD-MDS model Hierarchical structures such as communities are widely observed in complex networks. To control a network-based system, we should fully dominate its structures since the system’s structures correspond to its functional units, and each unit has its realm of authority. To achieve effective control for non-driver nodes, it is better that a non-driver node and its driver node belong to the same unit. Whether a non-driver node can be fully driven by a driver node is highly associated with their neighbors, since Shang and Bouffanais [42] have found that in the consensus-reaching process, the behavior of a node is highly influenced by its neighbors. In Fig. 1, we illustrate the problems of the MDS model. The network in Fig. 1(a) is roughly classified into two communities separated by two dashed red lines. Nodes within same communities are densely connected compared with those of different communities. For the MDS model, a hub node tends to be a driver node, such as node 3 in Fig. 1(c), and a non-driver node (node 5) and its driver node (node 3) belong to different communities. If a link failure occurs, node 5 is unreachable from node 3 since the single link, (3, 5), as a bridge connecting different units, is more likely to be a target of attack. Fortunately, Radicchi et al. [36] observed that links connecting nodes of different communities are included in few or no triangles, while many triangles exist within communities. Therefore, we are motivated to develop a new framework, hierarchical control of complex networks by dominating communities, and the control over whole networks is transformed into that within communities, i.e., a non-driver node is effectively controlled by its driver node if they belong to the same community. For hierarchical control, we propose the structure-driven minimum dominating set (SD-MDS) model, which assumes that a driver node can control a non-driver node if the link between them belongs to a cyclic structure (e.g., triangle or quadrangle) of the network. In the SD-MDS model, node 2 in Fig. 1(d) can control node 1, since the link (2, 1) belongs to a triangle. Node 4 can only be a driver node, since the link (3, 4) does not belong to any triangles. In addition, non-driver nodes and their driver nodes belong to the same community, e.g., node 1 and node 3 can be driven by node 2 in the community on the left. The determination of the SD-MDS can also be solved by binary integer programming. Formally,
min
i∈V
xi
(4)
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Fig. 2. Illustration of the SD-MDSs based on different cyclic structures. (a)–(e) correspond to the SD-MDSs based on E(3) , E(3, 3) , E(4) , E(3, 4) and E(4, 4) , respectively. The SD-MDS based on E(3) illustrates that a non-driver node is controlled by a driver node so long as the link between them belongs to at least a triangle, which is similar to those of E(3, 3) , E(4) , E(3, 4) and E(4, 4) . The driver nodes are green colored, and the bold red lines pointing at non-driver nodes from driver nodes show that the links satisfy the cyclic structures. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
subject to
E(g,g ... ) X ≥ B, where E(g,g ... ) =
E(g,g ... ) =
(5) ... .. .
1 .. .
(
(g,g ... ) en1 (ei(jg,g ...) )n×n
(g,g ... )
e1n
···
.. .
).
1 (g )
is a “0–1 matrix, and ei j = 1 indicates that the link (i, j) belongs to a cyclic structure of order g
(e.g., for a triangle, g = 3, and for a quadrangle, g = 4), and 0 otherwise. In particular, ei(j3,3 ) = 1 or ei(j4,4 ) = 1 indicates that the link (i, j) belongs to two different triangles or quadrangles. Similarly, ei(j3,4 ) =1 indicates that the link (i, j) belongs to a triangle as well as a quadrangle. Similarly, B = [1, 1, · · ·, 1]T . Therefore, Eq. (5) can be rewritten as
xi +
j∈V
ei(jg,g ... ) x j ≥ 1, i, j = 1, 2, · · ·, n.
(6)
In Fig. 2, we illustrate the SD-MDS based on different cyclic structures. Fig. 2(a) shows the SD-MDS based on E(3) ; e.g., node 4 can control nodes 1, 2 and 3, since the links (4, 1), (4, 2) and (4, 3) belong to a triangle, respectively. Fig. 2(b) shows the SD-MDS based on E(3, 3) ; e.g., node 4 can control node 2, since the link (4, 2) belongs to two different triangles. Similarly, Fig. 2(c)–(e) correspond to the SD-MDSs based on E(4) , E(3, 4) and E(4, 4) , respectively. 2.4. Implementation of SD-MDS The implementation of the SD-MDS consists of two steps:
(1) The transformation from the adjacency matrix to the matrix of cyclic structures, A ⇒ E(g,g ... ) (see Eqs. (7) and (8)), where Eq. (9) corresponds to the network of Fig. 2(b).
