arxiv: 2604.17151 · v1 · submitted 2026-04-18 · 🧬 q-bio.NC

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Causality as a Minimum Energy Principle

Moo K. Chung , D. Vijay Anand , Anass B El-Yaagoubi , Jae-Hun Jung , Anqi Qiu , Hernando Ombao

Authors on Pith no claims yet

Pith reviewed 2026-05-10 06:22 UTC · model grok-4.3

classification 🧬 q-bio.NC

keywords causalityHodge theoryvariational principlefMRIbrain networkscyclic interactionsnetwork flowsenergy minimization

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The pith

Causality is directional energy flow from high to low states that exposes stable cycles in brain networks.

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

The paper tries to show that causality can be reframed as a variational minimum energy principle, where flows move directionally along network connections from higher to lower energy states. Hodge theory then decomposes these flows into dissipative parts and a persistent harmonic component that encodes stable cyclic interactions. This matters because standard models like Granger causality assume acyclic graphs and miss feedback loops common in brain connectivity. Applied to resting-state fMRI, the approach detects robust cyclic causal patterns invisible to conventional methods, offering a route to model higher-order dynamics in complex networks.

Core claim

By grounding causality in a variational minimum energy principle, network flows are decomposed via Hodge theory into dissipative components and a harmonic component that captures stable cyclic causal interactions, as shown in resting-state fMRI connectivity where such patterns emerge robustly unlike in conventional causal models.

What carries the argument

Hodge decomposition of network flows into dissipative and persistent harmonic components under a minimum energy variational principle.

If this is right

Cyclic causal patterns become detectable in resting-state fMRI connectivity.
These patterns are not recovered by conventional models restricted to acyclic interactions.
The variational framework can represent higher-order dynamics beyond what structural equation models or Granger causality allow.
Stable harmonic components provide a signature for persistent interactions in network data.

Where Pith is reading between the lines

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

The same decomposition could be tested on non-brain networks such as gene regulation or traffic systems to check for analogous cyclic energy flows.
If the harmonic part tracks behavioral or cognitive states over time, it might serve as a biomarker for network stability.
Extending the energy principle to directed graphs with time delays could address dynamic causality in non-stationary data.

Load-bearing premise

Causality can be validly interpreted as directional energy flow from high- to low-energy states along network connections, and that the harmonic component from Hodge decomposition meaningfully represents causal cycles rather than mathematical artifacts.

What would settle it

Apply the decomposition to synthetic networks with known ground-truth cyclic causal structures; if the harmonic component fails to recover those cycles or detects cycles in purely acyclic data, the central claim would be falsified.

read the original abstract

Classical causal models, such as Granger causality and structural equation modeling, are largely restricted to acyclic interactions and struggle to represent cyclic and higher-order dynamics in complex networks. We introduce a causal framework grounded in a variational principle, interpreting causality as directional energy flow from high- to low-energy states along network connections. Using Hodge theory, network flows are decomposed into dissipative components and a persistent harmonic component that captures stable cyclic interactions. Applied to resting-state fMRI connectivity, our variational framework reveals robust cyclic causal patterns that are not detected by conventional causal models, highlighting the value of variational principles for causality.

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

The paper recasts causality as directional energy flow and uses Hodge decomposition to pull out a harmonic component for cycles in fMRI networks, but that component still lacks any demonstrated connection to temporal order or interventional effects.

read the letter

The central claim is that a variational minimum-energy principle plus Hodge theory can detect stable cyclic causality in brain networks where Granger or SEM methods fail because of their acyclic bias. They decompose network flows into gradient, curl, and harmonic parts, then interpret the harmonic subspace as the persistent cycles that matter for resting-state fMRI connectivity. That synthesis is the main new piece; separate energy models and Hodge applications exist, but the specific combination aimed at causal cycles in neuroimaging looks fresh on the abstract alone. The fMRI application is a reasonable place to test it, and they report finding patterns missed by standard tools. That is worth noting as a concrete illustration rather than pure theory. The soft spot is exactly the one the stress-test flags. The harmonic component is defined algebraically as the kernel orthogonal to the other subspaces; nothing in the linear algebra supplies time precedence, conditional independence, or response to external input. The energy-flow direction is motivated variationally but appears to rest on an interpretive step that is not derived from the data dynamics themselves. If the full methods include simulations that recover known directed cycles or comparisons against interventional benchmarks, that would close the gap; the abstract does not show those checks. The circularity concern is also live: if the energy landscape is built from the same connectivity matrix the model is trying to explain causally, the directionality may be assumed rather than discovered. This is for network neuroscientists who already work with topological tools and want to explore alternatives to acyclic causality. A reader looking for a new formal handle on loops in fMRI would find the framing useful even if the causal mapping needs tightening. I would send it to peer review. The idea is distinct enough and the domain application is relevant, so referees can press on validation and interpretation without the paper being obviously incoherent on its own terms.

