logo

SCIENTIA SINICA Informationis, Volume 50 , Issue 11 : 1714(2020) https://doi.org/10.1360/SSI-2020-0087

Spreading dynamics on complex dynamical networks

More info
  • ReceivedApr 9, 2020
  • AcceptedMay 21, 2020
  • PublishedOct 20, 2020

Abstract


Funded by

国家自然科学基金(61751301,61533001,11971405,61703082)

中央高校基本科研业务费(20720180005,N2004004)


References

[1] Qian X S, Yu J Y, Dai R W. A new discipline of science — The study of open complex giant system and its methodology. Chin J Nat, 1990, 13: 3--10. Google Scholar

[2] Guo L. What is systematology. J Syst Sci Math Sci, 2016, 36: 291--301. Google Scholar

[3] Mitchell M. Complexity: A Guided Tour. Oxford: Oxford University Press, 2009. Google Scholar

[4] Bar-Yam Y. Dynamics of Complex Systems. Reading: Addison-Wesley, 1997. Google Scholar

[5] Wiener N. Cybernetics. Paris: Hermann, 1948. Google Scholar

[6] Wiener N. The Human Use of Human Beings: Cybernetics and Society. Cambridge: MIT Press, 1950. Google Scholar

[7] Guo L, Cheng D Z, Feng D X. Introduction to Control Theory: From Basic Concept to Research Frontiers. Beijing: Scientific Press, 2005. Google Scholar

[8] Cheng D Z, Zhao Y, Xu T T. Dynamic games and optimal control of logical dynamic systems. J Syst Sci Math Sci, 2012, 32: 1226--1238. Google Scholar

[9] von Neumann J, Morgenstern O. Theory of Games and Economic Behavior. Princeton: Princeton University Press, 1944. Google Scholar

[10] Nash J F. Equilibrium Points in n-Person Games. Proc Natl Acad Sci USA, 1950, 36: 48-49 CrossRef ADS Google Scholar

[11] Nash J F. Non-cooperative games. Ann Math, 1951, 54: 286--295. Google Scholar

[12] Maynard Smith J. Evolution and the Theory of Games. Cambridge: Cambridge University Press, 1982. Google Scholar

[13] Weibull J W. Evolutionary Game Theory. Cambridge: The MIT Press, 1995. Google Scholar

[14] Cheng D, He F, Qi H. Modeling, Analysis and Control of Networked Evolutionary Games. IEEE Trans Automat Contr, 2015, 60: 2402-2415 CrossRef Google Scholar

[15] Lu K, Jing G, Wang L. A distributed algorithm for solving mixed equilibrium problems. Automatica, 2019, 105: 246-253 CrossRef Google Scholar

[16] Lu K, Jing G, Wang L. Distributed Algorithms for Searching Generalized Nash Equilibrium of Noncooperative Games. IEEE Trans Cybern, 2019, 49: 2362-2371 CrossRef Google Scholar

[17] Zhang R R, Guo L. Controllability of Nash Equilibrium in Game-Based Control Systems. IEEE Trans Automat Contr, 2019, 64: 4180-4187 CrossRef Google Scholar

[18] Zhang R, Guo L. Controllability of Stochastic Game-Based Control Systems. SIAM J Control Optim, 2019, 57: 3799-3826 CrossRef Google Scholar

[19] Wang L, Fu F, Chen X J, et al. Collective decision-making over complex networks. CAAI Trans Intell Syst, 2008, 3: 95--108. Google Scholar

[20] Cheng D Z, Chen H F. From swarm to social behavior control. Sci Technol Rev, 2004, 22: 4--7. Google Scholar

[21] Wang L, Tian Y, Du J M. Opinion dynamics in social networks. Sci Sin Inform, 2018, 48: 3--23. Google Scholar

[22] Watts D J, Strogatz S H. Collective dynamics of `small-world' networks. Nature, 1998, 393: 440-442 CrossRef ADS Google Scholar

[23] Olson M. The Logic of Collective Action: Public Goods and the Theory of Groups. Cambridge: Harvard University Press, 1965. Google Scholar

[24] Skyrms B. Evolution of the Social Contract. Cambridge: Cambridge University Press, 1996. Google Scholar

[25] Skyrms B. The Stag Hunt and the Evolution of Social Structure. Cambridge: Cambridge University Press, 2004. Google Scholar

