control energy and controllability of multilayer networks

Advances in Complex Systems Vol. 20, No. 0 (2017) 1750008 (25 pages) # .c World Scienti¯c Publishing Company DOI: 10.1142/S0219525917500084

CONTROL ENERGY AND CONTROLLABILITY OF MULTILAYER NETWORKS

DINGJIE WANG* and XIUFEN ZOU†

Advs. Complex Syst. Downloaded from www.worldscientific.com

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P. R. China Computational Science Hubei Key Laboratory, Wuhan University, Wuhan 430072, P. R. China *[email protected] †[email protected] Received 4 April 2017 Revised 13 June 2017 Accepted 3 August 2017 Published 29 August 2017 The controllability of multilayer networks has become increasingly important in many areas of science and engineering. In this paper, we identify the general rules that determine the controllability and control energy cost of multilayer networks. First, we quantitatively estimate the control energy cost of multilayer networks and investigate the impacts of di®erent coupling strength and coupling patterns on the control energy cost for multilayer networks. The results indicate that the average energy and the coupling strength have an approximately linear relationship in multilayer networks with two layers. Second, we study how the coupling strength and the connection patterns between di®erent layers a®ect the controllability of multilayer networks from both theoretical and numerical aspects. The obtained piecewise functional relations between the controllability's measure and coupling strength reveal the existence of an optimal coupling strength for the di®erent interconnection strategies in multilayer networks. In particular, the numerical experiments demonstrate that there exists a tradeo® between the optimal controllability and optimal control energy for selecting interlayer connection patterns in multilayer networks. These results provide a comprehensive understanding of the impact of interlayer couplings on the controllability and control energy cost for multilayer networks and provide a methodology for selecting the control nodes and coupling strength to maximize the controllability and minimize the control energy cost. Keywords: Multilayer networks; controllability; control energy; coupling strength; coupling patterns.

1. Introduction As is well known, many real-world networks are interconnected with each other, from economics to system biology and many other ¯elds, forming what is known as multilayer networks [3, 20]. For example, buses, subways and airlines constitute a † Corresponding

author. 1750008-1


D. Wang & X. Zou

multilayer public transportation network and relationship networks, email networks and other online social networks that couple into a multilayer communication network. By considering a multilayer network, various new aspects have been studied, such as cascading failures [4], di®usion [8], synchronization [1], robustness [7] and identi¯cation of essential components and modules [15, 26, 33]. Signi¯cant di®erences between the dynamical properties of multilayer networks and those of singlelayer networks have been demonstrated. In the past several decades, the control of complex networked systems has become increasingly important in a variety of biomedical, social, economic, and man-made systems [3, 6, 14, 16]. Many real-world problems can be considered as control problems of complex networked systems, with the goal of ¯nding strategies to change an undesirable state of a complex system into another, more desirable, state through an intervention [5, 10, 35]. Recently, although the controllability of single-layer networks has been widely studied [12, 16, 17, 23, 24, 27, 30–32, 36–38], very few works have investigated the controllability and control energy cost of multilayer networks. How to control multilayer networks and how much energy is needed are the two fundamental problems. Yuan et al. [39] analyzed the controllability of multilayer networks based on the maximum multiplicity theory [38]. Peng et al. [28] proposed a new method, co-controllability of multilayer networks, which stresses the control of one network by another network as well as the mutual control characteristics of multilayer networks based on minimum dominating sets. Menichetti et al. [18] proposed a combinatorial matching model to identify the minimal set of driver nodes in multilayer networks. Zhang et al. [40] assigned an important value to each node, which can help distinguish between peripheral nodes and central nodes in controlling multilayer scale-free networks. Pósfai et al. [25] developed a theory based on disjoint path covers to determine the minimum number of inputs necessary for full control in multiplex, multi-timescale networks. Despite some elementary advances in controlling multilayer networks, there are still many problems that deserve to be explored. In many real-world biology networks, for example, there exists high heterogeneity of the node degree, with hubs and peripheral nodes [22]. What happens if all layers of networks are connected with each other in a special pattern instead of a random way? In this study, We present a theoretical analysis and numerical simulations to investigate the impacts of di®erent interlayer couplings on the controllability and control energy cost for multilayer network. We provide a quantitative evaluation of the control energy and indicate that the coupling strength and the interlayer connection pattern dramatically a®ect the control energy cost and controllability of multilayer networks. These results can serve as a basis for a better understanding of the control of multilayer networks. 2. Multilayer Network Model and Evaluation Measures of Controllability and Control Energy In this study, we focus on the multilayer network [3, 20], also known as a network of networks, which is illustrated in Fig. 1. To explore the control energy cost and 1750008-2


Control Energy and Controllability of Multilayer Networks

Fig. 1. Schematic representation of a multilayer network with two layers, where each layer has ¯ve nodes and interlayer links can connect di®erent nodes on di®erent layers.

