Structural Robustness of Weighted Complex

ISSN: 0256 - 307 X

中国物理快报

Chinese Physics Letters

Volume 30 Number 10 October 2013

A Series Journal of the Chinese Physical Society Distributed by IOP Publishing Online: http://iopscience.iop.org/0256-307X http://cpl.iphy.ac.cn

C HINESE P HYSICAL S OCIET Y Institute of Physics PUBLISHING

CHIN. PHYS. LETT. Vol. 30, No. 10 (2013) 108901

Structural Robustness of Weighted Complex Networks Based on Natural Connectivity * ZHANG Xiao-Ke(张小可)1 , WU Jun(吴俊)1** , TAN Yue-Jin(谭跃进)1 , DENG Hong-Zhong(邓宏钟)1 , LI Yong(李勇) 2 1

College of Information Systems and Management, National University of Defense Technology, Changsha 410073 2 Department of Business Administration, Changsha University, Changsha 410073

(Received 13 May 2013) Natural connectivity has been recently proposed to efficiently characterize the structural robustness of complex networks. The natural connectivity, interpreted as the Helmholtz free energy of a network, can be derived from the graph spectrum. We extend the concept of natural connectivity to weighted complex networks, in which the weight represents the number of multiple edges. We prove that the weighted natural connectivity changes monotonically when the weights are increased or decreased. We investigate the influence of weight on the network robustness within scenarios of weight changing and show that the weighted natural connectivity allows a precise quantitative analysis of the structural robustness for weighted complex networks.

PACS: 89.75.Hc, 89.75.Fb

DOI: 10.1088/0256-307X/30/10/108901

In the past few years, the study of complex networks has attracted great attention. Complex networks rely on their structural robustness for their function and performance, i.e., the ability of a network to maintain its connectivity when the fraction of its vertices is damaged. As one of the most central topics in the field of complex networks, network robustness has received increasing attention.[1−11] Simple and effective measures of structural robustness are important for system design and optimization. A variety of measures have been proposed to measure the structural robustness of networks. For instance, super connectivity,[12] conditional connectivity,[13] restricted connectivity,[14] fault diameter,[15] toughness,[16] scattering number,[17] the expansion parameter,[18] and the isoperimetric number.[19] Unfortunately, all these robustness measures are time-consuming and not suitable for unfolding the robustness of weighted complex networks. The second smallest Laplacian eigenvalue (i.e., the algebraic connectivity)[20] is another remarkable measure of un-weighted network robustness. However the algebraic connectivity is equal to zero for all disconnected networks, which is also unsuitable for measuring weighted complex networks through the experiment validated in this study. Recently, the concept of natural connectivity was proposed as a spectral measure of structural robustness in complex networks.[21−24] The natural connectivity can be derived from the graph spectrum as an average eigenvalue, and can be interpreted as the Helmholtz free energy of a network.[25] The natural connectivity provides a sensitive and reliable measure of the struc-

tural robustness of complex networks and has received growing attention.[26−29] However, a basic assumption in most existing works on network robustness is that all edges and vertices in networks are identical in terms of their functional roles in the networks.[30] In fact, along with a complex topological structure, real networks display a large heterogeneity in the properties of the vertices or edges.[31] For instance, in a neural network, edges, which are dendritic connections, can have very different capabilities in terms of transmitting electrical signals. These features have not been considered in the past studies where edges are usually represented as binary states, i.e., either present or absent. It is therefore important to study the robustness of weighted complex networks in which vertices and edges are not treated on an equal footing. In this Letter, we extend the concept of natural connectivity to weighted complex networks. The weight of a vertex or an edge can be defined in different ways. Consider that redundancy backup is frequently used to enhance the robustness of technical networks, such as the presence of alternative water supply pipes in a seismically active region to protect water supply from being damaged by an earthquake.[32] Our study focuses on a class of weighted complex networks, in which the weight of an edge represents the number of multi edges. A complex network can be described by a undirected multigraph 𝐺 = (𝑉, 𝐸), where 𝑉 is the set of vertices, and 𝐸 ⊆ 𝑉 × 𝑉 is the multiset of unordered pairs of vertices, i.e., edges. A multigraph is a graph which is permitted to have multiple edges, that is, edges that have the same end vertices.[37] Let

