Quantifying information flow along a stochastic trajectory

Euijoon Kwon; Yongjae Oh; Yongjoo Baek

arxiv: 2605.13509 · v1 · pith:YTTTLAVEnew · submitted 2026-05-13 · ❄️ cond-mat.stat-mech

Quantifying information flow along a stochastic trajectory

Yongjae Oh , Euijoon Kwon , Yongjoo Baek This is my paper

Pith reviewed 2026-05-14 19:14 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech

keywords stochastic information flowdeep learningtime-series analysistrajectory-level measurescooperative structuresKuramoto oscillatorsmotile cells

0 comments

The pith

Deep learning enables estimation of information flow along individual stochastic trajectories from time-series data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Stochastic information flow (SIF) quantifies directed information transfer along specific paths in stochastic processes, unlike traditional ensemble averages. The authors develop a deep learning method to estimate SIF scalably from raw time-series observations. This is validated on exactly solvable models and applied to Kuramoto oscillators and real motile cell trajectories to identify cooperative interactions. The approach makes trajectory-level analysis practical for complex systems where direct computation is hard.

Core claim

Stochastic information flow (SIF) quantifies information flow at the trajectory level, overcoming the limitations of conventional symmetric, ensemble-averaged measures. In this work, we propose a scalable deep-learning method for estimating the SIF from general time-series data. Its applications to an exactly solvable two-particle model, Kuramoto oscillators, and empirical trajectories of interacting motile cells demonstrate the utility of SIF as a data-driven indicator of cooperative structures.

What carries the argument

Deep neural network estimator trained on trajectory data to compute stochastic information flow (SIF) at the individual path level.

If this is right

SIF becomes computable for arbitrary stochastic time series without closed-form solutions.
The method identifies cooperative structures from empirical data alone.
It captures asymmetric information exchanges missed by symmetric measures.
Scales to large datasets in physics and biology for trajectory analysis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This could allow monitoring information dynamics in real-time systems like biological networks.
Extensions might combine it with other causality tools for better prediction of system evolution.
Applying it to new domains like social or economic time series could reveal hidden leadership patterns.

Load-bearing premise

The deep learning model generalizes accurately to compute SIF for unseen stochastic processes beyond its training data.

What would settle it

Testing the estimator on a previously unencountered exactly solvable stochastic model and finding that its predictions deviate substantially from the analytical SIF values.

Figures

Figures reproduced from arXiv: 2605.13509 by Euijoon Kwon, Yongjae Oh, Yongjoo Baek.

**Figure 2.** Figure 2: (b), as N is varied, we compare the exact values of DJX1 (black solid line) with its estimated values (symbols with error bars) for different values of M, the number of infinitesimal trajectory fragments. For M = 104 , the estimation is reliable up to 4 oscillators. If M = 107 , the estimation stays in good agreement with the exact [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Mean (blue symbols) and variance (orange symbols) [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Sample trajectories and schematics of a pair of cells [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. (a,c) The scaled variance ( [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. SIF statistics of 32 noisy Kuramoto oscillators with [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗

read the original abstract

Stochastic information flow (SIF) quantifies information flow at the trajectory level, overcoming the limitations of conventional symmetric, ensemble-averaged measures. However, computational difficulties have hindered the empirical application of the SIF. In this work, we propose a scalable deep-learning method for estimating the SIF from general time-series data. Its applications to an exactly solvable two-particle model, Kuramoto oscillators, and empirical trajectories of interacting motile cells demonstrate the utility of SIF as a data-driven indicator of cooperative structures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Deep learning for single-trajectory SIF works on the three tested systems but rests on narrow validation without error metrics or out-of-distribution checks.

read the letter

The main takeaway is a neural network trained to estimate stochastic information flow along individual trajectories instead of ensemble averages. The authors apply it to an exactly solvable two-particle model, Kuramoto oscillators, and real motile-cell paths, showing it can flag cooperative structure in those cases. That is the concrete advance: a practical computational route to path-wise directed information that was previously hard to compute at scale. They get credit for including the solvable model, which at least lets them check against known values on one example. The cell-trajectory results also suggest the estimator can pick up real biological signals without obvious circularity. The soft spot is the limited testing. Only three systems appear, with no reported quantitative error bars, no systematic comparison to exact SIF on the non-solvable cases, and no trials on processes outside the training distribution such as non-Markovian or high-dimensional dynamics. The claim that one architecture generalizes to arbitrary unseen time series therefore rests on empirical success in a narrow slice; that assumption could fail when trajectory length or state-space complexity grows. This is for people working on non-equilibrium statistical mechanics or data-driven collective behavior who need a tool to extract trajectory-level information flow from simulations or experiments. A reader already comfortable with neural nets for stochastic processes would get the most out of it. The work shows clear thinking on the problem setup and honest engagement with the literature, so it deserves peer review to examine the training details, code, and to require broader validation tests before the generality claim can be trusted.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a scalable deep-learning method to estimate stochastic information flow (SIF) along individual trajectories from general time-series data, addressing limitations of symmetric ensemble-averaged measures. It validates the approach on an exactly solvable two-particle model, Kuramoto oscillators, and empirical trajectories of interacting motile cells, claiming SIF serves as a data-driven indicator of cooperative structures.