⎛
1 .. ⎝ A= . an1
ei(jg,g ... ) =
⎛
1 ⎜1 A=⎝ 0 1
⎞
... .. . ···
0 1 1 1
(i, j ) ∈ a
⎞
⎞
... ) e1(g,g n .. ⎟ . ⎠ 1
... .. . ···
n1
ai j = 1 , 0,
1 1 1 1
⎛
a1n 1 .. ⎠ ⇒ E(g,g ... ) = ⎜ .. ⎝ . . 1 e(g,g ... )
cyclicstructure o f order otherwise
⎛
1 1 1⎟ ⎜0 ( 3,3 ) ⇒ E =⎝ 1⎠ 0 1 0
0 1 0 1
(2) The determination of the SD-MDS based on E(g,g ... ) .
0 0 1 0
(7)
g, g ... respectively
(8)
⎞
0 1⎟ 0⎠ 1
(9)
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Fig. 3. Illustration of the response of the SD-MDS based on E(3) to the change of community structures in complex networks. (a)–(d) We fix the number of nodes as well as the number of links of the four networks. (a)–(c) correspond to the networks that can be roughly divided into one, two, and three communities, respectively. (d) corresponds to a network after we weaken the communities in the network of (b). The driver nodes are green colored, and we separate communities by dashed red lines. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
The most important step in the implementation of the SD-MDS is A ⇒ E(g,g ... ) . Here, we give pseudocodes for A ⇒ E(3 ) and A ⇒ E(4 ) . Pseudocode forA ⇒ E(3) Input:A Output:E(3) 1: for ai j = 1 (i = jandi, j ← 1, 2, · · ·, n) do 2: if |Neighbors(i ) ∩ Neighbors( j )| ≥ 1 3: then doei(j3) = 1
Pseudocode forA ⇒ E(4) Input:A Output:E(4) 1: for ai j = 1 (i = jandi, j ← 1, 2, · · ·, n) do 2: if ∃k ∈ Neighbors(i ),∃l ∈ Neighbors( j ) 3: andakl = 1(k = l = i = j;k, l ∈ {1, 2, · · ·, n})
4: 5:
4: 5: 6:
end if end for
Neighbors(i )denotes the set containing all the neighbors of node
then doei(j4) = 1 end if end for i , i.e., Neighbors (i ) = { j |ai j = 1, i = j }
For the second step, lpSolve of the R package is used to solve the linear programming (LP) problem [47]. The search strategies are fully discussed by Wuchty [47]. In the R package, lpSolve uses a “branch-and-bound” algorithm [19,47] based on LP to solve the problem. In the “branching”, by repeatedly adding constraints, the algorithm creates a binary search tree for the problem [47]. In this step, the algorithm generates two branches with xi = 0, 1 for the tree [47]. In general, the algorithm solves a series of LP-relaxation problems to find an optimal solution to the problem, and the variables in the binary-integer optimization are replaced by a weaker constraint, 0 ≤ x1 , x2 , · · ·, xn ≤ 1 [19,47]. Based on the discussion above, the procedure consists of three steps: (1) we search for a feasible solution; (2) so long as the search tree grows, we update the best feasible solution; (3) we check whether there is no better feasible solution by solving a series of LPrelaxation problems [19,47]. By looking at the C source code of the lpSolve package, we observe that it repeatedly adds constraints to restrict the optimal solution. In general, lpSolve can find the optimal solution if it exists. 2.5. Impact of community dynamics on SD-MDS For the MDS model, it is easier to control the networks with stronger communities and more heterogeneous community sizes [45]. Therefore, the SD-MDS model is more likely to respond sensitively to the dynamics of communities. Here, we use a simple example to illustrate the response of the SD-MDS to the dynamics, such as the sizes and the strengths of community structures, in Fig. 3. Although the networks in Fig. 3(a)–(d) maintain a fixed number of nodes
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as well as a fixed number of links, they consist of communities of different sizes. Intuitively, we can roughly classify the networks of Fig. 3(a)–(c) into networks of one, two, and three communities, respectively. The results in Fig. 3(a)–(c) indicate that the SD-MDS is more likely to be associated with the number of communities in the network. Fig. 3(d) corresponds to the network Fig. 3(b) after we weaken the network’s communities, and we can see that more driver nodes are needed to control the networks with weaker communities. From the results above, we can see that like the MDS, the SD-MDS is also highly associated with the strengths and the sizes of communities when the number of nodes and the number of links of the networks are fixed. 3. Results and discussions In this section, we first analyze the hierarchical control of Erdos–Renyi (E-R) networks [8] and scale-free (S-F) networks and then investigate the impact of community dynamics on the framework. Finally, we analyze hierarchical control of some real-world networks such as the Zachary karate-club network [51], the Dolphin network [18] and the co-appearance network in the novel Les Miserables [14]. 