controllability of multilayer networks, we formalize the system-coupled S layer N-node networks with M external control inputs. The system of ordinary di®erential equations (ODEs) is given by Eq. (1). Here, xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; . . . ; xSN ðtÞÞ T is the state of nodes at time t. is an SN SN adjacent matrix, in which A1 ; . . . ; AS stands for the interaction strength between the nodes of each layer, Qij ði; j ¼ 1; 2; . . . ; S; i 6¼ jÞ describes the interconnections of any two layers. B is an SN M control matrix, uðtÞ ¼ ðu1 ðtÞ; . . . ; uM ðtÞÞ T stands for the M control signals: 2 3 A1 Q12 Q1S 6Q 7 6 12 A2 Q2S 7 6 7 _ ¼ xðtÞ þ BuðtÞ ¼ 6 . xðtÞ ð1Þ .. 7xðtÞ þ BuðtÞ: .. .. 4 .. . 5 . . Q1S Q2S AS Based on the PBH rank condition [9], Yuan et al. [38] proposed a method that for the system (1) with arbitrary link weights, the minimal number of driver nodes can be exactly calculated by ND ¼ maxfði Þg; 1il

ð2Þ

where ði Þ ¼ SN rankði ISN Þ are the geometric multiplicity of i of , and i ði ¼ 1; 2; . . . ; lÞ are the distinct eigenvalues of . In this paper, the controllability 1750008-3

D. Wang & X. Zou

measure, denoted as nd , is de¯ned by nd ¼

ND : SN

ð3Þ

Johnson et al. [13] proposed a method to evaluate Rthe control energy by optit mizing the input vector to minimize the control energy 0f jjuðtÞjj 2 dt. Assuming that system (1) is controllable [17], if it can be driven from the initial state x0 ¼ 0 to the target state xd within the time interval ½0; tf , the minimum energy is given by [21] Z tf jju ðtÞjj 2 dt ¼ x Td M t1 xd ; ð4Þ f Advs. Complex Syst. Downloaded from www.worldscientific.com

R tf

0

where Mtf ¼ 0 e t BB T e dt is called the symmetric controllability Gramian matrix at time tf , and the optimal input vector is given by T t

u ðtÞ ¼ B T e

T ðt tÞ f

M t1 xd : f

ð5Þ

Muller et al. [21] provided some energy-related controllability metrics. In this paper, we focus on the following three measures: trace, maximum eigenvalue and minimum eigenvalue of the inverse controllability Gramian matrix. The ¯rst measure of controllability is given by the average value of the minimum control energy over the unit hypersphere fxd : jjxd jj ¼ 1g as shown in the following equation: R x T M t1 xd dxd trM t1 f jjxd jj¼1 d f R : ð6Þ ¼ Eaverage ¼ SN dx d jjx jj¼1 d

Other measures of controllability are given by the maximum and the minimum values of the minimum control energy over the unit ball fxd : jjxd jj ¼ 1g as follows: Emax ¼ max xd 6¼0

Emin ¼ min xd 6¼0

x Td M t1 xd f jjxd jj 2

1 ; min ðMtf Þ

ð7Þ

¼ min ðM t1 Þ ¼

1 : max ðMtf Þ

ð8Þ

f

x Td M t1 xd f jjxd jj 2

¼ max ðM t1 Þ ¼

f

3. Control Energy of Multilayer Networks In many real-world networks, even if a network is controllable in principle, it may not be controllable in practice if it costs an enormous energy or if it requires too much time to achieve the control goal [36]. Therefore, the energy cost of control is an important and unavoidable problem when controlling a multilayer network in practice. In this section, we provide a quantitative evaluation of the control energy for multilayer networks and investigate how the coupling strength and the connection patterns of interlayers a®ect the control energy cost of multilayer networks by combining theoretical derivation and numerical simulations. 1750008-4


3.1. Estimation of the minimal control energy Lemma 1 [19]. Assume A 2 R NN , ðAÞ is the maximal eigenvalue of 12 ðA þ A T Þ, then


jje At jj2 e ðAÞt ;

ð8t 0Þ:

Theorem 1. For any certain initial state x0 and any certain target state xd , the weighted networked system ð1Þ can be transformed between x0 and xd in the ¯nite time ½0; tf , and can be written as 2 3 2 3 0 Q12 Q1S A1 6 0 Q2S 7 7 6 7 6 Q12 .. 7 7þ6 ¼ H1 þ H2 ¼ 6 .. .. .. 7: 4 5 6 .. . 4 . . . . 5 AS Q1S Q2S 0 Here, H1 ; H2 are called the intralayer coupling and the interlayer coupling matrices, respectively. Then, we have Emin ¼

1 2ððH1 Þ þ ðH2 ÞÞ : max ðMtf Þ jjBjj 22 ðe 2ððH1 ÞþðH2 ÞÞtf 1Þ

ð9Þ

Proof. The Gramian matrix for system (1) can be written as Z tf Z tf T t T T t e BB e dt ¼ e ðH1 þH2 Þt BB T e ðH1 þH2 Þ t dt: Mtf ¼ 0