Referee Report

3 major / 1 minor

Summary. The manuscript introduces a variational framework interpreting causality as directional energy flow from high- to low-energy states along network connections. It applies Hodge theory to decompose network flows into dissipative (gradient) and persistent harmonic components, claiming the harmonic part captures stable cyclic causal interactions. Applied to resting-state fMRI connectivity matrices, the approach is said to reveal robust cyclic causal patterns undetected by conventional models such as Granger causality or structural equation modeling.

Significance. If the interpretive link between the harmonic subspace and causal semantics can be established, the work would offer a mathematically grounded alternative for modeling cyclic dynamics in undirected networks, which is relevant for neuroscience. The variational minimum-energy principle and use of Hodge decomposition provide a clean algebraic separation of flow components, representing a potential strength in formalizing persistent cycles. However, without validation tying the decomposition to temporal ordering or interventional effects, the significance for causal inference remains limited.

major comments (3)

Abstract: the claim of 'robust cyclic causal patterns' is asserted without any equations, validation metrics, error controls, or comparison details; the full methods must specify the exact variational principle, energy definition from fMRI data, and quantitative support for robustness.
Hodge decomposition section (application to fMRI connectivity): the persistent harmonic component is the kernel of the graph Laplacian (or higher-order analogue) and is orthogonal to gradient flows by construction; the manuscript must demonstrate how this algebraic object encodes temporal precedence or interventional causality rather than remaining a topological artifact, as the input matrices are undirected and the decomposition introduces no time-ordering.
Variational principle (core framework): interpreting causality as minimum-energy directional flow risks circularity if the energy landscape or flow directions are defined using the same network structure the model aims to discover; explicit equations showing independence from fitted parameters or implicit assumptions are required.

minor comments (1)

Notation for the energy function, flow decomposition, and harmonic subspace should be introduced with explicit equations early in the methods to improve readability.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the thoughtful and detailed report. The comments highlight important points regarding clarity, the interpretive link to causality, and potential circularity. We address each major comment below and indicate where revisions will be made to the manuscript.

read point-by-point responses

Referee: Abstract: the claim of 'robust cyclic causal patterns' is asserted without any equations, validation metrics, error controls, or comparison details; the full methods must specify the exact variational principle, energy definition from fMRI data, and quantitative support for robustness.

Authors: We agree that the abstract is overly concise and lacks supporting details. In the revised version we will expand the abstract to reference the variational energy functional (minimizing the squared norm of the coboundary of the flow), the construction of the energy from the absolute values of the fMRI correlation matrix entries as edge weights, and quantitative robustness measures such as the proportion of total flow variance captured by the harmonic subspace (approximately 18% across subjects) together with direct numerical comparisons against Granger causality on the same datasets. revision: yes
Referee: Hodge decomposition section (application to fMRI connectivity): the persistent harmonic component is the kernel of the graph Laplacian (or higher-order analogue) and is orthogonal to gradient flows by construction; the manuscript must demonstrate how this algebraic object encodes temporal precedence or interventional causality rather than remaining a topological artifact, as the input matrices are undirected and the decomposition introduces no time-ordering.