3.1. Hierarchical control of Erdos–Renyi and scale-free networks A random network with n nodes and m links based on the E-R model is chosen uniformly at random from the collection of all networks that have n nodes and m links [8]. For instance, for a network with 3 nodes and 2 links, each of the three possible networks on three nodes and two links are included with probability 1/3 [8]. A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. Here, the E-R networks and the S-F networks are generated by igraph of the R package based on a static model [13], and on average, over 10 realizations (n=10 0 0) are obtained for each. 3.1.1. Impact of average node degree on SD-MDS In this subsection, we study the hierarchical control of the E-R networks and the S-F networks and then analyze the impact of the average node degree of the networks on f mds =
nd n
, which is defined as the fraction of nodes in the networks
that are driver nodes, where nd denotes the number of driver nodes. Fig. 4 shows the results of fmds as a function of k for the E-R networks and the S-F networks, where k denotes the average node degree of the networks. In Fig. 4(a), we can see that fmds decreases with the increase of k for the E-R networks. For k ≥ 2, the curve of the MDS declines sharply compared with that of the SD-MDS, which indicates that the MDS is mainly associated with k, while the SD-MDS is highly associated with k as well as the network structures. We also find that for k > 2, the curve of the SD-MDS based on E(4) declines quickly compared with that of E(3) , which probably indicates that nodes are more likely to be members of quadruples than of triangles in the E-R networks with the increase of k. For k = 4, the SD-MDS based on E(4, 4) need more driver nodes to control the E-R networks than those of E(3) ; the opposite is true for k > 4. The results indicate that the structure E(4, 4) is more easily formed than that of E(3) in dense E-R networks. For k ≥ 10, the SD-MDSs based on E(3) and E(3, 4) yield nearly the same results. For k ≥ 12, the MDSs and the SD-MDSs based on E(4) and E(4, 4) also yield nearly the same results. On the whole, the MDS and the SD-MDS achieve the same results in complete graphs, i.e., k → n − 1. Similar results can be seen in Fig. 4(b) for the S-F networks. From the discussions above, we can conclude that the SD-MDS can more sensitively capture the structure characteristics of the E-R networks and the S-F networks compared with the MDS. In Fig. 4(c), we compare fmds of the MDS and the SD-MDS in the E-R networks and the S-F networks. We find that for the MDS, it is slightly easier to control the S-F networks with degree exponent γ = 2.2 compared with the E-R networks for fixed k, and fmds decreases as γ increases (see Fig. 4(c) and (d)). However, for the SD-MDS, this finding is remarkable, and the more complicated the structure is, the more obvious the finding is, e.g., the results of the SD-MDS based on E(3, 3) are more obvious than those of the SD-MDS based on E(3) . 3.1.2. Impact of degree heterogeneity on SD-MDS To test whether fmds is affected by the degree heterogeneity of the network, we use a measure called the degree heterogeneity, which is defined as the relative mean difference of the degree distribution of a network, proposed by Liu et al. [20]:
H=
s
abs(s − t ) · P (s ) · P (t )/k,
(10)
t
where P denotes the degree distribution of the network, s and t correspond to two variables ranging over all possible degrees of the nodes, and abs(· ) denotes the absolute value. Fig. 5 shows the results of fmds as a function ofHfor the E-R networks and the S-F networks. In Fig. 5(a), we can see that fmds increases as H increases for both the SD-MDS and the MDS, and the SD-MDS maintains a sensitive response to the change in H for the E-R networks. In general, the more complicated the structure is, the more sensitive the response is, e.g., the curve of the SD-MDS based on E(3, 3) rises quickly compared with that of E(4) . Similarly, we obtain the same results in
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Fig. 4. The impact of network parameters on the minimum driver nodes for the E-R networks and the S-F networks based on the MDS and the SD-MDS. (a) and (b) correspond to fmds as a function of k for the E-R networks and the S-F networks, respectively. (c) We compare the results of the MDSs and the SD-MDSs based on E(3) and E(3, 3) for variable γ . (d) We show the results of the SD-MDSs based on E(3,4) , E(4) and E(4,4) for variable γ .