[26] Wilson E O. Sociobiology: The New Synthesis. Cambridge: Harvard University Press, 1975. Google Scholar

[27] Minsky M. The Society of Mind. New York: Simon and Schuster, 1986. Google Scholar

[28] Diani M, McAdam D. Social Movements and Networks. Oxford: Oxford University Press, 2003. Google Scholar

[29] Etesami S R, Basar T. Game-Theoretic Analysis of the Hegselmann-Krause Model for Opinion Dynamics in Finite Dimensions. IEEE Trans Automat Contr, 2015, 60: 1886-1897 CrossRef Google Scholar

[30] Tian Y, Wang L. Opinion dynamics in social networks with stubborn agents: An issue-based perspective. Automatica, 2018, 96: 213-223 CrossRef Google Scholar

[31] Lin X, Jiao Q, Wang L. Opinion Propagation Over Signed Networks: Models and Convergence Analysis. IEEE Trans Automat Contr, 2019, 64: 3431-3438 CrossRef Google Scholar

[32] Wang L, Wu T, Zhang Y L. Feedback mechanism in coevolutionary games. Control Theory Appl, 2014, 31: 823--836. Google Scholar

[33] Wang L, Cong R, Li K. 合作演化中的反馈机制. Sci Sin-Inf, 2014, 44: 1495-1514 CrossRef Google Scholar

[34] Dorogovtsev S N, Mendes J F F. Evolution of Networks: From Biological Nets to the Internet and WWW. Oxford: Oxford University Press, 2003. Google Scholar

[35] Anderson R M. The Population Dynamics of Infectious Diseases: Theory and Applications. London: Chapman & Hall, 1982. Google Scholar

[36] Anderson R M, May R M. Infectious Diseases of Humans: Dynamics and Control. London: Oxford University Press, 1991. Google Scholar

[37] Adaptive Contact Networks Change Effective Disease Infectiousness and Dynamics. PLoS Comput Biol, 2010, 6: e1000895 CrossRef ADS Google Scholar

[38] Wang L, Wang J, Wu B. Quantum games: New methodologies and strategies. CAAI Trans Intell Syst, 2008, 3: 294--304. Google Scholar

[39] Lu Q, Chen L, Mei S. Typical applications and prospects of game theory in power system. Proc CSEE, 2014, 34: 5009--5017. Google Scholar

[40] Bullo F, Cortés J, Martínez S. Distributed Control of Robotic Networks: A Mathematical Approach to Motion Coordination Algorithms. Princeton: Princeton University Press, 2009. Google Scholar

[41] Kennedy J, Eberhart R C. Swarm Intelligence. San Francisco: Morgan Kaufmann Publishers, 2001. Google Scholar

[42] Hassanien A E, Emary E. Swarm Intelligence: Principles, Advances, and Applications. Boca Raton: CRC Press, 2015. Google Scholar

[43] Eguíluz V M, Zimmermann M G. Transmission of Information and Herd Behavior: An Application to Financial Markets. Phys Rev Lett, 2000, 85: 5659-5662 CrossRef ADS arXiv Google Scholar

[44] 郑大钟. 线性系统理论. 第2版. 北京: 清华大学出版社, 2002. Google Scholar

[45] 郑大钟, 赵千川. 离散事件动态系统. 北京: 清华大学出版社, 2000. Google Scholar

[46] Yang Y X, Niu X X. Hacker Cybernetics. Beijing: Electronics Industry Press, 2019. Google Scholar

[47] Barabási A-L. Network Science. Cambridge: Cambridge University Press, 2016. Google Scholar

[48] Strogatz S H. Exploring complex networks. Nature, 2001, 410: 268-276 CrossRef ADS Google Scholar

[49] Barabási A-L. Linked: The New Science of Networks. Cambridge: Perseus, 2002. Google Scholar

[50] Barabási A L, Albert R. Emergence of Scaling in Random Networks. Science, 1999, 286: 509-512 CrossRef ADS arXiv Google Scholar

[51] Albert R, Barabási A L. Statistical mechanics of complex networks. Rev Mod Phys, 2002, 74: 47-97 CrossRef ADS arXiv Google Scholar

[52] Barabási A L. Scale-Free Networks: A Decade and Beyond. Science, 2009, 325: 412-413 CrossRef ADS Google Scholar