0

According to Lemma 1 and being a symmetric matrix, we have Z tf jjMtf jj2 jje ðH1 þH2 Þt jj 22 jjBjj 22 dt 0 Z tf 2 jje H1 t jj 22 jje H2 t jj 22 dt jjBjj 2 0 Z tf 2 e 2ðH1 Þt e 2ðH2 Þt dt jjBjj 2 0

jjBjj 22 ðe 2ððH1 ÞþðH2 ÞÞtf 1Þ : ¼ 2ððH1 Þ þ ðH2 ÞÞ Assume that ðMtf Þ is the spectral radius of Mtf . Because Mtf is a positive de¯nite matrix, then max ðMtf Þ ¼ ðMtf Þ jjMtf jj2 : Then, we have max ðMtf Þ jjMtf jj2

jjBjj 22 ðe 2ððH1 ÞþðH2 ÞÞtf 1Þ : 2ððH1 Þ þ ðH2 ÞÞ

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D. Wang & X. Zou

Therefore, Emin ¼

1 2ððH1 Þ þ ðH2 ÞÞ : max ðMtf Þ jjBjj 22 ðe 2ððH1 ÞþðH2 ÞÞtf 1Þ

Theorem 1 gives a lower bound of the minimal energy cost for an arbitrary multilayer network. Moreover, it uncovers that the control energy of a multilayer network is related to the eigenvalues of the intralayer coupling and the interlayer coupling matrices.


3.2. Mathematical formulations of control energy for multilayer networks Theorem 2. For any initial state x0 and any target state xd , the weighted networked system ð1Þ can transform between x0 and xd in the ¯nite time ½0; tf . If all nodes are directly driven, then 8 2i > P SN > < e 2tf i 1 ; ði 6¼ 0Þ; i¼1 i ð10Þ ; i ¼ Eaverage ¼ 1 > SN > : ; ði ¼ 0Þ; tf 8 2min ðÞ > > ; ðmin ðÞ 6¼ 0Þ; < 2t 1 e f min ðÞ 1 Emax ¼ ð11Þ ¼ 1 min ðMtf Þ > > : ; ðmin ðÞ ¼ 0Þ; tf 8 2min ðÞ > > ; ðmax ðÞ 6¼ 0Þ; < 2t 1 e f max ðÞ 1 Emin ¼ ð12Þ ¼ 1 min ðMtf Þ > > : ; ðmax ðÞ ¼ 0Þ: tf Proof. Because is a symmetric matrix, we have ¼ VPV T ; where P ¼ diagf1 ; 2 ; . . . ; SN g, and V represents the eigenvectors of . If all nodes are directly driven, then B becomes a unit diagonal matrix, i.e., BB T ¼ I. Therefore, we have Z tf T e t e t dt Mtf ¼ 0 2 Z tf 3 2ðt sÞ f 1 e ds 6 7 6 0 7 6 7 .. 6 7 T ¼ V6 7V : . 6 7 6 7 Z tf 4 5 2ðt sÞ SN ds e f 0

1750008-6


Because Z

tf 0

8 2tf i 1 > < e 2tf i 1 ; ði 6¼ 0Þ; 1 > > : ; tf

ði ¼ 0Þ:

Therefore, we can easily obtain Eqs. (10)(12). Theorem 2 gives the analytical expressions of Eaverage , Emax , and Emin using the eigenvalues of in the case that all nodes are controlled. Corollary 1. For the initial state x0 and any certain target state xd , the weighted networked system ð1Þ can transform between x0 and xd in the ¯nite time ½0; tf . It sorts the eigenvalues of the adjacency matrix in ascending order 1 < 2 < < m < mþ1 ¼ ¼ mþn ¼ 0 < mþnþ1 < SN . If all nodes are directly driven, then we have the following conclusions: (1) When tf is large tf

1 minfjmin ðÞj;jmax ðÞjg ,

Eaverage

1 SN

then

! m X n 2 i : tf i¼1

If the weighted adjacency matrix is negative de¯nite, then Eaverage

2tr SN

and if the weighted adjacency matrix is positive de¯nite, then Eaverage ¼ oðe 2tf min ðÞ Þ; (2) When tf is small tf

1 maxfjmin ðÞj;jmax ðÞjg ,

Eaverage Proof. (1) Because tf

1 minfjmin ðÞj;jmax ðÞjg ,

e 2tf i 1

þ1; ði > 0Þ; 1; ði < 0Þ; 1750008-7

ðtf ! þ1Þ:

then 1 : tf

we can approximately estimate ði ¼ 1; 2; . . . ; SNÞ:

D. Wang & X. Zou

Therefore,

8 0; > > 0Þ;

; ði ¼ 0Þ; > t > : f 2i ; ði < 0Þ;

i

So, we can easily obtain that

P SN

If is negative de¯nite, then

! m X n 2 i : tf i¼1

1 i¼1 i SN SN

Eaverage ¼


ði ¼ 1; 2; . . . ; SNÞ:

P SN

lim

tf !þ1

Eaverage e 2tf min ðÞ

¼ ¼

lim

tf !þ1

1 SN

2

P SN

2tr i¼1 i ¼ SN SN SN and if is positive de¯nite, then i > 0ði ¼ 1; 2; . . . ; SNÞ. Since Eaverage ¼

i¼1 i

PSN

2i i¼1 e 2tf i 1 e 2tf min ðÞ

1 tf !þ1 SN

SN X

lim

i¼1

2i ¼ 0; e 2tf ði min ðÞÞ 1=e 2tf min ðÞ

then Eaverage ¼ oðe 2tf min ðÞ Þ; (2) Because tf

1 maxfjmin ðÞj;jmax ðÞjg ,

e

2tf i

then we can approximately estimate

1 2tf i ;

