Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Topological limits to the parallel processing capability of network architectures

An Author Correction to this article was published on 05 March 2021

This article has been updated

Abstract

The ability to learn new tasks and generalize to others is a remarkable characteristic of both human brains and recent artificial intelligence systems. The ability to perform multiple tasks simultaneously is also a key characteristic of parallel architectures, as is evident in the human brain and exploited in traditional parallel architectures. Here we show that these two characteristics reflect a fundamental tradeoff between interactive parallelism, which supports learning and generalization, and independent parallelism, which supports processing efficiency through concurrent multitasking. Although the maximum number of possible parallel tasks grows linearly with network size, under realistic scenarios their expected number grows sublinearly. Hence, even modest reliance on shared representations, which support learning and generalization, constrains the number of parallel tasks. This has profound consequences for understanding the human brain’s mix of sequential and parallel capabilities, as well as for the development of artificial intelligence systems that can optimally manage the tradeoff between learning and processing efficiency.

This is a preview of subscription content, access via your institution

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Graph-theoretic measures predict parallel processing capacity.
Fig. 2: Graph-theoretic results for ρα.
Fig. 3: Graph-theoretic results for Pγ, ϕγ and \({\tilde{\phi }}_{\gamma }\).

Similar content being viewed by others

Data availability

Example data files are available at https://github.com/lordgrilo/Multitasking_capacity. Source data are provided with this paper.

Code availability

Code to reproduce the simulations and analysis reported here is availabile at https://github.com/lordgrilo/Multitasking_capacity.

Change history

References

  1. McClelland, J. L., Rumelhart, D. E. & Hinton, G. E. The Appeal of Parallel Distributed Processing (MIT Press, 1986).

  2. Rogers, T. T. & McClelland, J. L. Semantic Cognition: A Parallel Distributed Processing Approach (MIT Press, 2004).

  3. Bengio, Y., Courville, A. & Vincent, P. Representation learning: a review and new perspectives. IEEE Trans. Pattern Anal. Mach. Intell. 35, 1798–1828 (2013).

    Article  Google Scholar 

  4. Caruana, R. Multitask learning. Mach. Learn. 28, 41–75 (1997).

    Article  Google Scholar 

  5. Baxter, J. Learning internal representations. In Proc. Eighth Annual Conference on Computational Learning Theory 311–320 (ACM, 1995).

  6. Gropp, W., Lusk, E., Doss, N. & Skjellum, A. A high-performance, portable implementation of the MPI message passing interface standard. Parallel Comput. 22, 789–828 (1996).

    Article  Google Scholar 

  7. Musslick, S. et al. Multitasking capability versus learning efficiency in neural network architectures. In Proc. 39th Annual Meeting of the Cognitive Science Society 829–834 (Cognitive Science Society, 2017).

  8. Posner, M. I. & Snyder, C. R. in Information Processing and Cognition: The Loyola Symposium 55–85 (Erlbaum, 1975).

  9. Shiffrin, R. M. & Schneider, W. Controlled and automatic human information processing: II. Perceptual learning, automatic attending and a general theory. Psychol. Rev. 84, 127–190 (1977).

    Article  Google Scholar 

  10. Wickens, C. D. in Multiple-Task Performance 1st edn (ed. Damos, D. L.) Ch. 1 (CRC Press, 1991).

  11. Allport, D. A. in New Directions in Cognitive Psychology (ed. Claxton, G. L.) 112–153 (Routledge, 1980).

  12. Meyer, D. E. & Kieras, D. E. A computational theory of executive cognitive processes and multiple-task performance: Part I. Basic mechanisms. Psychol. Rev. 104, 3–65 (1997).

    Google Scholar 

  13. Navon, D. & Gopher, D. On the economy of the human-processing system. Psychol. Rev. 86, 214–255 (1979).

    Article  Google Scholar 

  14. Feng, S. F., Schwemmer, M., Gershman, S. J. & Cohen, J. D. Multitasking vs. multiplexing: toward a normative account of limitations in the simultaneous execution of control-demanding behaviors. Cogn. Affect. Behav. Neurosci. 14, 129–146 (2014).

    Article  Google Scholar 

  15. Musslick, S. et al. Controlled vs. automatic processing: a graph-theoretic approach to the analysis of serial vs. parallel processing in neural network architectures. In Proc. 38th Annual Meeting of the Cognitive Science Society 1547–1552 (Cognitive Science Society, 2016).

  16. Stroop, J. R. Studies of interference in serial verbal reactions. J. Exp. Psychol. 18, 643–662 (1935).

    Article  Google Scholar 

  17. Gavril, F. Algorithms for a maximum clique and a maximum independent set of a circle graph. Networks 3, 261–273 (1973).

    Article  MathSciNet  Google Scholar 

  18. Cohen, J. D., Dunbar, K. & McClelland, J. L. On the control of automatic processes: a parallel distributed processing account of the stroop effect. Psychol. Rev. 97, 332–361 (1990).