[53] Arenas A, Díaz-Guilera A, Kurths J. Synchronization in complex networks. Phys Rep, 2008, 469: 93-153 CrossRef ADS arXiv Google Scholar

[54] Boccaletti S, Latora V, Moreno Y. Complex networks: Structure and dynamics. Phys Rep, 2006, 424: 175-308 CrossRef ADS Google Scholar

[55] Newman M E J. Networks: An Introduction. Oxford: Oxford University Press, 2010. Google Scholar

[56] Cohen R, Havlin S. Complex Networks: Structure, Robustness and Function. Cambridge: Cambridge University Press, 2010. Google Scholar

[57] Menache I, Ozdaglar A. Network Games: Theory, Models, and Dynamics. San Rafael: Morgan & Claypool Publishers, 2011. Google Scholar

[58] Bollobás B. Random Graphs. London: Academic Press, 1985. Google Scholar

[59] Godsil C, Royal G. Algebraic Graph Theory. New York: Springer, 2001. Google Scholar

[60] Szabó G, Fáth G. Evolutionary games on graphs. Phys Rep, 2007, 446: 97-216 CrossRef ADS arXiv Google Scholar

[61] Mesbahi M, Egerstedt M. Graph Theoretic Methods in Multiagent Networks. Princeton: Princeton University Press, 2010. Google Scholar

[62] Shakarian P, Roos P, Johnson A. A review of evolutionary graph theory with applications to game theory. BioSystems, 2011, 107: 66--80. Google Scholar

[63] Allen B, Nowak M. Games on graphs. EMS Surv Math Sci, 2014, 1: 113-151 CrossRef Google Scholar

[64] Newman M E J. The Structure and Function of Complex Networks. SIAM Rev, 2003, 45: 167-256 CrossRef ADS arXiv Google Scholar

[65] Liu Y-Y, Slotine J-J, Barabási A-L. Controllability of complex networks. Science, 2011, 473: 167--173. Google Scholar

[66] Li A-M, Wang L. Controlling temporal networks. J Syst Sci Math Sci, 2019, 39: 184--202. Google Scholar

[67] Ruths J, Ruths D. Control Profiles of Complex Networks. Science, 2014, 343: 1373-1376 CrossRef ADS Google Scholar

[68] Rohr R P, Saavedra S, Bascompte J. On the structural stability of mutualistic systems. Science, 2014, 345: 1253497 CrossRef Google Scholar

[69] Onnela J P J. Flow of Control in Networks. Science, 2014, 343: 1325-1326 CrossRef ADS Google Scholar

[70] Duan G, Li A, Meng T. Energy cost for controlling complex networks with linear dynamics. Phys Rev E, 2019, 99: 052305 CrossRef ADS arXiv Google Scholar

[71] Liu Y Y, Barabási A L. Control principles of complex systems. Rev Mod Phys, 2016, 88: 035006 CrossRef ADS arXiv Google Scholar

[72] Wu C W. Synchronization in Complex Networks of Nonlinear Dynamical Systems. Singapore: World Scientific, 2007. Google Scholar

[73] Guan Y, Wang L. Controllability of multi-agent systems with directed and weighted signed networks. Syst Control Lett, 2018, 116: 47-55 CrossRef Google Scholar

[74] Barrat A, Barthélemy M, Vespignani A. Dynamical Processes on Complex Networks. Cambridge: Cambridge University Press, 2008. Google Scholar

[75] Lin X, Jiao Q, Wang L. Competitive diffusion in signed social networks: A game-theoretic perspective. Automatica, 2020, 112: 108656 CrossRef Google Scholar

[76] Time-Ordered Networks Reveal Limitations to Information Flow in Ant Colonies. PLoS ONE, 2011, 6: e20298 CrossRef ADS Google Scholar

[77] Ren J, Sun W, Manocha D. Stable information transfer network facilitates the emergence of collective behavior of bird flocks. Phys Rev E, 2018, 98: 052309 CrossRef ADS Google Scholar

[78] Scott J. Social Network Analysis: A Handbook. London: Sage, 2000. Google Scholar

[79] Wasserman S, Faust K. Social Network Analysis: Methods and Applications. Cambridge: Cambridge University Press, 1994. Google Scholar