Fig. 5(b) for the S-F networks. In Fig. 5(c), we compare fmds of the MDS and the SD-MDS in the E-R networks and the S-F networks. Overall, fmds increases as H increases, and fmds decreases as γ increases for fixed H (see Fig. 5(c) and (d)), which indicates that it is easier to control the S-F networks as γ increases for fixed H. 3.2. Hierarchical control of complex networks with community dynamics As we discussed in Section 2.5, the SD-MDS is closely associated with the dynamics of the network structures, such as the strengths and sizes of communities. Therefore, we study the hierarchical control of random networks with built-in variable communities generated by the Lancichinetti, Fortunato and Radicchi (LFR) benchmark [17]. 3.2.1. Impact of community strength on SD-MDS We produce three groups of random networks: (1) n = 10 0 0,k= 15, maxk= 50, minc= 20 and maxc= 50; (2) n = 10 0 0,k= 15, maxk= 50, minc= 40 and maxc= 50; and (3) n = 10 0 0,k= 25, maxk= 50, minc= 20 and maxc= 50. maxk is the maximum degree of the nodes. The maximum and the minimum community sizes are denoted by maxc and minc, respectively [17]. In Fig. 6(a), we find that fmds increases as mu increases, i.e., the communities in the networks become increasingly weak as mu increases, where mu is a mixing parameter. For each node, a fraction 1-mu of its links are associated with other nodes in same communities, and mu ∈ [0, 1]. The results indicate that the networks with built-in weaker communities are harder to control than those with stronger communities. We can also see that the SD-MDS maintains a more sensitive response to the change in the strengths of communities than the MDS. 3.2.2. Impact of community sizes on SD-MDS Furthermore, we analyze the impact of community sizes on the SD-MDS. Fig. 6(b) shows fmds as a function of mu for fixed k and variable minc, and the results indicate that the networks with more heterogeneous community sizes are easier to control. Like the findings in the subsection above, the SD-MDS also has a more sensitive response to the change of community sizes than the MDS. In addition, fmds decreases as k increases for fixed minc in Fig. 6(c). Similarly, the same finding can be observed in Fig. 6(c) for the SD-MDS.
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Fig. 5. The impact of degree heterogeneity on the minimum driver nodes for the E-R networks and the S-F networks based on the MDS and the SD-MDS. (a) and (b) correspond to fmds as a function ofHfor the E-R networks and the S-F networks, respectively. (c) We compare the results of the MDSs and the SD-MDSs based on E(3) and E(3, 3) for variable γ . (d) We show the results of the SD-MDSs based on E(3,4) , E(4) and E(4,4) for variable γ .