* Supported by the National Science Foundation of China under Grant Nos 60904065, 71031007 and 71101013, and the Program for New Century Excellent Talents in University under Grant No NCET-12-0141. ** Corresponding author. Email: [email protected] © 2013 Chinese Physical Society and IOP Publishing Ltd

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𝑊 (𝐺) = (𝑤𝑖𝑗 )𝑁 ×𝑁 be the weighted adjacency matrix of 𝐺, where 𝑤𝑖𝑗 = 𝑤𝑗𝑖 ∈ 𝑍 + denotes the number of multi edges between vertices 𝑣𝑖 and 𝑣𝑗 if they are adjacent, and 𝑤𝑖𝑗 = 𝑤𝑗𝑖 = 0 otherwise. Let 𝑁 = |𝑉 | and 𝑀 = |𝐸| be the number of vertices and edges, respectively. It immediately follows that 𝑊 (𝐺) is a real symmetric matrix with real eigenvalues 𝜆1 ≥ 𝜆2 ≥ . . . ≥ 𝜆𝑁 . The set {𝜆1 , 𝜆2 , . . . , 𝜆𝑁 } is called the graph spectrum of 𝐺. A walk of length 𝑘 in a graph 𝐺 is an alternating sequence of vertices and edges 𝑣0 𝑒1 𝑣1 𝑒2 . . . 𝑒𝑘 𝑣𝑘 , where 𝑣𝑖 ∈ 𝑉 and 𝑒𝑖 = (𝑣𝑖−1 , 𝑣𝑖 ) ∈ 𝐸. A walk is closed if 𝑣0 = 𝑣𝑘 . Closed walks are directly related to the subgraphs of a graph. For instance, a closed walk of length 𝑘 = 2 corresponds to an edge and a closed walk of length 𝑘 = 3 represents a triangle. The number of closed walks is an important index for complex networks. Recently, we have proposed that the number of closed walks of all lengths quantify the redundancy of alternative paths in the graph and can therefore serve as a measure of network robustness.[21,22] Consider that shorter closed walks have more influence on the redundancy than longer closed walks, we define ∑︀∞ a weighted sum of numbers of closed walks 𝑆 = 𝑘=0 𝑛𝑘 /𝑘!, where 𝑛𝑘 is the number of closed walks of length 𝑘 > 0 and 𝑛0 = 𝑁 . This scaling ensures that the weighted sum does not diverge. (𝑘) Denote by 𝑛𝑖𝑗 the number of walks from 𝑣𝑖 to 𝑣𝑗 of ∑︀𝑁 (𝑘) length 𝑘 > 0. Then we have 𝑛𝑘 = 𝑖=1 𝑛𝑖𝑖 . Denote (𝑘) by 𝑤𝑖𝑗 the element in the 𝑘th power of 𝑊 . In a simple graph with no multi edges or self-loops, it follows from the definition (𝑘)

(𝑘)

𝑛𝑖𝑗 = 𝑤𝑖𝑗 .

(1)

Then we obtain 𝑛𝑘 =

∑︁𝑁 𝑖=1

(𝑘)

𝑛𝑖𝑖 = 𝑡𝑟𝑎𝑐𝑒(𝑊 𝑘 ) =

𝑁 ∑︁

𝜆𝑘𝑖 .

(2)

𝑖=1

Therefore, we have 𝑆=

∞ ∑︁ 𝑛𝑘 𝑘=0

𝑘!

=

∞ ∑︁ 𝑁 ∑︁ 𝜆𝑘 𝑖

𝑘=0 𝑖=1

𝑘!

=

𝑁 ∑︁ ∞ ∑︁ 𝜆𝑘 𝑖

𝑖=1 𝑘=0

𝑘!

=

𝑁 ∑︁

𝑒𝜆𝑖 . 𝑖=1 (3)

Note that 𝑆 will be a large number for large 𝑁 , the natural connectivity is then defined as an average eigenvalue of the graph as follows:[21] 𝑁 (︁ ∑︁ )︁ ¯ = ln( 𝑆 ) = ln 1 𝜆 𝑒𝜆𝑖 . 𝑁 𝑁 𝑖=1

(4)

Next we will prove that Eq. (1) is still tenable for a multigraph using the recursion method. When 𝑘 = 1, (1) we know that 𝑛𝑖𝑗 is just the number of multi edges from 𝑣𝑖 to 𝑣𝑗 . It follows from the definition of 𝑊 that (1)

𝑛𝑖𝑗 = 𝑤𝑖𝑗 .