Significance. If the neural-network estimator reliably recovers trajectory-level SIF values without model-specific retraining, the work would provide a practical tool for analyzing non-equilibrium dynamics and cooperation in physical and biological systems, extending beyond conventional information-theoretic measures. The inclusion of an exactly solvable model offers a potential anchor for validation, though broader generalization remains unproven.

major comments (2)

[Results] Results section on Kuramoto oscillators and cell trajectories: no quantitative error metrics, validation details, or direct comparisons to ground-truth SIF values are supplied, so the accuracy of the learned estimator cannot be assessed and the central claim of reliable estimation for general time-series data is unsupported.
[Method and validation] Method and validation sections: the claim that a single architecture trained on simulated or limited data recovers accurate SIF for arbitrary unseen stochastic processes lacks theoretical guarantees, approximation bounds, or systematic out-of-distribution tests (e.g., non-Markovian or high-dimensional cases); empirical success on only three specific systems does not establish the required generality, especially given that SIF estimation error can grow with trajectory length and state-space complexity.

minor comments (2)

[Abstract] Abstract: the term 'scalable' is used without accompanying analysis of computational complexity or scaling behavior with trajectory length or dimensionality.
[Introduction] Notation: the definition of SIF and its relation to the neural-network output should be stated more explicitly to avoid ambiguity in how the estimator approximates the path-dependent functional.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their thorough review and valuable comments. We address each of the major comments point by point below. Where appropriate, we have revised the manuscript to incorporate additional validation details and clarifications.

read point-by-point responses

Referee: [Results] Results section on Kuramoto oscillators and cell trajectories: no quantitative error metrics, validation details, or direct comparisons to ground-truth SIF values are supplied, so the accuracy of the learned estimator cannot be assessed and the central claim of reliable estimation for general time-series data is unsupported.

Authors: We appreciate this observation. The manuscript does include direct comparisons to ground-truth SIF values for the exactly solvable two-particle model in the Results section. For the Kuramoto oscillators and motile cell trajectories, where no analytical ground truth exists, we have added quantitative error metrics based on cross-validation, consistency with physical expectations, and comparisons to ensemble-averaged information measures in the revised manuscript. Additional validation details, including the training dataset composition and performance on held-out trajectories, have been included to better support the estimator's accuracy for these systems. revision: yes
Referee: [Method and validation] Method and validation sections: the claim that a single architecture trained on simulated or limited data recovers accurate SIF for arbitrary unseen stochastic processes lacks theoretical guarantees, approximation bounds, or systematic out-of-distribution tests (e.g., non-Markovian or high-dimensional cases); empirical success on only three specific systems does not establish the required generality, especially given that SIF estimation error can grow with trajectory length and state-space complexity.

Authors: We agree that the method lacks formal theoretical guarantees and approximation bounds, as it is an empirical deep-learning approach. The paper demonstrates the estimator on three distinct systems to illustrate its applicability, but does not claim it works for arbitrary processes without retraining. In the revision, we have included additional out-of-distribution tests on non-Markovian dynamics and higher-dimensional cases, along with an analysis of error scaling with trajectory length. We have also tempered the language regarding generality in the abstract and discussion sections to reflect the empirical nature of the validation. revision: partial

standing simulated objections not resolved

Provision of theoretical guarantees or approximation bounds for the performance of the neural network estimator on arbitrary stochastic processes.

Circularity Check

0 steps flagged

No circularity: SIF defined independently; NN estimator is a learned approximation, not a re-derivation

full rationale

The manuscript defines stochastic information flow (SIF) as a trajectory-level functional independent of the estimator. The proposed deep-learning method is presented as a scalable approximation technique trained on data to recover this pre-defined quantity. No equation reduces the output SIF value to a fitted parameter by construction, no self-citation supplies a uniqueness theorem that forces the architecture, and no ansatz is smuggled in. Validation on three concrete systems is empirical testing rather than tautological prediction. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, invented entities, or detailed axioms beyond the domain assumption that SIF is a well-defined trajectory-level quantity.

axioms (1)

domain assumption Stochastic information flow is a well-defined quantity that can be estimated from observed time series
Implicit in the proposal to estimate SIF from general time-series data.

pith-pipeline@v0.9.0 · 5374 in / 1070 out tokens · 25936 ms · 2026-05-14T19:14:44.078437+00:00 · methodology

discussion (0)

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Relation between the paper passage and the cited Recognition theorem.

we propose a scalable deep-learning method for estimating the SIF from general time-series data... NESIF... MINE... variational representation I(X,Y) ≥ ⟨iθ(x,y)⟩pX,Y − ⟨eiθ(x,y)−1⟩pXpY

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Reference graph

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