Universality in randomly connected neural networks: old and new on a conjecture of Geman and Hwang
Abstract
Randomly connected networks are ubiquitous in computer science, physics or biology. A central question is whether, and how, their collective dynamics depend on the choice of the distribution of the random couplings. Forty years ago, Geman and Hwang formulated a precise version of this \emph{universality} question for randomly connected neural networks. We come back to these conjectures, review what was proven since, provide a few new extensions and propose new conjectures in this paper. For n neuron networks with couplings with a mean scaling as 1/n and light enough tails, two classes of limits are expected. For \emph{weakly heterogeneous} couplings, with variance o(1/n), the network is predicted to converge to a McKean--Vlasov equation, and a broad literature has established this under general assumptions. For \emph{strongly heterogeneous} couplings, with variance of order 1/n, the limit is a non-Markovian equation driven by a self-consistent Gaussian process. After landmark results for Gaussian couplings, universality was only recently established for linear interactions, while the continuous-time nonlinear case relevant to neuroscience remained open. Here, in the weakly heterogeneous regime, we prove quenched convergence on every time interval under a fourth-moment assumption only, and convergence in probability under an assumption slightly stronger than a second moment. In the strongly heterogeneous regime, we prove convergence for nonlinear interactions in continuous time, averaged over sub-Gaussian couplings and on every time interval, by a relative-entropy comparison with Gaussian couplings. We then extend the conjecture to heavy-tailed couplings, and propose a limit driven by a self-consistent α-stable process, for which we prove existence for every α∈(0,2) and uniqueness for α∈[1,2), and provide extensive numerical evidence in all three cases.
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BibTeX
@misc{touboul2026universality,
title = {{Universality in randomly connected neural networks: old and new on a conjecture of Geman and Hwang}},
author = {Jonathan Touboul},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38893},
eprint = {2610.38893},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38893},
}
RIS
TY - UNPB
TI - Universality in randomly connected neural networks: old and new on a conjecture of Geman and Hwang
AU - Jonathan Touboul
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38893
UR - https://arxiv.org/abs/2610.38893
AB - Randomly connected networks are ubiquitous in computer science, physics or biology. A central question is whether, and how, their collective dynamics depend on the choice of the distribution of the random couplings. Forty years ago, Geman and Hwang formulated a precise version of this \emph{universality} question for randomly connected neural networks. We come back to these conjectures, review what was proven since, provide a few new extensions and propose new conjectures in this paper. For n neuron networks with couplings with a mean scaling as 1/n and light enough tails, two classes of limits are expected. For \emph{weakly heterogeneous} couplings, with variance o(1/n), the network is predicted to converge to a McKean--Vlasov equation, and a broad literature has established this under general assumptions. For \emph{strongly heterogeneous} couplings, with variance of order 1/n, the limit is a non-Markovian equation driven by a self-consistent Gaussian process. After landmark results for Gaussian couplings, universality was only recently established for linear interactions, while the continuous-time nonlinear case relevant to neuroscience remained open. Here, in the weakly heterogeneous regime, we prove quenched convergence on every time interval under a fourth-moment assumption only, and convergence in probability under an assumption slightly stronger than a second moment. In the strongly heterogeneous regime, we prove convergence for nonlinear interactions in continuous time, averaged over sub-Gaussian couplings and on every time interval, by a relative-entropy comparison with Gaussian couplings. We then extend the conjecture to heavy-tailed couplings, and propose a limit driven by a self-consistent α-stable process, for which we prove existence for every α∈(0,2) and uniqueness for α∈[1,2), and provide extensive numerical evidence in all three cases.
ER -