(5)

Therefore, Eq. (1) is also tenable for 𝑘 = 1. Assuming that Eq. (1) is tenable for 𝑘 = 𝑚 ≥ 1, (𝑚) (𝑚) i.e., 𝑛𝑖𝑗 = 𝑤𝑖𝑗 , we consider the case of 𝑘 = 𝑚 + 1. We obtain (𝑚+1)

𝑛𝑖𝑗

=

𝑁 ∑︁

(𝑚) (1)

𝑛𝑖𝑡 𝑛𝑡𝑗 =

𝑡=1

𝑁 ∑︁

(𝑚)

𝑤𝑖𝑡 𝑤𝑡𝑗 .

(6)

𝑡=1

Note that 𝑊 𝑚 · 𝑊 = 𝑊 𝑚+1 , thus we obtain (𝑚+1)

𝑤𝑖𝑗

(𝑚)

(𝑚)

= 𝑤𝑖1 𝑤1𝑗 + 𝑤𝑖2 𝑤2𝑗 (𝑚)

+ . . . + 𝑤𝑖𝑁 𝑤𝑁 𝑗 =

𝑁 ∑︁

(𝑚)

𝑤𝑖𝑡 𝑤𝑡𝑗 .

(7)

𝑡=1

Using Eqs. (6)and (7), we obtain (𝑚+1)

𝑛𝑖𝑗

(𝑚+1)

= 𝑤𝑖𝑗

(𝑘)

.

(8) (𝑘)

Consequently, we prove that 𝑛𝑖𝑗 = 𝑤𝑖𝑗 is still tenable for a multigraph 𝐺 = (𝑉, 𝐸). It means that the concept of natural connectivity can be extended to weighted complex networks, in which the weight represents the number of multi edges. In Ref. [21], we have proved that the natural connectivity changes strictly monotonically when edges are added or deleted. Now we will prove that it also changes strictly monotonically when the weights are increased or decreased. Consider a pair of vertices 𝑣𝑖 ∈ 𝑉 and 𝑣𝑗 ∈ 𝑉 , where 𝑤𝑖𝑗 = 𝑤𝑗𝑖 > 0. Denote by 𝐺′ the graph obtained by adding a redundancy edge 𝑒𝑖𝑗 between 𝑣𝑖 and 𝑣𝑗 , i.e., 𝑤𝑖𝑗 = 𝑤𝑖𝑗 + 1 and ′ ′′ 𝑤𝑗𝑖 = 𝑤𝑗𝑖 + 1. Let 𝑛 ̂︁𝑘 = 𝑛 ̂︁𝑘 + 𝑛 ̂︁𝑘 be the number ′ of closed walks of length 𝑘 in 𝐺′ , where 𝑛 ̂︁𝑘 is the number of closed walks of length 𝑘 containing 𝑒𝑖𝑗 and ′′ 𝑛 ̂︁𝑘 is the number of closed walks of length 𝑘 without ′′ ′ containing 𝑒𝑖𝑗 . Note that 𝑛 ̂︁𝑘 = 𝑛𝑘 and 𝑛 ̂︁𝑘 ≥ 0, thus we can obtain that 𝑛 ̂︁𝑘 ≥ 𝑛𝑘 and then 𝜆(𝐺′ ) ≥ 𝜆(𝐺). ′′ It is easy to show 𝑛 ̂︁𝑘 > 𝑛 ̂︁𝑘 = 𝑛𝑘 for some 𝑘, e.g., 𝑛 ̂︁2 = 𝑛2 + 2. Consequently, 𝜆(𝐺′ ) > 𝜆(𝐺), indicating that the natural connectivity increases strictly monotonically as the weight is increased. Similarly, we can prove that the natural connectivity changes strictly monotonically when the weight is decreased. To investigate the influence of weight increasing on the network robustness, we consider a scenario of weight increasing, i.e., adding the redundancy backup. An initial network with power-law degree distributions is generated using the BA model,[33] where 𝑁 = 1000 and average degree ⟨𝑘⟩ = 6. Four weight increasing strategies are considered: (i) random strategy, increasing the weights randomly; (ii) rich-rich strategy, i.e., increasing the weights between high-degree and highdegree vertices in descending order of 𝑑𝑖 · 𝑑𝑗 , where 𝑑𝑖 and 𝑑𝑗 are the degrees of the end vertices. (iii) poor-poor strategy, increasing the weights between low-degree and low-degree in ascending order of 𝑑𝑖 ·𝑑𝑗 ;