    Article  Google Scholar 

  19. Cohen, J. D., Servan-Schreiber, D. & McClelland, J. L. A parallel distributed processing approach to automaticity. Am. J. Psychol. 105, 239–269 (1992).

    Article  Google Scholar 

  20. Botvinick, M. M., Braver, T. S., Barch, D. M., Carter, C. S. & Cohen, J. D. Conflict monitoring and cognitive control. Psychol. Rev. 108, 624–652 (2001).

    Article  Google Scholar 

  21. Newman, M. Networks: An Introduction (Oxford Univ. Press, 2010).

  22. Lucibello, C. & Ricci-Tersenghi, F. The statistical mechanics of random set packing and a generalization of the Karp–Sipser algorithm. Int. J. Stat. Mech. 2014, 1–13 (2014).

    Article  Google Scholar 

  23. Newman, M. E. J. Random graphs with clustering. Phys. Rev. Lett. 103, 058701 (2009).

    Article  ADS  Google Scholar 

  24. Alon, N. et al. A graph-theoretic approach to multitasking. In Proc. 31st Annual Conference on Neural Information Processing Systems 2101–2110 (NIPS, 2017).

  25. Alon, N. et al. Multitasking capacity: hardness results and improved constructions. SIAM J. Discrete Math. 34, 885–903 (2020).

    Article  MathSciNet  Google Scholar 

  26. Pósfai, M. & Hövel, P. Structural controllability of temporal networks. New J. Phys. 16, 123055 (2014).

    Article  ADS  MathSciNet  Google Scholar 

  27. Li, A., Cornelius, S. P., Liu, Y.-Y., Wang, L. & Barabási, A.-L. The fundamental advantages of temporal networks. Science 358, 1042–1046 (2017).

    Article  ADS  Google Scholar 

  28. Townsend, J. T. & Wenger, M. J. A theory of interactive parallel processing: new capacity measures and predictions for a response time inequality series. Psychol. Rev. 111, 1003–1035 (2004).

    Article  Google Scholar 

  29. Townsend, J. T. & Wenger, M. J. The serial–parallel dilemma: a case study in a linkage of theory and method. Psychon. Bull. Rev. 11, 391–418 (2004).

    Article  Google Scholar 

  30. Wenger, M. J. & Townsend, J. T. On the costs and benefits of faces and words: process characteristics of feature search in highly meaningful stimuli. J. Exp. Psychol. Human 32, 755–779 (2006).

    Article  Google Scholar 

  31. Musslick, S. & Cohen, J. D. A mechanistic account of constraints on control-dependent processing: shared representation, conflict and persistence. In Proc. 41st Annual Meeting of the Cognitive Science Society 849–855 (Cognitive Science Society, 2019).

  32. Bernardi, S. et al. The geometry of abstraction in hippocampus and prefrontal cortex. Cell 183, 954–967 (2020).

    Article  Google Scholar 

  33. Cohen, U., Chung, S. Y., Lee, D. D. & Sompolinsky, H. Separability and geometry of object manifolds in deep neural networks. Nat. Commun. 11, 746 (2020).

    Article  ADS  Google Scholar 

  34. Hinton, G. E., Rumelhart, D. E. & Williams, R. J. Learning representations by back-propagating errors. Nature 323, 533–536 (1986).

    Article  ADS  Google Scholar 

  35. Usher, M. & McClelland, J. L. The time course of perceptual choice: the leaky, competing accumulator model. Psychol. Rev. 108, 550–592 (2001).

    Article  Google Scholar 

Download references

Acknowledgements

G.P. has received funding support from Fondazione Compagnia San Paolo and from Intesa Sanpaolo Innovation Center. S.M. and J.D.C. acknowledge support from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.

Author information

Authors and Affiliations

Authors

Contributions

G.P., S.M., B.D., K.Ö., N.K.A., T.L.W. and J.D.C. designed the research. G.P. developed and performed analytical and numerical calculations. S.M. and D.T. designed, implemented and performed the neural network simulations. S.M., K.Ö., B.D. and N.K.A. provided tools and performed neural network analysis. J.D.C. and T.L.W. conceptualized research and provided advice for all parts of the work. G.P., S.M., B.D., K.Ö., N.K.A., T.L.W. and J.D.C. wrote the manuscript.

Corresponding author

Correspondence to Giovanni Petri.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Physics thanks Hartmut Lentz and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Figs. 1–15 and Sections 1–10.

Source data

Source Data Fig. 1

MIS from neural network simulation data.

Source Data Fig. 2

Degree distribution prediction, MIS size simulation data and predictions for interference graphs with Gaussian degree distribution, MIS size simulation data and predictions for task structure graph with Gaussian degree distribution.

Source Data Fig. 3

Data for effective capacity simulated and predicted.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Petri, G., Musslick, S., Dey, B. et al. Topological limits to the parallel processing capability of network architectures. Nat. Phys. 17, 646–651 (2021). https://doi.org/10.1038/s41567-021-01170-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-021-01170-x

This article is cited by

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing