SciPost Submission Page
Nonequilibrium steady-state dynamics of Markov processes on graphs
by Stefano Crotti, Thomas Barthel, Alfredo Braunstein
Submission summary
| Authors (as registered SciPost users): | Alfredo Braunstein |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202501_00038v1 (pdf) |
| Code repository: | https://github.com/stecrotti/EternalDynamicCavity |
| Date submitted: | Jan. 20, 2025, 4:20 p.m. |
| Submitted by: | Braunstein, Alfredo |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approaches: | Theoretical, Computational |
Abstract
We propose an analytic approach for the steady-state dynamics of Markov processes on locally tree-like graphs. It is based on time-translation invariant probability distributions for edge trajectories, which we encode in terms of infinite matrix products. For homogeneous ensembles on regular graphs, the distribution is parametrized by a single d × d × r² tensor, where r is the number of states per variable, and d is the matrix-product bond dimension. While the method becomes exact in the large-d limit, it typically provides highly accurate results even for small bond dimensions d. The d² r² parameters are determined by solving a fixed point equation, for which we provide an efficient belief-propagation procedure. We this approach to a variety of models, including Ising-Glauber dynamics with symmetric and asymmetric couplings, as well as the SIS model. Even for small d, the results are compatible with Monte Carlo estimates and accurately reproduce known exact solutions. The method provides access to precise temporal correlations, which, in some regimes, would be virtually impossible to estimate by sampling.
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
Current status:
Reports on this Submission
Strengths
1- Combines cavity theory with matrix product representation for inference about steady states 2- Good range of test cases both with and without detailed balance (and absorbing states) 3- Method with potential wide applicability
Weaknesses
1- Derivations in the main text are somewhat terse 2- Discussion of continuous time limit (important for many applications) is particularly short
Report
Requested changes
1- After eq(10), please make clear why complex conjugation (*) is used to define inner products, given that all messages (as probabilities) are real - is the idea to use the method also for complex measures as in calculation of spectral densities? 2- Please comment (at least briefly in the main text, otherwise in an appendix) how the optimisation in (10) is carried out, e.g. is it done by brute force optimisation over A or is there a more efficient method; similarly please comment on how the actual objective function in (10) is evaluated 3- In discussion of continuous time dynamics, please comment on whether the limit \Delta t -> 0 can be taken in the EDC equations (and if not, why not) 4- Please fix typos: Abstract: we apply? this approach p7 princip_al_ eigenvalues Caption fig 6a refers to an error of 0.005, while text says 10^{-3} After (C3b), w -> W App D: an_ iMP distribution (5)
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Strengths
- This paper proposes eternal dynamical cavity method to resolve the challenge of describing the non-equilibrium steady state dynamics for a broad range of dynamics models, in particular on sparse random graphs.
- The topic is very relevant in practical applications, e.g., disease spreading.
- The paper is well-written, giving a very nice introduction of the motivation, sufficient technical details to understand the relevant concepts.
Weaknesses
- The continuous dynamics part is very concise. I wonder if the method can be applied to a continuous asymmetric spin model? Could the authors give a concrete experiment?
- In Figure 1, the one-time-delayed correlation is compared. But the equilibrium model can be captured by a Boltzmann distribution. I wonder how the one-time-delayed correlation can be computed by a static cavity method. Could the authors offer more details about this?
Report
Requested changes
**Below Eq. 5, A is explained as a matrix, depending on i-th and j-th spin values, which confused me for its explicit form.
**In Eq. 7, the symbol may be a tensor product, please specify.
**In Figure 1, the one-time-delayed correlation is compared. But the equilibrium model can be captured by a Boltzmann distribution. I wonder how the one-time-delayed correlation can be computed by a static cavity method. Could the authors offer more details about this?
***Below Eq. 13, the threshold may be 0.055, as checked from Fig.4.
***The continuous dynamics part is very concise. I wonder if the method can be applied to a continuous asymmetric spin model? Could the authors give a concrete experiment?
Recommendation
Ask for minor revision