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1.55

Algebraic connectivity

(a)

1.5 1.45 1.4 Random strategy Rich-rich strategy Poor-poor strategy Rich-poor strategy

1.35 1.3

25

Random strategy Rich-rich strategy Poor-poor strategy Rich-poor strategy

1 0.8 0.6 0.4 0.2 0 10 (b) 8 6 4 2 0

200

400

600

800

1000

Number of decreased weights

Fig. 2. The network robustness measured by algebraic connectivity (a) and natural connectivity (b) as a function of decreased weights for four strategies: random strategy (squares), rich-rich strategy (circus), poor-poor strategy (diamonds) and rich-poor strategy (cross). Each quantity is an average over 100 realizations.

(b)

20

15

10 5 0

1.2

-0.2

(a)

1.25

Natural connectivity

gies mentioned above, the rich-rich strategy seems to be the best rich-rich strategy. However, there must exist other backup strategies better than the rich-rich strategy. Therefore, the optimal strategy of redundancy backup leading to the best network robustness is significant and interesting to research. To study the influence of weight on the network robustness in depth, we consider a similar scenario of weight decreasing, i.e., edge eliminating the redundancy backup. An initial network 𝐺 with a powerlaw degree distributions is generated using the BA model,[33] where 𝑁 = 1000 and average degree ⟨𝑘⟩ = 12. In this network, every edge has a redundancy backup edge. The same four strategies are adopted in this experiment.

Natural connectivity

Algebraic connectivity

(iv) rich-poor strategy, increasing the weights between high-degree and low-degree vertices in descending order of |𝑑𝑖 − 𝑑𝑗 |. For comparison, we first show in Fig. 1(a) the algebraic connectivity as a function of increased weights for the four weight increasing strategies. We find that algebraic connectivity increases as the weight increases. However, the network robustness grows most slowly for rich-to-rich strategy but most rapidly for poor-to-poor strategy. It is generally believed that the edges between high degree vertices are more important than the edges between low degree vertices. For the example, in the water supply network, the failure of a supply pipe between core water sources would be disastrous. Hence the network robustness measured by algebraic connectivity does not agree with our general intuition. On the other hand, there is no obvious difference in robustness growing for four different strategies. The four growing lines even cross together as the weight increases. This means that the network robustness based on algebraic connectivity cannot clearly reflect the effect on robustness under different strategies, which is not helpful for redundancy design and system optimization.

200

400

600

800

1000

Number of increased weights

Fig. 1. The network robustness measured by algebraic connectivity (a) and natural connectivity (b) as a function of increased weights for four weight increasing strategies: random strategy (squares), rich-rich strategy (circus), poor-poor strategy (diamonds), and rich-poor strategy (cross). Each quantity is an average over 100 realizations.

Figure 1(b) shows the result for the natural connectivity corresponding to the four strategies. We find a clear variation of the natural connectivity with distinct differences between the four weight increasing strategies, showing a clear ranking for the four edge elimination strategies: rich-rich strategy ≻ rich-poor strategy ≻ random strategy ≻ poor-poor strategy, which agrees with our intuition. In the four strate-

In Fig. 2(a), the algebraic connectivity decreases as the weights decrease. The network robustness drops most rapidly for the poor-to-poor strategy, but drops most slowly for the rich-to-rich strategy, which does not agree with our intuition. Moreover, we find that, for all four strategies, algebraic connectivity is equal to zero after particular edges are deleted, even in the case where only a few edges are deleted. This means that the algebraic connectivity loses discrimination when the network is disconnected. Compared with algebraic connectivity, the decrease of natural connectivity along with weights decreasing in Fig. 2(b) agrees with our intuition. In addition, natural connectivity shows its discrimination in the whole edge elimination process. Therefore, the natural connectivity can well reflect the effect of edge redundancy backup on structural robustness of weighted complex networks rather than algebraic connectivity. It is significant and interesting to research how to protect critical redundancy

