, Volume 104, Issue 3, pp 715–735 | Cite as

Independent simultaneous discoveries visualized through network analysis: the case of linear canonical transforms

  • Sofia Liberman
  • Kurt Bernardo Wolf


We describe the structural dynamics of two groups of scientists in relation to the independent simultaneous discovery (i.e., definition and application) of linear canonical transforms. This mathematical construct was built as the transfer kernel of paraxial optical systems by Prof. Stuart A. Collins, working in the ElectroScience Laboratory in Ohio State University. At roughly the same time, it was established as the integral kernel that represents the preservation of uncertainty in quantum mechanics by Prof. Marcos Moshinsky and his postdoctoral associate, Dr. Christiane Quesne, at the Instituto de Física of the Universidad Nacional Autónoma de México. We are interested in the birth and parallel development of the two follower groups that have formed around the two seminal articles, which for more than two decades did not know and acknowledge each other. Each group had different motivations, purposes and applications, and worked in distinct professional environments. As we will show, Moshinsky–Quesne had been highly cited by his associates and students in Mexico and Europe when the importance of his work started to permeate various other mostly theoretical fields; Collins’ paper took more time to be referenced, but later originated a vast following notably among Chinese applied optical scientists. Through social network analysis we visualize the structure and development of these two major coauthoring groups, whose community dynamics shows two distinct patterns of communication that illustrate the disparity in the diffusion of theoretical and technological research.


Psychology of science Simultaneous discoveries Network analysis Scientific communication Linear canonical transforms 

Mathematical Subject Classification

91D30 01A65 65R10 



The work of the first author has been supported by the project DGAPA-PAPIIT Nº IN305013_2014, of the Universidad Nacional Autónoma de México; the work of the second author has been supported by DGAPA-PAPIIT Nº101115. The authors also acknowledge the very useful comments of two anonymous reviewers.


