SciPost Submission Page
Gaussian approximation of dynamic cavity equations for linearly-coupled stochastic dynamics
by Mattia Tarabolo, Luca Dall'Asta
Submission summary
| Authors (as registered SciPost users): | Mattia Tarabolo |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202502_00024v2 (pdf) |
| Code repository: | https://github.com/Mattiatarabolo/GaussianExpansionCavityMethod.jl |
| Date accepted: | June 12, 2025 |
| Date submitted: | May 19, 2025, 3:47 p.m. |
| Submitted by: | Tarabolo, Mattia |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approaches: | Theoretical, Computational |
Abstract
Stochastic dynamics on sparse graphs and disordered systems often lead to complex behaviors characterized by heterogeneity in time and spatial scales, slow relaxation, localization, and aging phenomena. The mathematical tools and approximation techniques required to analyze these complex systems are still under development, posing significant technical challenges and resulting in a reliance on numerical simulations. We introduce a novel computational framework for investigating the dynamics of sparse disordered systems with continuous degrees of freedom. Starting with a graphical model representation of the dynamic partition function for a system of linearly-coupled stochastic differential equations, we use dynamic cavity equations on locally tree-like factor graphs to approximate the stochastic measure. Here, cavity marginals are identified with local functionals of single-site trajectories. Our primary approximation involves a second-order truncation of a small-coupling expansion, leading to a Gaussian form for the cavity marginals. For linear dynamics with additive noise, this method yields a closed set of causal integro-differential equations for cavity versions of one-time and two-time averages. These equations provide an exact dynamical description within the local tree-like approximation, retrieving classical results for the spectral density of sparse random matrices. Global constraints, non-linear forces, and state-dependent noise terms can be addressed using a self-consistent perturbative closure technique. The resulting equations resemble those of dynamical mean-field theory in the mode-coupling approximation used for fully-connected models. However, due to their cavity formulation, the present method can also be applied to ensembles of sparse random graphs and employed as a message-passing algorithm on specific graph instances.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
In response to the reports, we have revised the manuscript accordingly. The new version includes extensions of the original analysis, additional numerical results, clarifications of key assumptions, and several textual and graphical improvements. These changes are aimed at enhancing the accessibility and rigor of our presentation.
We believe that the revised manuscript now better communicates the ideas and results of our study, and we hope it will be of interest to researchers working on stochastic processes, complex networks, and disordered systems.
We gratefully acknowledge the valuable input of the reviewers and the editorial team in the preparation of this revised version.
List of changes
1) Added a comparison with related works in the introduction (page 4), specifically discussing Refs. [47] and [51].
2) Corrected the left panel of Figure 1.
3) Clarified the role of nonlinear interactions in the derivation of the equations at the end of page 6.
4) Standardized the notation for functionals by consistently using square brackets instead of round brackets.
5) Fixed various typographical errors throughout the manuscript.
6) Explained more clearly why hatted averages are expected to vanish, at the end of page 8.
7) Included the derivation of the equilibrium equations within the Gaussian Expansion Cavity Method (GECaM) framework at the end of page 13.
8) Clarified that stable solutions for linearly coupled Ornstein-Uhlenbeck processes exist only when all eigenvalues of the interaction matrix have negative real parts, and stated explicitly that this assumption is adopted throughout the paper (page 14, between Eqs. (54) and (55)).
9) Removed the sentence “We can therefore safely substitute z with his real part x = \text{Re}(z)” preceding Eq. (57).
10) Revised the notation in Eqs. (57) and (58) by using different indices for the eigenvalues.
11) Rewrote the introduction of Section 3 to improve clarity and context.
12) Added a quantitative comparison between equilibrium correlation functions obtained using GECaM and those obtained from Monte Carlo (MC) simulations on Random Regular Graphs (RRGs) with varying degrees. These results are also compared with the Fully Connected (FC) regime. The discussion is included at the end of Section 3.1 (page 19), and the data are presented in the new Figure 2.
13) Introduced a new Section 3.2, where GECaM is applied to study linear dynamics with thermal noise on heterogeneous graphs. This section examines the effects of topological heterogeneity and finite-size corrections. The results are summarized in the new Figure 3.
14) Justified the assumption of homogeneity in the analysis of the 2-spin model on RRGs, at the end of page 27.
15) Added a comment at the end of Section 3.5 pointing the reader to the Supplemental Material, where further numerical results on the 2-spin model are provided. These include a comparison between GECaM predictions and MC simulations, and validation of the homogeneity assumption.
16) Added a code availability statement in the Acknowledgments section.
17) Included additional references.
18) Added Section S2 in the Supplemental Material, providing a detailed derivation and implementation strategy for the equilibrium GECaM equations.
19) Added Section S5 in the Supplemental Material, presenting additional numerical results for the spherical 2-spin model. These include: (i) MC validation of the homogeneity assumption on RRGs (Figure 8); (ii) a direct comparison between MC and GECaM results (Figure 9); and (iii) MC simulations of the spherical ferromagnet (Figure 10), which indicate the presence of aging for small degree and its disappearance at higher connectivity.
Current status:
Editorial decision:
For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)
Reports on this Submission
Report
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Report
paper has been revised accordingly. In particular, the resubmitted
version includes results for the correlation function in the case of
sparse heterogeneous graphs, offering an even more
compelling demonstration of the potential applications of the
Gaussian cavity method.
This paper develops an efficient approach to studying the dynamics
of coupled differential equations on sparse random graphs, with
illutrative applications in different models. Although the method is
currently restricted to linear interactions, the authors introduce a
valuable and original theoretical framework for analyzing the dynamics
of continuous systems on complex networks. I therefore recommend
the paper for publication in SciPost Physics.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)