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A Spatial Small-World Graph Arising from Activity-Based Reinforcement

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Algorithms and Models for the Web Graph (WAW 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11631))

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Abstract

In the classical preferential attachment model, links form instantly to newly arriving nodes and do not change over time. We propose a hierarchical random graph model in a spatial setting, where such a time-variability arises from an activity-based reinforcement mechanism. We show that the reinforcement mechanism converges, and prove rigorously that the resulting random graph exhibits the small-world property. A further motivation for this random graph stems from modeling synaptic plasticity.

This work is supported by The Danish Council for Independent Research—Natural Sciences, grant DFF – 7014-00074 Statistics for point processes in space and beyond, and by the Centre for Stochastic Geometry and Advanced Bioimaging, funded by grant 8721 from the Villum Foundation.

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References

  1. Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)

    Article  MathSciNet  Google Scholar 

  2. Bengio, Y., Lee, D., Bornschein, J., Lin, Z.: Towards biologically plausible deep learning. CoRR abs/1502.04156 (2015). http://arxiv.org/abs/1502.04156

  3. Bollobás, B., Riordan, O.: The diameter of a scale-free random graph. Combinatorica 24(1), 5–34 (2004)

    Article  MathSciNet  Google Scholar 

  4. Bonacich, P., Liggett, T.M.: Asymptotics of a matrix valued Markov chain arising in sociology. Stoch. Process. Appl. 104(1), 155–171 (2003)

    Article  MathSciNet  Google Scholar 

  5. Couzinié, Y., Hirsch, C.: Infinite WARM graphs I. Weak reinforcement regime (in preparation)

    Google Scholar 

  6. Delattre, S., Fournier, N., Hoffmann, M.: Hawkes processes on large networks. Ann. Appl. Probab. 26(1), 216–261 (2016)

    Article  MathSciNet  Google Scholar 

  7. Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events. Springer, Berlin (1997). https://doi.org/10.1007/978-3-642-33483-2

    Book  MATH  Google Scholar 

  8. Feller, W.: An Introduction to Probability Theory and its Applications, vol. II, 2nd edn. Wiley, New York (1971)

    MATH  Google Scholar 

  9. Hirsch, C., Holmes, M., Kleptsyn, V.: Absence of WARM percolation in the very strong reinforcement regime, preprint available at https://christian-hirsch.github.io/publications.html

  10. Van Der Hofstad, R., Holmes, M., Kuznetsov, A., Ruszel, W.: Strongly reinforced Pólya urns with graph-based competition. Ann. Appl. Probab. 26(4), 2494–2539 (2016)

    Article  MathSciNet  Google Scholar 

  11. Holmes, M., Kleptsyn, V.: Infinite WARM graphs. Critical regime (in preparation)

    Google Scholar 

  12. Holmes, M., Kleptsyn, V.: Proof of the WARM whisker conjecture for neuronal connections. Chaos 27(4), 043104 (2017)

    Article  MathSciNet  Google Scholar 

  13. Jacob, E., Mörters, P.: A spatial preferential attachment model with local clustering. In: Bonato, A., Mitzenmacher, M., Prałat, P. (eds.) WAW 2013. LNCS, vol. 8305, pp. 14–25. Springer, Cham (2013). https://doi.org/10.1007/978-3-319-03536-9_2

    Chapter  Google Scholar 

  14. Kalisman, N., Silberberg, G., Markram, H.: The neocortical microcircuit as a tabula rasa. Proc. Natl. Acad. Sci. 102(3), 880–885 (2005)

    Article  Google Scholar 

  15. Liggett, T.M., Rolles, S.W.W.: An infinite stochastic model of social network formation. Stoch. Process. Appl. 113(1), 65–80 (2004)

    Article  MathSciNet  Google Scholar 

  16. Montague, P.R., Dayan, P., Sejnowski, T.J.: A framework for mesencephalic dopamine systems based on predictive Hebbian learning. J. Neurosci. 16(5), 1936–1947 (1996)

    Article  Google Scholar 

  17. Pemantle, R.: A survey of random processes with reinforcement. Probab. Surv. 4, 1–79 (2007)

    Article  MathSciNet  Google Scholar 

  18. Pemantle, R., Skyrms, B.: Network formation by reinforcement learning: the long and medium run. Math. Soc. Sci. 48(3), 315–327 (2004)

    Article  MathSciNet  Google Scholar 

  19. Pittel, B.: On a random graph evolving by degrees. Adv. Math. 223(2), 619–671 (2010)

    Article  MathSciNet  Google Scholar 

  20. Skyrms, B., Pemantle, R.: A dynamic model of social network formation. Proc. Natl. Acad. Sci. USA 97(16), 9340–9346 (2000)

    Article  Google Scholar 

  21. Stanovich, K.E.: Matthew effects in reading: some consequences of individual differences in the acquisition of literacy. J. Educ. 189(1–2), 23–55 (2009)

    Article  Google Scholar 

  22. Sutton, R.S., Barto, A.G.: Reinforcement Learning: An Introduction, 2nd edn. MIT Press, Cambridge (2018)

    MATH  Google Scholar 

  23. Zhu, T.: Nonlinear Pólya urn models and self-organizing processes. Ph.D. thesis, University of Pennsylvania (2009)

    Google Scholar 

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Acknowledgments

The authors thank all anonymous referees. We also thank C. Leibold for interesting discussions on the neuro-scientific background of synaptic plasticity and comments on an earlier version of the manuscript.

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Authors and Affiliations

  1. Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstraße 39, 80333, München, Germany

    Markus Heydenreich

  2. Institut für Mathematik, Universität Mannheim, B6 26, 68161, Mannheim, Germany

    Christian Hirsch

Authors
  1. Markus Heydenreich
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  2. Christian Hirsch
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Correspondence to Christian Hirsch .

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Editors and Affiliations

  1. Inria, Sophia Antipolis, France

    Konstantin Avrachenkov

  2. Ryerson University, Toronto, ON, Canada

    Paweł Prałat

  3. The University of Queensland, Brisbane, QLD, Australia

    Nan Ye

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Heydenreich, M., Hirsch, C. (2019). A Spatial Small-World Graph Arising from Activity-Based Reinforcement. In: Avrachenkov, K., Prałat, P., Ye, N. (eds) Algorithms and Models for the Web Graph. WAW 2019. Lecture Notes in Computer Science(), vol 11631. Springer, Cham. https://doi.org/10.1007/978-3-030-25070-6_8

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  • DOI: https://doi.org/10.1007/978-3-030-25070-6_8

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-25069-0

  • Online ISBN: 978-3-030-25070-6

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