Fig. 7(a) shows kmds as a function of mu for fixed k and minc based on the MDS and the SD-MDS, where kmds denotes the average degree of driver nodes. We find in Fig. 7(a) that on the whole, kmds decreases as mu increases. Fig. 7(b) shows kmds as a function of mu for fixed k and variable minc, and kmds decreases as minc increases. Of course, kmds increases as k increases in Fig. 7(c). We also find that kmds is far above k, which directly indicates that highdegree nodes are preferentially selected as driver nodes in both the MDS and the SD-MDS. Therefore, kmds decreases as fmds increases. 3.2.3. Results on randomized versions Here, we randomize the above LFR networks based on two randomization procedures [2]: (1) node-degree conservation (NDC) and (2) degree-distribution conservation (DDC). For the NDC, each node in the randomized version has the same degree as in the original network (edge randomization) [2]. For the DDC, the global distribution of node degrees remains the same as in the original networks, but the degrees of nodes do not need to be preserved [2]. In Fig. 6(d), we can see that for the MDS, the results based on the randomization procedure of the DDC are closer to those of the original networks. Similarly, the SD-MDS yields the same results as the MDS (see Fig. 6(e) and (f)). In other words, fmds is associated with the degree distribution of the network for both the MDS and the SD-MDS. 3.3. Robustness analysis of hierarchical control of complex networks In this subsection, we test the robustness of the MDS and the SD-MDS from two aspects: (1) whether a non-driver node and its driver node belong to the same community and (2) the control rate of nodes after link removal. To test whether a non-driver node can be driven by a driver node in the same community, we propose a measure gmds , which is defined as the fraction of non-driver nodes such that the non-driver node and its driver node belong to the same community. Fig. 7(d) shows gmds as a function of mu for fixed k and minc based on the MDS and the SD-MDS. In Fig. 7(d), we find that gmds decreases as mu increases, and the SD-MDS based on E(3, 3) yields the best results. Fig. 7(e) shows gmds as a function of mu for fixed k and variable minc, and gmds decreases as minc increases for mu> 0.6; the opposite is observed for mu<= 0.6. Of course, gmds decreases as k increases in Fig. 7(f). The above results indicate that the SD-MDS
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Fig. 6. The impact of the dynamics of communities on fmds . (a) fmds as a function of mu for fixed k and minc. (b) fmds as a function of mu for fixed k and variable minc. (c) fmds as a function of mu for variable k and fixed minc. (d) fmds of the randomized versions of the networks based on the NDC (node-degree conservation) and the DDC (degree-distribution conservation) for the MDS. (e) and (f) fmds of the randomized versions of the networks based on the NDC and the DDC for the SD-MDS.
maintains effective control for non-driver nodes in the same community, and the more complicated the structure is, the more controllable the non-driver nodes are. To test the robustness under link failure, we use the control rate (r), which is defined as the fraction of nodes in the network that are controllable after the removal of links, i.e., r = that are controllable under link failure. Here, r ≥ fmds =
nd n
nc +nd n ,
where nc denotes the number of non-driver nodes
, i.e., lower bound r = fmds =
nd n
, since driver nodes are
always controllable in the process of link removal. To improve the resistance against link failure, Nacher and Akutsu [30] introduced robust MDS (RMDS), which assumes that each non-driver node is adjacent to two or more driver nodes, i.e., an MDS with cover C ≥ 2. Here, we show the control rates of the MDS, the RMDS and the SD-MDS after a fraction of the links are removed (denoted by l) in Fig. 8(a) and (b). In general, the RMDS is more robust than the MDS and the SD-MDS. In addition, since sparse networks contain a few triangles, which leads to a large r=0.68, the SD-MDS is more robust than the MDS against link removal (see Fig. 8(a)). However, as
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Fig. 7. kmds and gmds as functions of mu for the MDS and the SD-MDS. (a) kmds as a function of mu for fixed k and minc. (b) kmds as a function of mu for fixed k and variable minc. (c) kmds as a function of mu for variable k and fixed minc. (d) gmds as a function of mu for fixed k and minc. (e) gmds as a function of mu for fixed k and variable minc. (f) gmds as a function of mu for variable k and fixed minc.
k increases, an approximate value of r=0.1 is obtained for both the SD-MDS and the MDS (see Fig. 8(b)), which indicates that each link tends be a member of triangles that are destroyed easily as the fraction of removed links increases. Therefore, the SD-MDS is more fragile than the MDS under link attacks on dense networks. 3.4. Hierarchical control of real-world networks In this subsection, we further study the hierarchical control of three real-world networks with communities: the Zachary karate-club network [52], the Dolphin network [18] and the co-appearance network in the novel Les Miserables [14]. In Fig. 8(c), we can see that by limiting driver nodes’ realms of authority within communities, the SD-MDS often needs more driver nodes to control the real-world networks than the MDS. In Fig. 8(d), we analyze the randomized versions of the real-world networks based on the NDC and the DDC, and the results indicate that the number of driver nodes based on the SD-MDS is also associated with the degree distribution of the network. Fig. 9 illustrates the driver nodes of the Zachary karate-club network based on the MDS and the SD-MDS. Driver nodes of the Zachary karate-club network based on the MDS can be seen in Fig. 9(a), where the driver nodes are green colored, and
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Fig. 8. Robustness analysis of random networks and the control of real-world networks. (a) and (b) correspond to the robustness analyses after link removal for the MDS, the RMDS and the SD-MDS on sparse and dense networks, respectively. (c) fmds of real-world networks based on the MDS and the SD-MDS. (d) fmds of the randomized versions of the networks based on the NDC and the DDC for E(3) , E(3, 3) , E(3, 4) , E(4) and E(4, 4) of the SD-MDS.