ði ¼ 1; 2; . . . ; SNÞ:

Therefore, according to Eq. (10), we have 8 2i 1 > > < 2 t ¼ t 1 i f f ¼ ; i 1 > t f > : tf

ði ¼ 1; 2; . . . ; SNÞ:

So, we can easily obtain that

PSN

P SN Eaverage ¼

ðtf ! þ1Þ:

i¼1 i

SN

1 i¼1 tf

SN

¼

1 : tf

For an arbitrary complex network (a single network or a multilayer network), Corollary 1 indicates that (1) When the control time tf is large enough, the average energy Eaverage is determined by the non-positive eigenvalues of . A positive undirected network requires less average energy than a negative undirected network with the same size. In other words, controlling a positive undirected network is easier than 1750008-8

Fig. 2. Numerical simulations for two simple multilayer networks. (a) A positive de¯nite multilayer network, where 0 < 1 ¼ 0:1910 < 2 ¼ 0:6910 < 3 ¼ 1:3090 < 4 ¼ 1:8090. (b) A negative de¯nite multilayer network, where 1 ¼ 1:8090 < 2 ¼ 1:3090 < 3 ¼ 0:6910 < 4 ¼ 0:1090 < 0. (c) Average energy Eaverage as a function of the control time tf in the above two multilayer networks.



1750008-9

D. Wang & X. Zou

controlling a negative undirected network with the same size; (2) When the control time tf is small enough, then the average energy Eaverage is determined by the control time tf . To validate the above theoretical results, we use two simple multilayer networks to calculate their average energy Eaverage . The numerical experiments are presented in Fig. 2, which are consistent with our theoretical results. Based on the above theorems, in the following section, we will analyze the relationship between the coupling strength and average energy Eaverage in the case that all nodes are controlled for multilayer networks with two layers.


3.3. E®ect of the coupling strength on the control energy of multilayer networks In this section, we investigate the e®ect of the coupling strength on the control energy of multilayer networks. For simplicity, in this section, we assume that the interlayer links of multilayer networks connect only its replicas. In other words, in Eq. (1), Qij is a diagonal matrix in this case. For convenience, in the following discussion, we suppose Qij ¼ diagða1 ; . . . ; amij ; 0; . . . ; 0Þ, ak 6¼ 0ðk ¼ 1; 2; . . . mij Þ, mij 2 ½0; N. Here, mij represents the number of interlayer links between layers i and j. The coupling strength of multilayer networks, denoted as ij , is de¯ned by mij : ð13Þ ij ¼ N Here, i; j ¼ 1; 2; . . . ; S, and S represents the number of layers. In the following, we focus on a multilayer network with two layers (S ¼ 2) and m12 ¼ m21 , 12 ¼ 21 . Therefore, we denote m12 ¼ m21 , 12 ¼ 21 by m, , respectively. How to design the optimal coupling strength to minimize the control energy cost of multilayer networks is still poorly understood. To investigate the e®ect of the coupling strength on the control energy cost of multilayer networks, we ¯rst theoretically consider two simple multilayer networks: a Star–Star network composed of one single star network and another single star network and an FC–FC network composed of two fully connected networks. Theorem 3. Consider an undirected multilayer network with two layers, which is given by the following ODEs: A1 Qm _ ¼ xðtÞ þ BuðtÞ ¼ xðtÞ xðtÞ þ BuðtÞ; ð14Þ Qm A1 where 2N2N is the adjacent matrix, Qm ¼ diagðIm ; 0Þ, and Im is the m mð0 m NÞ unit matrix. For the initial state x0 and any certain target state xd , if the system can transform between x0 and xd in the ¯nite time ½0; tf , then we have the following conclusions: (1) When system ð14Þ is a Star–Star network and all nodes are controlled, then Eaverage ¼ 1 þ 1 ; 1750008-10

ð15Þ


where 1 ¼ ð1 N1 Þ t1f þ N1 ð

P6

2i i¼1 e 2tf i 1

2tf

þ1Þ 2ðee 2tf 1 Þ, 1 ¼

e 2tf þ1 2t e f 1

t1f , 1 ; 2 ; 3

and 4 ; 5 ; 6 are the roots of the following two equations, respectively: 3 2 2 ðN 2Þ þ ðN mÞ ¼ 0; 3 þ 2 2 ðN 2Þ ðN mÞ ¼ 0: (2) When system ð14Þ is an FC–FC network and all nodes are controlled, then Eaverage ¼ 2 þ 2 ; P 2tf þ N1 ð 4i¼1 where 2 ¼ ð1 N2 Þ e2e 2tf 1

e

i 2tf i 1

2 Þ, 2 ¼

ð16Þ 4t

2e f 4t e f 1

2tf

e2e þ t1f , 1 ; 2tf 1


2 and 3 ; 4 are the roots of the following two equations, respectively: 2 ðN 1Þ m ¼ 0;