[80] Girvan M, Newman M E J. Community structure in social and biological networks. Proc Natl Acad Sci USA, 2002, 99: 7821-7826 CrossRef ADS arXiv Google Scholar

[81] Vicsek T, Czirók A, Ben-Jacob E. Novel Type of Phase Transition in a System of Self-Driven Particles. Phys Rev Lett, 1995, 75: 1226-1229 CrossRef ADS arXiv Google Scholar

[82] Reynolds C W. Flocks, herds and schools: A distributed behavioral model. SIGGRAPH Comput Graph, 1987, 21: 25-34 CrossRef Google Scholar

[83] Hatano Y, Mesbahi M. Agreement over random networks. IEEE Trans Automat Contr, 2005, 50: 1867-1872 CrossRef Google Scholar

[84] Wang L, Fu F, Chen X, et al. Evolutionary games on complex networks. CAAI Trans Intell Syst, 2007, 2: 1--10. Google Scholar

[85] Wang L, Fu F, Chen X, et al. Evolutionary games and self-organizing cooperation. J Syst Sci Math Sci, 2007, 27: 330--343. Google Scholar

[86] Tahbaz-Salehi A, Jadbabaie A. A Necessary and Sufficient Condition for Consensus Over Random Networks. IEEE Trans Automat Contr, 2008, 53: 791-795 CrossRef Google Scholar

[87] Altafini C. Consensus Problems on Networks With Antagonistic Interactions. IEEE Trans Automat Contr, 2013, 58: 935-946 CrossRef Google Scholar

[88] Pastor-Satorras R, Castellano C, Van Mieghem P. Epidemic processes in complex networks. Rev Mod Phys, 2015, 87: 925-979 CrossRef ADS arXiv Google Scholar

[89] Long Wang , Feng Xiao . Finite-Time Consensus Problems for Networks of Dynamic Agents. IEEE Trans Automat Contr, 2010, 55: 950-955 CrossRef Google Scholar

[90] Jing G, Zhang G, Lee H W J. Angle-based shape determination theory of planar graphs with application to formation stabilization. Automatica, 2019, 105: 117-129 CrossRef Google Scholar

[91] Proskurnikov A V, Matveev A S, Cao M. Opinion Dynamics in Social Networks With Hostile Camps: Consensus vs. Polarization. IEEE Trans Automat Contr, 2016, 61: 1524-1536 CrossRef Google Scholar

[92] Ma J, Zheng Y, Wang L. Nash Equilibrium Topology of Multi-Agent Systems With Competitive Groups. IEEE Trans Ind Electron, 2017, 64: 4956-4966 CrossRef Google Scholar

[93] Etesami S R, Ba?ar T. Price of anarchy and an approximation algorithm for the binary-preference capacitated selfish replication game. Automatica, 2017, 76: 153-163 CrossRef Google Scholar

[94] Chen X, Liu J, Belabbas M A. Distributed Evaluation and Convergence of Self-Appraisals in Social Networks. IEEE Trans Automat Contr, 2017, 62: 291-304 CrossRef Google Scholar

[95] Duan G, Xiao F, Wang L. Asynchronous Periodic Edge-Event Triggered Control for Double-Integrator Networks With Communication Time Delays. IEEE Trans Cybern, 2018, 48: 675-688 CrossRef Google Scholar

[96] Wang L, Du J M. Evolutionary game theoretic approach to coordinated control of multi-agent systems. J Syst Sci Math Sci, 2016, 36: 302--318. Google Scholar

[97] Parsegov S E, Proskurnikov A V, Tempo R. Novel Multidimensional Models of Opinion Dynamics in Social Networks. IEEE Trans Automat Contr, 2017, 62: 2270-2285 CrossRef Google Scholar

[98] Jing G, Zheng Y, Wang L. Consensus of Multiagent Systems With Distance-Dependent Communication Networks. IEEE Trans Neural Netw Learning Syst, 2017, 28: 2712-2726 CrossRef Google Scholar

[99] Ohtsuki H, Hauert C, Lieberman E. A simple rule for the evolution of cooperation on graphs and social networks. Nature, 2006, 441: 502-505 CrossRef ADS Google Scholar

[100] Kerr B, Neuhauser C, Bohannan B J M. Local migration promotes competitive restraint in a host-pathogen 'tragedy of the commons'. Nature, 2006, 442: 75-78 CrossRef ADS Google Scholar

