Topological limits to the parallel processing capability of network architectures


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.

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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 \({\tilde{\phi }}_{\gamma }\).

Data availability

Example data files are available at Source data are provided with this paper.

Code availability

Code to reproduce the simulations and analysis reported here is availabile at


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G.P. has received funding support from Fondazione Compagnia San Paolo and from Intesa Sanpaolo Innovation Center.

Author information




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.

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Correspondence to Giovanni Petri.

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

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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.

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Petri, G., Musslick, S., Dey, B. et al. Topological limits to the parallel processing capability of network architectures. Nat. Phys. (2021).

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