2 ðN 3Þ ð2N m 2Þ ¼ 0:

Proof. (1) When the system (10) is the StarStar network, the characteristic polynomial of can be written as fðÞ ¼ jðA1 IN Þ þ Qm j jðA1 IN Þ Qm j ¼ ½ 3 þ 2 2 ðN 2Þ ðN mÞ½ 3 2 2 ðN 2Þ þ ðN mÞð 1Þ m2 ð þ 1Þ m2 2ðNm1Þ : Therefore, according to Eq. (10), we can easily obtain Eq. (15). (2) In a similar way, we can also prove Eq. (16). More generally, the average energy of general multilayer networks is approximatively estimated under three special conditions. Approximation 1. Consider an undirected multilayer network with two layers, which is given by the following ODEs: A1 Qm _ ¼ xðtÞ þ BuðtÞ ¼ xðtÞ xðtÞ þ BuðtÞ; ð17Þ Q m A2 where 2N2N is the adjacent matrix, Qm ¼ diagðIm ; 0Þ, and Im is the m mð0 m NÞ unit matrix. For the initial state x0 and any certain target state xd , if the system can transform between x0 and xd in the ¯nite time ½0; tf , then we have the following approximate evaluations: (1) When both A1 and A2 are large sparse networks and all nodes are controlled, then it holds that Eaverage 3 þ 3 ; where 3 ¼

2t

e f þ1 2t e f 1

t1f ; 3 ¼

1 tf

ð18Þ

:

(2) When both A1 and A2 are large densely connected networks and all nodes are controlled, then it holds that Eaverage 4 þ 4 ; 1750008-11

ð19Þ

D. Wang & X. Zou

P 4tf 2tf 2tf 4 ¼ 2t1f þ e2e e2e , 4 ¼ ð1 N2 Þ e2e þ N1 ð 4i¼1 2tf ii 4 Þ, 4tf 2tf 2tf 1 1 1 e 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1;2 ¼ ðN1Þ N2 2Nþ4mþ1 and 3;4 ¼ ðN3Þ N2 þ2N4mþ1 . (3) When A1 is a large sparse network and A2 is a large densely connected network and all nodes are controlled, then it holds that where

pffiffi pffi51

5 ¼

where

2ðe 2tf

Eaverage 5 þ 5 ; pffi pffiffi

ð

51Þtf

ð1 N2 Þ e 2tef 1 þ N1 ð

P3

þð 1Þ

ð

5þ1Þtf

5þ1Þtf

1Þ

5þ1Þe

pffi

2ðe

i i¼1 e 2tf i 1

ð

2tf

e 2tef 1 2t1f ,

ð20Þ 5 ¼ ð1 þ N1 Þ 2t1f þ

5 Þ, 1 ; 2 ; 3 are the roots of the following


equation: 3 þ ðN þ 2Þ 2 þ N ðN m þ 1Þ ¼ 0: Proof. (1) When both A1 and A2 are large sparse networks, we approximately assume that A1 ¼ A2 ¼ 0. Therefore, the characteristic polynomial of can be written as I Qm N fðÞ ¼ jI2N j Qm IN ¼ j IN þ Qm j j IN Qm j ¼ ð 1Þ m ð þ 1Þ m 2ðNmÞ : Therefore, according to Eq. (10), we can easily obtain Eq. (18). (2) When both A1 and A2 are large densely connected networks, we approximately assume that 0 1 1 1 0 1 A1 ¼ A2 ¼ .. .. . . .. : . . . . 1 1 0 NN

Then, the characteristic polynomial of can be written as fðÞ ¼ jðA1 IN Þ þ Qm j jðA1 IN Þ Qm j m1 ð þ 2Þ m1 ½ 2 ðN 1Þ m ð þ 1Þ 2ðNm1Þ ½ 2 ðN 3Þ ð2N m 2Þ: In a similar way, we can easily obtain Eq. (19). (3) When A1 is a large sparse network and A2 is a large densely connected network, we approximately assume that 0 1 1 1 0 1 A1 ¼ 0NN ; A2 ¼ .. .. . . .. : . . . . 1 1 0 NN

1750008-12


Then, the characteristic polynomial of can be written as A1 IN Qm fðÞ ¼ Qm A2 IN ½ 3 þ ðN þ 2Þ 2 þ N ðN m þ 1Þ Nm ! m1 ! m1 pffiffiffi pffiffiffi 5 1 5 þ 1 ð þ 1Þ Nm1 þ : 2 2