[101] Reichenbach T, Mobilia M, Frey E. Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games. Nature, 2007, 448: 1046-1049 CrossRef ADS arXiv Google Scholar

[102] Helbing D, Yu W. MIGRATION AS A MECHANISM TO PROMOTE COOPERATION. Advs Complex Syst, 2008, 11: 641-652 CrossRef Google Scholar

[103] Wu T, Fu F, Zhang Y. Expectation-driven migration promotes cooperation by group interactions. Phys Rev E, 2012, 85: 066104 CrossRef ADS Google Scholar

[104] Chen X, Szolnoki A, Perc M. Risk-driven migration and the collective-risk social dilemma. Phys Rev E, 2012, 86: 036101 CrossRef ADS arXiv Google Scholar

[105] Fu F, Nowak M A. Global Migration Can Lead to Stronger Spatial Selection than Local Migration. J Stat Phys, 2013, 151: 637-653 CrossRef ADS Google Scholar

[106] Limdi A, Pérez-Escudero A, Li A. Asymmetric migration decreases stability but increases resilience in a heterogeneous metapopulation. Nat Commun, 2018, 9: 2969 CrossRef ADS Google Scholar

[107] Li A, Cornelius S P, Liu Y Y. The fundamental advantages of temporal networks. Science, 2017, 358: 1042-1046 CrossRef ADS arXiv Google Scholar

[108] Wang W, Liu Q H, Liang J. Coevolution spreading in complex networks. Phys Rep, 2019, 820: 1-51 CrossRef ADS arXiv Google Scholar

[109] Gross T, Blasius B. Adaptive coevolutionary networks: a review. J R Soc Interface, 2008, 5: 259-271 CrossRef Google Scholar

[110] Perc M, Szolnoki A. Coevolutionary games-A mini review. Biosystems, 2010, 99: 109-125 CrossRef Google Scholar

[111] Evolution of Cooperation on Stochastic Dynamical Networks. PLoS ONE, 2010, 5: e11187 CrossRef ADS Google Scholar

[112] Moving Away from Nasty Encounters Enhances Cooperation in Ecological Prisoner's Dilemma Game. PLoS ONE, 2011, 6: e27669 CrossRef ADS Google Scholar

[113] Wu B, Mao S, Wang J. Control of epidemics via social partnership adjustment. Phys Rev E, 2016, 94: 062314 CrossRef ADS arXiv Google Scholar

[114] Feng X, Wang L, Levin S A. Dynamic analysis and decision-making in disease-behavior systems with perceptions. In: Proceedings of 2019 Chinese Control and Decision Conference (CCDC), 2019. Google Scholar

[115] Wei Y, Lin Y, Wu B. Vaccination dilemma on an evolving social network. J Theor Biol, 2019, 483: 109978 CrossRef Google Scholar

[116] Fu F, Hauert C, Nowak M A. Reputation-based partner choice promotes cooperation in social networks. Phys Rev E, 2008, 78: 026117 CrossRef ADS Google Scholar

[117] Cooperation Prevails When Individuals Adjust Their Social Ties. PLoS Comput Biol, 2006, 2: e140 CrossRef ADS Google Scholar

[118] Van Segbroeck S, Santos F C, Nowé A. The evolution of prompt reaction to adverse ties. BMC Evol Biol, 2008, 8: 287 CrossRef Google Scholar

[119] Cheng D Z, Qi H S. The Semi-Tensor Product of Matrices: Theory and Applications. Beijing: Scientific Press, 2017. Google Scholar

[120] Cheng D Z, Xia Y Q, Ma H B, et al. Matrix Algebra, Control and Game Theory. Beijing: Beijing Institute of Technology Press, 2016. Google Scholar

[121] Cheng D, Xu T, Qi H. Evolutionarily Stable Strategy of Networked Evolutionary Games. IEEE Trans Neural Netw Learning Syst, 2014, 25: 1335-1345 CrossRef Google Scholar

[122] Cheng D Z, Liu T, Wang Y H. Matrix approach to game theory. J Syst Sci Math Sci, 2014, 34: 1291--1305. Google Scholar

[123] Taylor P D, Jonker L B. Evolutionary stable strategies and game dynamics. Math Biosci, 1978, 40: 145-156 CrossRef Google Scholar