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backup in complex networks. In summary, we have studied the structural robustness of weighted complex networks based on natural connectivity. Our result confirms that the concept of natural connectivity is also suitable for weighted complex networks. The natural connectivity shows its advantages in measuring the structural robustness of weighted complex networks: (1) its algorithm is of lower complexity, which implies that natural connectivity is of great practical use for large-scale weighted complex networks; (2) it changes strictly monotonically when the weights are increased or decreased, which implies that the weighted natural connectivity allows a precise and sensitive quantitative analysis of the structural robustness for weighted complex networks. Hence, natural connectivity is a simple and efficient structural robustness measure for weighted complex networks, and it is of great interest for the problem of how to allocate redundancies in a networked system so as to optimize the system performance in reliability engineering and system security. However, our study only focuses on a class of weighted complex networks, in which the weight of an edge represents the number of multi edges. Structural robustness measure of other weighted complex networks is worth researching in future. In addition, the optimal strategy of redundancy backup in a networked system based on natural connectivity is an interesting topic in future research.

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CONDENSED MATTER: STRUCTURE, MECHANICAL AND THERMAL PROPERTIES 106101 Ground States of Silicon-Multisubstituted Fullerene: First-Principles Calculations and Monte Carlo Simulations FAN Bing-Bing, SHI Chun-Yan, ZHANG Rui, JIA Yu 106102 Correlation between the Local Atomic Structure of Melts and Glass Forming Ability in Zr-Cu-Ni-al Alloys WU Chen, HUANG Yong-Jiang, SHEN Jun 106103 Biaxiality of Liquid Crystal Formed by Bent-Core Molecules with a Transverse Dipole Moment Deviating from their Angular Bisector YE Xiao-Fang, MERLITZ Holger, WU Chen-Xu

106801 Surface Oxidation Properties in a Topological Insulator Bi2 Te3 Film GUO Jian-Hua, QIU Feng, ZHANG Yun, DENG Hui-Yong, HU Gu-Jin, LI Xiao-Nan, YU Guo-Lin, DAI Ning 106802 Topography Multiplicity of Titanyl Phthalocyanine on Ultrathin Insulating Films Observed by STM YUAN Bing-Kai, CHEN Peng-Cheng, ZHANG Jun, DENG Ke, CHENG Zhi-Hai, WANG Chen

CONDENSED MATTER: ELECTRONIC STRUCTURE, ELECTRICAL, MAGNETIC, AND OPTICAL PROPERTIES 107201 Thick-Film Negative-Temperature-Coefficient Thermistors with a Linear Resistance-Temperature Relation LING Zhi-Yuan, HE Lin 107301 Nonlinear Optical Properties in a Quantum Dot of Some Polar Semiconductors A. Azhagu Parvathi, A. John Peter, Chang Kyoo Yoo 107302 Bipolar Resistive Switching Characteristics of TiN/HfOx /ITO Devices for Resistive Random Access Memory Applications TAN Ting-Ting, CHEN Xi, GUO Ting-Ting, LIU Zheng-Tang 107303 Effects of Contact Geometry on the Transport Properties of a Silicon Atom LIU Fu-Ti, CHENG Yan, YANG Fu-Bin, CHEN Xiang-Rong 107304 Low Bias Negative Differential Resistance Behavior in Carbon/Boron Nitride Nanotube Heterostructures WU Qiu-Hua, ZHAO Peng, LIU De-Sheng 107305 Dirac Points in Two-Dimensional Inverse Opals G. D. Mahan 107306 Hydrogen Storage Capacity Study of a Li+Graphene Composite System with Different Charge States SUI Peng-Fei, ZHAO Yin-Chang, DAI Zhen-Hong, WANG Wei-Tian 107401 The Observation of Small Polaron Tunnelling in the ab-Plane of K0.85 Fe1.66 Se2.0 MA Yong-Chang, YAN Qian, ZHAO Jie, LU Cui-Min 107501 X-Ray Magnetic Linear Dichroism of Co35 Fe65 Alloy Films Annealed with and without Oxygen Gas WANG De-Lai, CUI Ming-Qi, YANG Dong-Liang, XI Shi-Bo, LIU Li-Juan 107701 Dynamic Control of Tunneling Conductance in Ferroelectric Tunnel Junctions ZOU Ya-Yi, CHEW Khian-Hooi, ZHOU Yan 107702 Effective Electromagnetic Parameters and Absorbing Properties for Honeycomb Sandwich Structures with a Consideration of the Disturbing Term HU Ji-Wei, HE Si-Yuan, RAO Zhen-Min, ZHU Guo-Qiang, YIN Hong-Cheng 107801 Bulk Heterojunction Photovoltaic Devices Based on a Poly(2-Methoxy, 5-Octoxy)-1, 4-Phenylenevinylene-Single Walled Carbon Nanotube-ZnSe Quantum Dots Active Layer QU Jun-Rong, ZHENG Jian-Bang, WU Guang-Rong, CAO Chong-De