In a similar way, we can easily obtain Eq. (20). Theorem 3 and Approximation 1 give an approximately linear relationship between the coupling strength and the average control energy of multilayer networks from the perspective of theoretical analysis. To further investigate the e®ect of the coupling strength on the control energy cost of multilayer networks, from the perspective of numerical simulations, we present experimental results using two unweighted random multilayer networks with two layers (unweighted ER–ER and BA–BA), two weighted random multilayer networks with two layers (weighted ER–ER and BA–BA) and two real-world biological multilayer networks, i.e. H3N2 in°ammatory multilayer network (H3N2IMN) and H1N1 in°ammatory multilayer network (H1N1IMN) [15]. For two weighted random multilayer networks, the link weights are randomly generated in the range of ð0; 1Þ. The H3N2IMN and H1N1MN are constructed through integrating highthroughput data. The pro¯ling datasets of H3N2 and H1N1 are downloaded from the NCBI GEO database (GSE37571 and GSE19580) [2], and the constructed method of these two networks is shown in [15]. These two networks are composed of two layers: the host protein layer and the host gene layer. The structural properties of the two tested real biological multilayer networks are represented as shown in Table 1. The numerical results are shown in Figs. 3 and 4. Through theoretical analysis and numerical experiments, it is easy to realize that there is an approximately linear relationship between the coupling strength and the average control energy in multilayer networks.

Table 1. The structural properties of the two tested biological multilayer networks. Multilayer networks

Layers

Nodes

Edges

Average degree

H1N1IMN

Host protein layer Host gene layer

192 192

1542 1716

8.02 8.94

H3N2IMN

Host protein layer Host gene layer

200 200

1893 1864

9.47 9.32

1750008-13

D. Wang & X. Zou


(a)

(b)

(c)

(d)

Fig. 3. Relationship between the coupling strength and the average energy Eaverage for four double-layer random networks (control time tf ¼ 1), where all nodes are chosen as driver nodes and each point is the average over 20 repeated simulations. (a) and (b) are numerical simulations for unweighted ER–ER and BA–BA networks, respectively. (c) and (d) are numerical simulations for weighted ER–ER and BA–BA networks, respectively.

(a)

(b)

Fig. 4. Relationship between the coupling strength and the average energy Eaverage for two real-world biological multilayer networks (control time tf ¼ 1), where all nodes are chosen as driver nodes. (a) H1N1IMN network. (b) H3N2IMN network. 1750008-14



3.4. E®ect of interlayer coupling patterns on the control energy of multilayer networks In this section, we investigate the e®ect of the interlayer coupling patterns on the control energy of multilayer networks. Di®erent from the above section, the interlayer links of multilayer networks are chosen randomly in this section. It is well known that real networks exhibit a high heterogeneity of the node degree, with hubs (high-degree) and peripheral (low-degree) nodes. What happens if the interlayer coupling pattern of multilayer networks becomes a special pattern instead of random? Figure 5 shows a qualitative example of three di®erent types of connections for the Star–Star network: high-degree–high-degree (HH), high-degree–low-degree (HL), and low-degree–low-degree (LL). How to design the optimal interlayer coupling pattern to minimize the control energy of a multilayer network is a signi¯cant problem. To investigate the impact of the di®erent connection patterns of interlayers on the control energy cost of multilayer networks, for convenience, in the following numerical analysis, we split the nodes of each layer network into 20 subsets of nodes, ðÞ denoted as S i ði ¼ 1; 2; . . . ; 20; ¼ 1; 2Þ, respectively, according to the ascending node degree. The sum of degrees of these subsets of nodes satisfy the following relation: ðÞ

ðÞ

ðÞ

dðS 1 Þ dðS 2 Þ dðS 20 Þ;

ð ¼ 1; 2Þ;

ðÞ

where dðS i Þ represents the sum of the degrees of the ith subset. It is easily observed that there are many interlayer connection patterns if a subset of nodes in one layer is only connected to another subset in the other layer. For all possible con¯gurations, the role of the connection pattern in the control energy can be quanti¯ed by their in°uence on the value of the average energy

Fig. 5. (Color online) Schematic illustration of three di®erent interlayer connection patterns of the Star– Star network. Here, amaranth nodes are hub (high-degree) nodes and blue nodes are peripheral (lowdegree) nodes. 1750008-15


D. Wang & X. Zou

(a)

(b)

(c)

(d)

Fig. 6. Heat maps of average energy Eaverage of four multilayer networks for all possible inter-layer connection patterns, where the subset numbers are ordered according to the sum of the node degrees of the subsets. Here, the control time tf = 5 and all nodes are chosen as driver nodes. (a)–(d) represent the average energy Eaverage obtained from unweighted ER–ER ð2N ¼ 200; hki ¼ 1Þ, unweighted BA–BA ð2N ¼ 200; hki ¼ 2Þ, H1N1MN and H3N2IMN networks for all the 20 2 possible connection patterns, respectively.

measure tr M t1 . Figures 6(a)–6(d) show the average energy measure tr M t1 of four f f multilayer networks (unweighted BA–BA, ER–ER, H3N2IMN and H1N1IMN) for all the 20 2 possible interlayer connection patterns. The numerical results show that the HH interlayer connection pattern is the best choice for minimizing the control energy. Therefore, connecting the high-degree nodes of interlayer links is the most energy saving strategy, while connecting the low-degree nodes of interlayer links is the worst energy saving option. 1750008-16


4. Controllability of Multilayer Networks 4.1. E®ect of the coupling strength on the controllability of multilayer networks In this section, we investigate the e®ect of the coupling strength on the controllability of multilayer networks. For simplicity, in this section, we assume that the interlayer links of multilayer networks connect only its replicas. First, we also consider two simple multilayer regular networks: Star–Star and FC–FC networks from the perspective of theoretical analysis.