[124] Hofbauer J, Sigmund K. Evolutionary Games and Population Dynamics. Cambridge: Cambridge University Press, 1998. Google Scholar

[125] Matsuda H, Ogita N, Sasaki A. Statistical Mechanics of Population — The Lattice Lotka-Volterra Model —. Prog Theor Phys, 1992, 88: 1035-1049 CrossRef ADS Google Scholar

[126] Gintis H. Game Theory Evolving: A Problem-Centered Introduction to Modeling Strategic Interaction. Princeton: Princeton University Press, 2000. Google Scholar

[127] Nowak M A. Evolutionary Dynamics: Exploring the Equations of Life. Cambridge: Harvard University Press, 2006. Google Scholar

[128] Webb J N. Game Theory: Decisions, Interaction and Evolution. New York: Springer, 2006. Google Scholar

[129] Tadelis S. Game Theory: An Introduction. Princeton: Princeton University Press, 2013. Google Scholar

[130] Nowak M A. Evolutionary Dynamics of Biological Games. Science, 2004, 303: 793-799 CrossRef ADS Google Scholar

[131] Schuster H G. Deterministic Chaos: An Introduction. 3rd ed. Weinheim: Wiley-VCH, 1995. Google Scholar

[132] Sato Y, Akiyama E, Farmer J D. Chaos in learning a simple two-person game. Proc Natl Acad Sci USA, 2002, 99: 4748-4751 CrossRef ADS Google Scholar

[133] Sato Y, Crutchfield J P. Coupled replicator equations for the dynamics of learning in multiagent systems. Phys Rev E, 2003, 67: 015206 CrossRef ADS arXiv Google Scholar

[134] Imperfect Vaccine Aggravates the Long-Standing Dilemma of Voluntary Vaccination. PLoS ONE, 2011, 6: e20577 CrossRef ADS Google Scholar

[135] Wu B, Zhou D, Wang L. Evolutionary dynamics on stochastic evolving networks for multiple-strategy games. Phys Rev E, 2011, 84: 046111 CrossRef ADS Google Scholar

[136] Pennisi E. How Did Cooperative Behavior Evolve?. Science, 2005, 309: 93-93 CrossRef Google Scholar

[137] Axelrod R, Hamilton W. The Evolution of Cooperation. Science, 1981, 211: 1390-1396 CrossRef ADS Google Scholar

[138] Axelrod R. The Evolution of Cooperation. New York: Basic Books, 1984. Google Scholar

[139] Axelrod R. The Complexity of Cooperation: Agent-Based Models of Competition and Collaboration. Princeton: Princeton University Press, 1997. Google Scholar

[140] Fehr E, G?chter S. Altruistic punishment in humans. Nature, 2002, 415: 137-140 CrossRef Google Scholar

[141] Nowak M A, Sasaki A, Taylor C. Emergence of cooperation and evolutionary stability in finite populations. Nature, 2004, 428: 646-650 CrossRef ADS Google Scholar

[142] Nowak M A. Five Rules for the Evolution of Cooperation. Science, 2006, 314: 1560-1563 CrossRef ADS Google Scholar

[143] Pennisi E. On the Origin of Cooperation. Science, 2009, 325: 1196-1199 CrossRef Google Scholar

[144] Nowak M A. Evolving cooperation. J Theor Biol, 2012, 299: 1-8 CrossRef Google Scholar

[145] Rand D G, Nowak M A. Human cooperation. Trends Cognitive Sci, 2013, 17: 413-425 CrossRef Google Scholar

[146] Traulsen A, Nowak M A, Pacheco J M. Stochastic dynamics of invasion and fixation. Phys Rev E, 2006, 74: 011909 CrossRef ADS arXiv Google Scholar

[147] Wu B, Arranz J, Du J. Evolving synergetic interactions. J R Soc Interface, 2016, 13: 20160282 CrossRef Google Scholar

[148] Tang C B, Wu B, Wang J B. Evolutionary Origin of Asymptotically Stable Consensus. Sci Rep, 2015, 4: 4590 CrossRef ADS Google Scholar

[149] Wu B, Park H J, Wu L. Evolution of cooperation driven by self-recommendation. Phys Rev E, 2019, 100: 042303 CrossRef ADS arXiv Google Scholar