Heterogeneity of central nodes explains the benefits of time-varying control scheduling in complex dynamical networks

Despite extensive research and remarkable advancements in the control of complex dynamical networks, most studies and practical control methods limit their focus to time-invariant control schedules (TICS). This is both due to their simplicity and the fact that the benefits of time-varying control schedules (TVCS) have remained largely uncharacterized. In this paper we study networks with linear and discrete-time dynamics and analyze the role of network structure in TVCS. First, we show that TVCS can significantly enhance network controllability over TICS both in small and large networks. Through the analysis of a scale-dependent notion of nodal centrality, we then show that optimal TVCS involves the actuation of the most central nodes at appropriate spatial scales at all times. Consequently, it is the scale-heterogeneity of the central nodes in a network that determine whether, and to what extent, TVCS outperforms conventional policies based on TICS. Here, scale-heterogeneity of a network refers to how diverse the central nodes of the network are at different spatial (local vs. global) scales. Several analytical results and case studies support and illustrate this relationship.

[1]  J. Coleman,et al.  The Diffusion of an Innovation Among Physicians , 1957 .

[2]  R. Kálmán.  Mathematical description of linear dynamical systems , 1963 .

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

[4]  P. Bonacich.  Factoring and weighting approaches to status scores and clique identification , 1972 .

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

[6]  David E. Boyce,et al.  IMPLEMENTATION AND EVALUATION OF COMBINED MODELS OF URBAN TRAVEL AND LOCATION ON A SKETCH PLANNING NETWORK , 1985 .

[7]  Neo D. Martinez.  Artifacts or Attributes? Effects of Resolution on the Little Rock Lake Food Web , 1991 .

[8]  Chi-Tsong Chen,et al.  Linear System Theory and Design , 1995 .

[9]  E. V. Dam.  Graphs with few eigenvalues. An interplay between combinatorics and algebra , 1996 .

[10]  Dušan Teodorović,et al.  Traffic Control and Transport Planning:: A Fuzzy Sets and Neural Networks Approach , 1998 .

[11]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[12]  Robert J. Bursik,et al.  The Informal Control of Crime Through Neighborhood Networks , 1999 .

[13]  Ronald P. Barry,et al.  Monte Carlo estimates of the log determinant of large sparse matrices , 1999 .

[14]  Jon Kleinberg,et al.  Authoritative sources in a hyperlinked environment , 1999, SODA '98.

[15]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[16]  Arnold Reusken,et al.  Approximation of the Determinant of Large Sparse Symmetric Positive Definite Matrices , 2000, SIAM J. Matrix Anal. Appl..

[17]  Ramon Huerta,et al.  Contact tracing and epidemics control in social networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  D. Lusseau,et al.  The bottlenose dolphin community of Doubtful Sound features a large proportion of long-lasting associations , 2003, Behavioral Ecology and Sociobiology.

[19]  D. Bu,et al.  Topological structure analysis of the protein-protein interaction network in budding yeast. , 2003, Nucleic acids research.

[20]  Ilse C. F. Ipsen,et al.  Determinant Approximations , 2011, 1105.0437.

[21]  Andrew Parker,et al.  The Hidden Power of Social Networks: Understanding How Work Really Gets Done in Organizations , 2004 .

[22]  T. Geisel,et al.  Forecast and control of epidemics in a globalized world. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[23]  H. Lehrach,et al.  A Human Protein-Protein Interaction Network: A Resource for Annotating the Proteome , 2005, Cell.

[24]  S. L. Wong,et al.  Towards a proteome-scale map of the human protein–protein interaction network , 2005, Nature.

[25]  T. Tossavainen.  On the zeros of finite sums of exponential functions , 2006 .

[26]  Alessandro Vespignani,et al.  Reaction–diffusion processes and metapopulation models in heterogeneous networks , 2007, cond-mat/0703129.

[27]  Christos Faloutsos,et al.  Graph evolution: Densification and shrinking diameters , 2006, TKDD.

[28]  M. Moran,et al.  Large-scale mapping of human protein–protein interactions by mass spectrometry , 2007, Molecular systems biology.

[29]  Phillip Bonacich,et al.  Some unique properties of eigenvector centrality , 2007, Soc. Networks.

[30]  Shlomo Havlin,et al.  Finding a better immunization strategy. , 2008, Physical review letters.

[31]  Ernesto Estrada,et al.  Communicability in complex networks. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  Francesco Bullo,et al.  Distributed Control of Robotic Networks , 2009 .

[33]  James J. Collins,et al.  Next-Generation Synthetic Gene Networks , 2009, Nature Biotechnology.

[34]  Jorge Cortes,et al.  Distributed Control of Robotic Networks: A Mathematical Approach to Motion Coordination Algorithms , 2009 .

[35]  Tore Opsahl,et al.  Clustering in weighted networks , 2009, Soc. Networks.

[36]  D. Bernstein.  Matrix Mathematics: Theory, Facts, and Formulas , 2009 .

[37]  Bo Chen,et al.  A Review of the Applications of Agent Technology in Traffic and Transportation Systems , 2010, IEEE Transactions on Intelligent Transportation Systems.

[38]  Lee E. Miller,et al.  Frontiers in Systems Neuroscience Systems Neuroscience , 2022 .

[39]  Lev Muchnik,et al.  Identifying influential spreaders in complex networks , 2010, 1001.5285.