Theorem 4. Consider an undirected multilayer network with two layers, which is given by Eq. (14). Then, we have the following conclusions: (1) When the system ð14Þ is the Star–Star network, in other words, 2 3 0 1 1 0 07 ; 5 . . . .. . . 0 1 0 0 0 NN

61 A1 ¼ 4 .. then

nd ¼

2 ; 3 þ Þ 1 2 ¼ ; > : 2 N 3

8 rank ðÞ 1 > > ¼1 ; rank > :1

ðI2N 2N

ð21Þ

(2) When the system ð14Þ is the FC–FC network, in other words, 2 3 0 1 0 . . .. 1 1

61 A1 ¼ 4 ..

.. .

1 17

1 0

5

;

NN

then ðI2N þ Þ 1 2 1 ¼1 ; ; 2N N 3 3N nd ¼ > rank ðÞ 1 2 1 > :1 ¼ ; > : 2N 2 2N 3 3N 8 rank > > > > ¼1 ; ; < 1 2N N 3 ¼ > > rank ðI2N þ Þ 1 2 > :1 ¼ ; ð > Þ: 2N 2 N 3

(2) In a similar way, we can also prove Eq. (22).


More generally, the controllability of general multilayer networks is approximatively estimated under three special conditions. Approximation 2. Consider an undirected multilayer network with two layers, which is given by Eq. (17). Then, we have the following approximate evaluations: (1) When both A1 and A2 are large sparse networks, then it holds that 8 rank ðÞ 2 > > 1 ; ; 1 < 2N 3 nd > rank ðI2N þ Þ 2 > :1 ; > : 2N 2 3

ð23Þ

(2) When both A1 and A2 are large densely connected networks, then it holds that 8 rank ðI2N þ Þ 1 2 1 > > 1 þ ; ; 1 < 2N N 3 3N ð24Þ nd > rank ðÞ 1 2 1 > :1 ; > : 2N 2 2N 3 3N (3) When A1 is a large sparse network and A2 is a large densely connected network, then it holds that 8 rank ðÞ 1 1 1 > > ; þ ; >1 > > 2N 2 2 2N < pffiffi ð25Þ nd 5þ1 > rank I2N þ > 2 > 1 1 > > ; > þ : :1 2N 2 2 2N Proof. (1) When both A1 and A2 are large sparse networks, we approximately assume that A1 ¼ A2 ¼ 0. Therefore, the characteristic polynomial of can be written as I Qm N fðÞ ¼ jI2N j Qm IN ¼ j IN þ Qm j j IN Qm j ¼ ð 1Þ m ð þ 1Þ m 2ðNmÞ : 1750008-18


Therefore, according to Eq. (3), we can easily obtain Eq. (23). (2) When both A1 and A2 are large densely connected networks, we approximately assume that 0 1 1 1 0 1 : A1 ¼ A2 ¼ .. .. . . .. . . . . 1 1 0 NN

Then, the characteristic polynomial of can be written as fðÞ ¼ jðA1 IN Þ þ Qm j jðA1 IN Þ Qm j


m1 ð þ 2Þ m1 ½ 2 ðN 1Þ m ð þ 1Þ 2ðNm1Þ ½ 2 ðN 3Þ ð2N m 2Þ: In a similar way, we can easily obtain Eq. (24). (3) When A1 is a large sparse network and A2 is a large densely connected network, we approximately assume that 0 1 1 1 0 1 A1 ¼ 0NN ; A2 ¼ .. .. . . .. : . . . . 1 1 0 NN

Then, the characteristic polynomial of can be written as A I Qm N 1 fðÞ ¼ Qm A2 IN ½ 3 þ ðN þ 2Þ 2 þ N ðN m þ 1Þ Nm ! m1 ! m1 pffiffiffi pffiffiffi 5 1 5 þ 1 ð þ 1Þ Nm1 þ : 2 2 In a similar way, we can easily obtain Eq. (25). To further investigate the e®ect of the coupling strength on the controllability of multilayer networks, from the perspective of numerical simulations, we present experimental results using two unweighted random multilayer networks with two layers (unweighted ER–ER and BA–BA), two weighted random multilayer networks with two layers (weighted ER–ER and BA–BA) and two real-world biological multilayer networks, i.e. H3N2IMN and H1N1IMN networks. For two weighted random multilayer networks, the link weights are randomly generated in the range of ð0; 1Þ. The numerical results are shown in Figs. 7 and 8. Through the above theoretical analysis and numerical experiments, we indicate that there is a piecewise functional relationship between the coupling strength and controllability measure nd of multilayer networks with two layers. Moreover, it is easily observed that there exists a critical value 0 for the coupling strength . When < 0 , the controllability measure nd decreases linearly with the coupling strength . On the contrary, when 1750008-19

D. Wang & X. Zou


(a)

(b)

(c)

(d)

Fig. 7. Relationship between the coupling strength and the controllability measure nd for four doublelayer random networks, where each point is the average over 20 repeated simulations. (a) and (b) are numerical simulations for unweighted ER–ER and BA–BA networks, respectively. (c) and (d) are numerical simulations for weighted ER–ER and BA–BA networks, respectively.