Authors: The referee is correct that the input matrices are symmetric and that the decomposition itself is purely algebraic. Our interpretation rests on the variational principle: the harmonic component is the unique flow that is divergence-free and curl-free, hence non-dissipative, and therefore persists under the observed connectivity without external driving. We will add an explicit subsection deriving the causal semantics from the minimum-energy condition and will include a limitations paragraph acknowledging that the current data do not contain explicit temporal ordering or interventional perturbations. The revised text will therefore present the harmonic component as capturing stable cyclic interactions consistent with the variational principle rather than claiming direct interventional causality. revision: partial
Referee: Variational principle (core framework): interpreting causality as minimum-energy directional flow risks circularity if the energy landscape or flow directions are defined using the same network structure the model aims to discover; explicit equations showing independence from fitted parameters or implicit assumptions are required.

Authors: We maintain that circularity is avoided because the graph and its weights are fixed by the observed fMRI connectivity matrix; no causal parameters are fitted. The variational problem is then solved on this fixed weighted graph by finding the flow f that minimizes the Dirichlet energy ||df||^2 subject to the harmonic conditions d*f = 0 and df = 0. This decomposition is parameter-free and depends only on the algebraic structure of the graph. We will insert the explicit Euler-Lagrange equations and the orthogonal decomposition formula into the methods section to make this independence transparent. revision: no

standing simulated objections not resolved

Direct validation of the causal interpretation via interventional experiments or time-resolved directed data, which would be required to move beyond the algebraic and variational arguments currently presented.

Circularity Check

0 steps flagged

No circularity: Hodge decomposition and variational interpretation are independent of target causal claims

full rationale

The paper defines causality interpretively as directional energy flow and applies standard Hodge decomposition to connectivity matrices. The harmonic subspace is the kernel of the appropriate Laplacian by linear algebra; this algebraic fact is not derived from or equivalent to the causal interpretation. No equations reduce the discovered 'cyclic causal patterns' to a fit or redefinition of the input connectivity. No load-bearing self-citations or uniqueness theorems from the same authors are invoked to force the result. The framework is self-contained against external benchmarks (graph theory, variational principles) and does not smuggle the target conclusion into the premises.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; the framework rests on the unproven interpretive step that energy minimization equates to causality and on the applicability of Hodge theory to causal flows.

axioms (2)

domain assumption Causality corresponds to directional energy flow from high- to low-energy states along network edges
This is the foundational reinterpretation stated in the abstract.
domain assumption Hodge decomposition separates network flows into dissipative and persistent harmonic components that carry causal meaning
Invoked to capture stable cyclic interactions.

pith-pipeline@v0.9.0 · 5407 in / 1263 out tokens · 50200 ms · 2026-05-10T06:22:18.728619+00:00 · methodology

discussion (0)

Reference graph

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INTRODUCTION Understanding causality is fundamental to scientific inquiry, as it enables the identification of mechanisms that govern system dynamics beyond mere statistical associations. In neuroscience, causal modeling is essential for elucidating how brain regions influence one another and how information propagates through neural circuits to support c...

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Causality as a Minimum Energy Principle

MATERIALS AND METHODS 2.1. Imaging Data and Preprocessing We analyzed rs-fMRI data from 400 healthy adults (168 males, 232 females; age22–36years, mean29.24±3.39) from the Human Connectome Project (HCP). Data were ac- arXiv:2604.17151v1 [q-bio.NC] 18 Apr 2026 quired on Siemens 3T Connectome Skyra scanners, with acquisition parameters and the HCP minimal p...

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V ALIDATION Most widely used causal models are built on pairwise in- teractions and acyclic graph structures. Methods such as Granger causality, structural equation models (SEMs), and many DAG-based approaches posit explicitly on a directed acyclic graph, thereby precluding feedback and circulation by design [4, 5]. Consequently, cyclic structure is often...
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Emergence of Stable Cyclic Structures in rs-fMRI We discarded fMRI frames before 72 seconds and after 720 seconds, and estimated Hodge flow at every 3.6-second in- terval (5 TRs)

RESULTS 4.1. Emergence of Stable Cyclic Structures in rs-fMRI We discarded fMRI frames before 72 seconds and after 720 seconds, and estimated Hodge flow at every 3.6-second in- terval (5 TRs). For each time window, the edge flowXwas decomposed into dissipativeX D and harmonicX H compo- nents. Across 400 subjects and all time frames, the harmonic component...
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