[40]  Olaf Sporns,et al.  Complex network measures of brain connectivity: Uses and interpretations , 2010, NeuroImage.

[41]  Marcel Salathé,et al.  Dynamics and Control of Diseases in Networks with Community Structure , 2010, PLoS Comput. Biol..

[42]  Ira B Schwartz,et al.  Enhanced vaccine control of epidemics in adaptive networks. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[44]  Jinde Cao,et al.  Adaptive bridge control strategy for opinion evolution on social networks. , 2011, Chaos.

[45]  Markus Diesmann,et al.  CoCoMac 2.0 and the future of tract-tracing databases , 2012, Front. Neuroinform..

[46]  Noah J. Cowan,et al.  Nodal Dynamics, Not Degree Distributions, Determine the Structural Controllability of Complex Networks , 2011, PloS one.

[47]  Jure Leskovec,et al.  Learning to Discover Social Circles in Ego Networks , 2012, NIPS.

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

[49]  Maryam Kamgarpour,et al.  A Hierarchical Flight Planning Framework for Air Traffic Management , 2012, Proceedings of the IEEE.

[50]  Cheng Wu,et al.  Exploration of the Association Rules Mining Technique for the Signal Detection of Adverse Drug Events in Spontaneous Reporting Systems , 2012, PloS one.

[51]  David J. Guggenmos,et al.  Restoration of function after brain damage using a neural prosthesis , 2013, Proceedings of the National Academy of Sciences.

[52]  John Lygeros,et al.  Optimal Sensor and Actuator Placement in Complex Dynamical Networks , 2013, ArXiv.

[53]  Tore Opsahl.  Triadic closure in two-mode networks: Redefining the global and local clustering coefficients , 2013, Soc. Networks.

[54]  D. Vecchio,et al.  Biomolecular Feedback Systems , 2014 .

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

[56]  Alexander Olshevsky,et al.  Minimal Controllability Problems , 2013, IEEE Transactions on Control of Network Systems.

[57]  Lin Zhao,et al.  On the Optimal Solutions of the Infinite-Horizon Linear Sensor Scheduling Problem , 2014, IEEE Trans. Autom. Control..

[58]  G Albi,et al.  Boltzmann-type control of opinion consensus through leaders , 2014, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[59]  Christine Klymko,et al.  Centrality and Communicability Measures in Complex Networks: Analysis and Algorithms , 2014 .

[60]  Tengyao Wang,et al.  A useful variant of the Davis--Kahan theorem for statisticians , 2014, 1405.0680.

[61]  Jinwoo Shin,et al.  Large-scale log-determinant computation through stochastic Chebyshev expansions , 2015, ICML.

[62]  Jung Eun Shim,et al.  EcoliNet: a database of cofunctional gene network for Escherichia coli , 2015, Database J. Biol. Databases Curation.

[63]  Rajeev K. Varshney,et al.  CicArVarDB: SNP and InDel database for advancing genetics research and breeding applications in chickpea , 2015, Database J. Biol. Databases Curation.

[64]  Christos Boutsidis,et al.  A Randomized Algorithm for Approximating the Log Determinant of a Symmetric Positive Definite Matrix , 2015, ArXiv.

[65]  Jean M. Vettel,et al.  Controllability of structural brain networks , 2014, Nature Communications.

[66]  George J. Pappas,et al.  Minimum number of probes for brain dynamics observability , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[67]  Stephen L. Smith,et al.  Submodularity and greedy algorithms in sensor scheduling for linear dynamical systems , 2015, Autom..

[68]  George J. Pappas,et al.  Analysis and Control of Epidemics: A Survey of Spreading Processes on Complex Networks , 2015, IEEE Control Systems.

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

[70]  Mohamed-Ali Belabbas,et al.  Geometric methods for optimal sensor design , 2015, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[71]  George J. Pappas,et al.  Minimal Actuator Placement With Bounds on Control Effort , 2014, IEEE Transactions on Control of Network Systems.

[72]  Jorge Cortés,et al.  Scheduling of control nodes for improved network controllability , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[73]  Alexey S. Matveev,et al.  Opinion Dynamics in Social Networks With Hostile Camps: Consensus vs. Polarization , 2015, IEEE Transactions on Automatic Control.

[74]  Hamsa Balakrishnan,et al.  Control and optimization algorithms for air transportation systems , 2016, Annu. Rev. Control..

[75]  Ling Shi,et al.  Optimal sensor scheduling for multiple linear dynamical systems , 2017, Autom..

[76]  Shreyas Sundaram,et al.  Sensor selection for Kalman filtering of linear dynamical systems: Complexity, limitations and greedy algorithms , 2017, Autom..

[77]  Jure Leskovec,et al.  Local Higher-Order Graph Clustering , 2017, KDD.

[78]  Stephen J. Roberts,et al.  Bayesian Inference of Log Determinants , 2017, UAI.

[79]  Jorge Cortés,et al.  Time-invariant versus time-varying actuator scheduling in complex networks , 2017, 2017 American Control Conference (ACC).

[80]  George J. Pappas,et al.  Selecting Sensors in Biological Fractional-Order Systems , 2018, IEEE Transactions on Control of Network Systems.

[81]  Y. J. Lee,et al.  Equilibrium Traffic Assignment on an Aggregated Highway Network for Sketch Planning , 2022 .