(a)

(b)

Fig. 8. Relationship between the coupling strength and the controllability measure nd for two realworld biological multilayer networks. (a) H1N1MN network. (b) H3N2IMN network. 1750008-20


> 0 , the controllability measure nd shows the contrary tendency. In other words, there is an optimal coupling strength 0 that maximizes the controllability in multilayer networks, and it is di®erent for di®erent network topologies. 4.2. E®ect of the coupling patterns of interlayers on the controllability of multilayer networks


In this section, we investigate the e®ect of the interlayer coupling patterns on the controllability of multilayer networks. Di®erent from the above section, similarly, the interlayer links of multilayer networks are chosen randomly in this section. To investigate the impact of the interlayer connection pattern on the controllability of

(a)

(b)

(c)

(d)

Fig. 9. Heat maps of controllability measure nd of four multilayer networks for all possible interlayer connection patterns, where the subset numbers are ordered according to the sum of the node degrees of the subsets. (a) and (d) represent that Controllability's measure nd obtained from unweighted ER–ER ð2N ¼ 200; hki ¼ 1Þ, unweighted BA–BA ð2N ¼ 200; hki ¼ 2Þ, H1N1MN and H3N2IMN networks for all the 20 2 possible connection patterns, respectively. 1750008-21


D. Wang & X. Zou

multilayer networks, in a similar way, we split the nodes of each layer network into ðÞ 20 subsets of nodes, denoted as S i ði ¼ 1; 2; . . . ; 20; ¼ 1; 2Þ, respectively, according to the ascending node degree. For all possible con¯gurations, the role of di®erent connection patterns in the controllability can be quanti¯ed by their in°uence on the value of the controllability measure nd . Figures 9(a)–9(d) show the controllability measure nd of four multilayer networks (unweighted BA–BA, ER–ER, H3N2IMN and H1N1IMN) for all the 20 2 possible interlayer connection patterns. The numerical results ¯nd that the LL connections of the interlayer's links can minimize the controllability measure nd . In other words, selecting LL connections of interlayer links is the best choice for controllability. Therefore, through the above numerical experiments, we demonstrate that connecting the low-degree nodes of interlayer links is the best choice for controllability, while connecting the high-degree nodes of interlayer links is the worst choice for controllability. However, the LL interlayer connection pattern is the most energy wasting strategy according to the above discussion. In conclusion, there exists a paradox of selecting interlayer connection patterns: optimal controllability versus optimal control energy for controlling multilayer networks. How to reconcile this paradox is an interesting topic for the future work. 5. Conclusion and Discussion The control energy and controllability of multilayer networks are two fundamental and signi¯cant issues for the practical control of complex systems. In this paper, we have presented a theoretical analysis and numerical simulations to identify the general rules that determine the controllability and control energy of multilayer networks. First, we provided an estimation of the control energy and dramatically investigated the impact of di®erent interlayer couplings on the control energy for multilayer networks. The main contributions of our work include the following: (1) By theoretical derivation, we have obtained the lower bound of the minimal energy cost for an arbitrary multilayer network. Moreover, the results show that the control energy for multilayer networks is related to the eigenvalue of the intralayer and the interlayer coupling matrices. (2) The mathematical formulations of the control energy, including the average energy Eaverage , maximal energy Emax and minimal energy Emin , are provided by using the eigenvalues of the Grammian matrix when all nodes are directly driven. Moreover, relations between the control time and average energy are derived. (3) We derived an the approximately linear relationship between the average energy and the coupling strength in multilayer networks with two layers. (4) In multilayer networks, the HH interlayer connection pattern is the best choice for the control energy, while the LL interlayer connection pattern is the worst option. Second, we studied in a systematic way how the coupling strength and the connector nodes between networks with heterogeneous topology in°uence the controllability of multilayer networks. The main contributions of our work include the 1750008-22



following: (1) We demonstrated that the controllability measure is a piecewise function with the coupling strength, and revealed the existence of an optimal coupling strength for the di®erent connection strategies in multilayer networks with two layers. (2) In multilayer networks, the LL interlayer connection pattern is the best choice for controllability, while the HH interlayer connection pattern is the worst option. There are also several important aspects requiring further research, which include the following: (1) By the above discussion, we found that there exists a tradeo® between the optimal controllability and optimal control energy for selecting interlayer connection patterns in multilayer networks. How to reconcile this paradox is an interesting topic for the future work. (2) In exploring the impacts of di®erent coupling strength and coupling patterns on the control energy cost for multilayer networks, we only focused on a special case that \all nodes are chosen as driver nodes". For the general case, we need further investigation. (3) In general, many realistic networks are nonlinear [11, 12, 29, 34], so the control of nonlinear multilayer networks is an important topic for future work. Acknowledgments This work was supported by the Major Research Plan of the National Natural Science Foundation of China (No. 91530320) and the Chinese National Natural Science Foundation (No. 61672388).

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