CROSS-DISCIPLINARY PHYSICS AND RELATED AREAS OF SCIENCE AND TECHNOLOGY 108101 Structure and Absorbability of a Nanometer Nonmetallic Polymer Thin Film Coating on Quartz Used in Piezoelectric Sensors GU Yu, LI Qiang, TIAN Fang-Fang, XU Bao-Jun 108102 Fabrication and Characterization of Single-Crystalline AgSbTe2 Nanowire Arrays YANG You-Wen, LI Tian-Ying, ZHU Wen-Bin, MA Dong-Ming, CHEN Dong 108103 Effect of Hydrogen and Nitrogen Carrier Gas Ratio on the Structural and Optical Properties of AlInGaN Alloy FENG Xiang-Xu, LIU Nai-Xin, ZHANG Lian, ZHANG Ning, ZENG Jian-Ping, WEI Xue-Cheng, LIU Zhe, WEI Tong-Bo, WANG Jun-Xi, LI Jin-Min

108104 The Geometry-Induced Superhydrophobic Property of Carpet-like Zinc Films LIANG Li-Xing, DENG Yuan, WANG Yao 108401 The Effect of Using 1-Dodecanethiol as a Processing Additive on the Performances of Polymer Solar Cells YANG Shao-Peng, WANG Tie-Ning, SHI Jiang-Bo, ZHANG Ye, LI Xiao-Wei, FU Guang-Sheng 108501 Temperature Characteristics of Monolithically Integrated Wavelength-Selectable Light Sources HAN Liang-Shun, ZHU Hong-Liang, ZHANG Can, MA Li, LIANG Song, WANG Wei 108502 Electrical Characteristics of High Mobility Si/Si0.5 Ge0.5 /SOI Quantum-Well p-MOSFETs with a Gate Length of 100 nm and an Equivalent Oxide Thickness of 1.1 nm MU Zhi-Qiang, YU Wen-Jie, ZHANG Bo, XUE Zhong-Ying, CHEN Ming 108503 Thermal and Structural Study of Mono- and Multi-Layered Thin Films Composed of CuAlS2 Chalcogenide Taher Ghrib, Rawdha Brini, Amel Lafi Al-otaibi, Muneera Abdullah Al-messiere 108701 Osmolyte Effects on the Unfolding Pathway of β-Lactoglobulin MENG Wei, PAN Hai, QIN Meng, CAO Yi, WANG Wei 108801 Analysis of Electron Recombination in Dye Sensitized Solar Cells Based on the Forward Bias Dependence of Dark Current and Electroluminescence Characterization XIAO Wen-Bo, LIU Wei-Qing, HE Xing-Dao 108901 Structural Robustness of Weighted Complex Networks Based on Natural Connectivity ZHANG Xiao-Ke, WU Jun, TAN Yue-Jin, DENG Hong-Zhong, LI Yong 108902 Cooperation of a Dissatisfied Adaptive Prisoner’s Dilemma in Spatial Structures ZHANG Wen, LI Yao-Sheng, DU Peng, XU Chen

GEOPHYSICS, ASTRONOMY, AND ASTROPHYSICS 109701 The Relation between the Magnetic Field and Spin Period of a Millisecond Pulsar PAN Yuan-Yue, WANG Na, ZHANG Cheng-Min