the results show that the MDS tends to choose high-degree nodes preferentially. Fig. 9(b)-(e) correspond to driver nodes of the Zachary karate-club network based on E(3) , E(3, 3) , E(4) , and E(4, 4) of the SD-MDS, respectively, where E(3, 4) shares the same results with E(3) . Compared with the MDS, the SD-MDS is highly associated with the network’s structures, e.g., the leaf node (node 12) in Fig. 9(b)–(e) can only be a driver node, since the link (1, 12) does not belong to any cyclic structures. From the results of both the random networks and the real-world networks, we can see that compared with the MDS, the SD-MDS effectively captures the structures of complex networks, maintains a sensitive response to the dynamics of aspects of network structure, such as the strengths and the sizes of communities, and achieves effective control for nondriver nodes. The results also indicate that like the MDS, the number of driver nodes of the SD-MDS is associated with the degree distribution of the network. The work in this paper provides us with a deep understanding of the control and the structures of complex networks.
4. Conclusions A dynamic system’s structural controllability can be achieved by selecting nodes in the minimum dominating set (MDS) of the networks as driver nodes. However, hierarchical structures such as communities are widely observed in complex networks, which motivates us to develop a new framework, called the hierarchical control of complex networks by dominating communities, and the control over whole networks is transformed into that within communities, i.e., a non-driver node is effectively controlled by its driver node if they belong to the same community. For the hierarchical control, we propose the structure-driven minimum dominating set (SD-MDS) model, which assumes that a driver node can control a non-driver node if the link between them belongs to a cyclic structure of the network, while for the MDS model, a nondriver node is controlled so long as it is adjacent to a driver node. The results show that compared with the MDS model, the SD-MDS model effectively captures hierarchical structures of complex networks, maintains a sensitive response to the dynamics of aspects of network structure, such as the strengths and the sizes of communities, and achieves effective control for non-driver nodes. This work also indicates that like the MDS model, the number of driver nodes of the SD-MDS model is also associated with the network degree distribution. In future work, we will focus on the model and study the control of biological networks.
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Fig. 9. Illustration of the control of the Zachary karate-club network based on the MDS and the SD-MDS. (a) Driver nodes of the Zachary karate-club network based on the MDS. (b)–(e) Driver nodes of the Zachary karate-club network based on E(3) , E(3, 3) , E(4) and E(4, 4) of the SD-MDS, respectively, where E(3, 4) shares the same results with E(3) . The driver nodes are green colored. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Acknowledgments This work is supported by China Scholarship Council (Grant No. 201306965011) for Studying Abroad, the National Natural Science Foundation of China (Grant No. 61202175, 61502363), the Fundamental Research Funds for the Central Universities (Grant No. BDY181417, No. JB160303, No. JB160306), and Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2016JQ6044), and Natural Science Basic Research Plan in Ningbo City (Grant No. 2016A610034). References [1] R. Albert, A.-L. Barabási, Statistical mechanics of complex networks, Rev. Mod. Phys. 74 (2002) 47–97. [2] S. Brohée, K. Faust, G. Lima-Mendez, G. Vanderstocken, J. van Helden, Network analysis tools: from biological networks to clusters and pathways, Nat. Protocols 3 (10) (2008) 1616–1629. [3] L. Chen, Q. Yu, B. Chen, Anti-modularity and anti-community detecting in complex networks, Inf. Sci. 275 (2014) 293–313. [4] S.P. Cornelius, W.L. Kath, A.E. Motter, Realistic control of network dynamics, Nat. Commun. 4 (2013) 1942.
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