  1. Bargmann, V. (1970). Group representations in Hilbert spaces of analytic functions. In P. Gilbert & R. G. Newton (Eds.), Analytical methods in mathematical physics (pp. 27–63). New York: Gordon and Breach.Google Scholar
  2. Bastian, M., Heymann, S. & Jacomy, M. (2009). Gephi: an open source software for exploring and manipulating networks. In International AAAI conference on weblogs and social media. Association for the Advancement of Artificial Intelligence.Google Scholar
  3. Bavelas, A. (1948). A mathematical model for group structures. Human Organization, 7, 16–30.CrossRefGoogle Scholar
  4. Bavelas, A. (1968). Communication patterns in task-oriented groups. In D. Cartwright & A. F. Zander (Eds.), Group dynamics: Research and theory. New York: Harper & Row.Google Scholar
  5. Blondel, V. D., Guillaume J., Lambiotte R., & Lefebvre E. (2008). Fast unfolding of communities in large networks. Journal of Statistical Mechanics: Theory and Experiment, 10, 1–12.Google Scholar
  6. Brannigan, A., & Wanner, R. A. (1983). Multiple discoveries in science: A test of the communication theory. Canadian Journal of Sociology, 8, 135–151.CrossRefGoogle Scholar
  7. Brouwer, W. (1964). Matrix methods in optical instrument design. New York: Benjamin.Google Scholar
  8. Collins, S. A. (1970). Lens-system diffraction integral written in terms of matrix optics. Journal of Optical Society of America, 60, 1168–1177.CrossRefGoogle Scholar
  9. Freeman, L. C. (1978). Centrality in social networks conceptual clarification. Social Networks, 1(3), 215–239.MathSciNetCrossRefGoogle Scholar
  10. Freeman, L. C. (2004). The development of social network analysis: A study in the sociology of science. Vancouver, BC: Empirical Press.Google Scholar
  11. Galton, F. (1874). English men of science: Their nature and nurture. London: Macmillan & Co.CrossRefGoogle Scholar
  12. Gerrard, A., & Burch, B. (1975). Introduction to matrix methods in optics. New York: Wiley.Google Scholar
  13. Hagstrom, W. O. (1974). Competition in science. American Sociological Review, 39(1), 1–18.Google Scholar
  14. Homans, G. C. (1950). The human group. New York: Harcourt, Brace & Co.Google Scholar
  15. Hu, Y. (2006). Efficient, high-quality force-directed graph drawing. Mathematica Journal, 10(1), 37–71.Google Scholar
  16. Infeld, L., & Plebañski, J. (1955). On a certain class of unitary transformations. Acta Physica Polononica, 14, 41–75.MATHGoogle Scholar
  17. Itzykson, C. (1969). Group representation in a continuous basis: An example. Journal of Mathematical Physics, 10, 1109–1114.CrossRefMATHGoogle Scholar
  18. Jacomy, M., Venturini, T., Heymann, S., & Bastian, M. (2014). ForceAtlas2, a continuous graph layout algorithm for handy network visualization designed for the Gephi software. PLoS ONE, 9, 6.CrossRefGoogle Scholar
  19. Kroeber, A. L. (1917). The superorganic. American Anthropologist, 19, 163–213.CrossRefGoogle Scholar
  20. Liberman, S., & Wolf, K. B. (2013). Scientific communication in the process to coauthorship. In G. J. Feist & M. Gorman (Eds.), Handbook of the psychology of science. Berlin: Springer.Google Scholar
  21. Merton, R. K. (1961). Singletons and multiples in scientific discovery. Proceedings of the American Philosophical Society, 105(5), 470–486.Google Scholar
  22. Merton, R. K. (1973). The sociology of science: Theoretical and empirical investigations. Chicago: University of Chicago Press.Google Scholar
  23. Moreno, J. L. (1953). Who shall survive?. New York: Beacon House.Google Scholar
  24. Moshinsky, M., & Quesne, C. (1971). Linear canonical transformations and their unitary representations. Journal of Mathematical Physics. (N. Y.), 12(8), 1772–1780.MathSciNetCrossRefMATHGoogle Scholar
  25. Moshinsky, M. & Quesne, C. (1974) Oscillator systems. In Proceedings of the 15th solvay conference in physics. Gordon and Breach, New York.Google Scholar
  26. Newman, M. E. J. (2006). Modularity and community structure in networks. Proceedings of the National Academy of Sciences, 103(23), 8577–8582. doi: 10.1073/pnas.0601602103.
  27. Ogburn, W. F., & Thomas, D. (1922). Are inventions inevitable? A note on social evolution. Political Science Quarterly, 37, 83–98.CrossRefGoogle Scholar
  28. Quesne, C., & Moshinsky, M. (1971). Linear canonical transformations and matrix elements. Journal of Mathematical Physics. (N. Y.), 12(8), 1780–1783.MathSciNetCrossRefMATHGoogle Scholar
  29. Sci2 Team. (2009). Science of Science (Sci2) Tool: Indiana University and SciTech Strategies, Inc.
  30. Simonton, D. K. (1978). Independent discovery in science and technology: A closer look at the Poisson distribution. Social Studies of Science, 8, 521–532.CrossRefGoogle Scholar
  31. Simonton, D. K. (1979). Multiple discovery and invention: Zeitgeist, genius, or chance? Journal of Personality and Social Psychology, 37, 1603.CrossRefGoogle Scholar
  32. Simonton, D. K. (1986). Multiple discovery: Some Monte Carlo simulations and Gedanken experiments. Scientometrics, 9, 269–280.Google Scholar
  33. Simonton, D. K. (2010). Creative thought as blind-variation and selective-retention: Combinatorial models of exceptional creativity. Physics of Life Reviews, 7(2), 190–194.CrossRefGoogle Scholar
  34. Small, H. G. (1978). Cited documents as concept symbols. Social Studies of Science, 8(3), 327–340.CrossRefGoogle Scholar
  35. Stavroudis, O. (1972). The optics of rays, wavefronts, and caustics. New York: Academic Press.Google Scholar
  36. Travers, J., & Milgram, S. (1969). An experimental study of the small world problem. Sociometry, 32(4), 425–443.CrossRefGoogle Scholar
  37. Weingart, S., Hanning, G., Börner, K., Boyac, K. W., Linnemeier, M., Duhon, R. J., Phillips, P. A., et al. (2010) Science of Science (Sci2) Tool User Manual. Accessed 16 July 2013.
  38. Wolf, K. B. (1979). Integral transforms in science and engineering. New York: Plenum Publ. Corp.
  39. Wolf, K. B. (2004). Geometric optics on phase space. Heidelberg: Springer.MATHGoogle Scholar
  40. Wolf, K. B. (2015). Developments of linear canonical transforms—a historical sketch. In M. Alper Kutay, J. J. Healy, H. M. Ozaktas, & J. T. Sheridan (Eds.), Linear canonical transforms: theory and applications. Heidelberg: Springer.Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2015

Authors and Affiliations

  1. 1.Facultad de PsicologíaUniversidad Nacional Autónoma de MéxicoMexicoMexico
  2. 2.Instituto de Ciencias FísicasUniversidad Nacional Autónoma de MéxicoMexicoMexico

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