Optimal control of weighted networks based on node connection strength

Exploring how connection strengths between nodes affect the cost in controlling complex networks with fixed topological structure is an important issue both in theory and applications. In this paper, by considering optimal control of the networks, a matrix function optimization model is proposed to address such an issue. With pre-given input matrix, a normalized gradient descent method (NPGM) is developed to solve the optimization problem, so as to obtain the optimal connection strength matrix. With proposed NPGM, we find that a network adaptively changes its connection strength such that several control-flow subnetworks are self-formed. Moreover, we further point out that the control cost with optimal weight matrix is smallest when pre-located controller sources distribute evenly. These findings provide a comprehensive understanding of the impact of connection link weight on control cost for complex networks.

[1]  W. Rugh.  Linear System Theory , 1992 .

[2]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[3]  Jie Sun,et al.  Controllability transition and nonlocality in network control. , 2013, Physical review letters.

[4]  P. Lancaster,et al.  The theory of matrices : with applications , 1985 .

[5]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[6]  Alessandro Vespignani,et al.  Weighted evolving networks: coupling topology and weight dynamics. , 2004, Physical review letters.

[7]  F. Garofalo,et al.  Controllability of complex networks via pinning. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Francesco Bullo,et al.  Controllability Metrics, Limitations and Algorithms for Complex Networks , 2013, IEEE Transactions on Control of Network Systems.

[9]  Albert-László Barabási,et al.  Controllability of complex networks , 2011, Nature.

[10]  Albert-László Barabási,et al.  Evolution of Networks: From Biological Nets to the Internet and WWW , 2004 .

[11]  S. Shen-Orr,et al.  Network motifs: simple building blocks of complex networks. , 2002, Science.

[12]  D. Mason,et al.  Compartments revealed in food-web structure , 2003, Nature.

[13]  S. Strogatz.  Exploring complex networks , 2001, Nature.

[14]  I. E. Leonard.  The Matrix Exponential , 1996, SIAM Rev..

[15]  A. Barabasi,et al.  Weighted evolving networks. , 2001, Physical review letters.

[16]  Pan Di,et al.  Weighted complex network analysis of travel routes on the Singapore public transportation system , 2010 .

[17]  T. Vicsek,et al.  Uncovering the overlapping community structure of complex networks in nature and society , 2005, Nature.

[18]  Francesco Bullo,et al.  Controllability Metrics, Limitations and Algorithms for Complex Networks , 2014, IEEE Trans. Control. Netw. Syst..

[19]  Wen-Xu Wang,et al.  Exact controllability of complex networks , 2013, Nature Communications.

[20]  Jie Ren,et al.  Controlling complex networks: How much energy is needed? , 2012, Physical review letters.

[21]  John Lygeros,et al.  Submodularity of energy related controllability metrics , 2014, 53rd IEEE Conference on Decision and Control.

[22]  W. Marsden.  I and J , 2012 .

[23]  J. Slotine,et al.  Spectrum of controlling and observing complex networks , 2015, Nature Physics.

[24]  Kaare Brandt Petersen,et al.  The Matrix Cookbook , 2006 .

[25]  John Lygeros,et al.  On Submodularity and Controllability in Complex Dynamical Networks , 2014, IEEE Transactions on Control of Network Systems.

[26]  H. Weber,et al.  Analysis and optimization of certain qualities of controllability and observability for linear dynamical systems , 1972 .

[27]  Guoqi Li,et al.  Minimum-cost control of complex networks , 2015 .

[28]  A. Barabasi,et al.  Global organization of metabolic fluxes in the bacterium Escherichia coli , 2004, Nature.

[29]  Derek Ruths,et al.  Control Profiles of Complex Networks , 2014, Science.