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arxiv: 2506.18498 · v2 · submitted 2025-06-23 · 💻 cs.IT · math.IT

A scalable estimator of higher-order information in complex dynamical systems

Alberto Liardi , George Blackburne , Hardik Rajpal , Fernando E. Rosas , Pedro A.M. Mediano This is my paper

Pith reviewed 2026-05-19 08:03 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords M-informationhigher-order informationcomplex dynamical systemsconvex optimizationinformation integrationmultivariate time seriesinformation decompositionscalability
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The pith

M-information measures higher-order information integration in complex dynamical systems through a scalable convex optimization routine.

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

The paper introduces M-information as a quantity that captures how collective patterns beyond pairwise interactions integrate information across a system. Existing higher-order measures become impractical for large systems because their computation grows too quickly with the number of variables. By recasting the problem as a convex optimization task, the authors obtain an algorithm whose runtime scales gracefully and that can be applied directly to multivariate time series. Tests on simulated neuronal populations and on macaque and mouse neuroimaging recordings show that the measure remains stable under noise, tracks critical regimes, and distinguishes different states of consciousness and task engagement. The same quantity can be slotted into existing information-decomposition schemes to produce a fuller taxonomy of information dynamics.

Core claim

M-information quantifies the higher-order integration of information in complex dynamical systems. It is obtained by solving a convex optimisation problem whose solution yields a robust estimator that scales with system size, remains resilient to noise, indexes critical behaviour in artificial populations, and distinguishes states of consciousness and task performance in real neuroimaging data from macaques and mice. The measure can also be embedded inside standard information-decomposition frameworks.

What carries the argument

M-information, obtained by solving a convex optimisation problem that isolates higher-order interdependencies among the variables of a dynamical system.

If this is right

  • Analysis of collective computation becomes feasible in systems with hundreds of variables where earlier measures were intractable.
  • The same estimator can be inserted into existing partial-information-decomposition pipelines to produce a more complete taxonomy of information dynamics.
  • Empirical studies of neuronal populations and whole-brain recordings can now routinely track higher-order coordination rather than stopping at pairwise statistics.
  • The measure supplies a practical index for critical behaviour and for changes in conscious state without requiring ad-hoc parameter tuning.

Where Pith is reading between the lines

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

  • The same convex formulation could be adapted to detect emergent collective states in non-neural domains such as climate or financial time series.
  • Because the algorithm is efficient, it opens the door to online monitoring of higher-order structure in streaming data from large sensor arrays.
  • Embedding M-information in decomposition frameworks may reveal previously hidden trade-offs between pairwise and higher-order contributions to overall system capacity.

Load-bearing premise

The convex optimisation problem accurately encodes the intended higher-order interdependencies without introducing systematic bias that would alter the reported empirical patterns.

What would settle it

If M-information values computed on larger simulated networks fail to peak at the known critical transition points or lose their ability to separate conscious from unconscious states in additional brain datasets, the central claim would be undermined.

Figures

Figures reproduced from arXiv: 2506.18498 by Alberto Liardi, Fernando E. Rosas, George Blackburne, Hardik Rajpal, Pedro A.M. Mediano.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The estimation of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: BROJA- [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p028_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p029_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p030_9.png] view at source ↗
read the original abstract

Our understanding of complex systems rests on our ability to characterise how they perform distributed computation and integrate information. Advances in information theory have introduced several quantities to describe complex information structures, where collective patterns of coordination emerge from higher-order (i.e. beyond-pairwise) interdependencies. Unfortunately, the use of these approaches to study large complex systems is severely hindered by the poor scalability of existing techniques. Moreover, there are relatively few measures specifically designed for multivariate time series data. Here we introduce a novel measure of information about macroscopic structures, termed M-information, which quantifies the higher-order integration of information in complex dynamical systems. We show that M-information can be calculated via a convex optimisation problem, and we derive a robust and efficient algorithm that scales gracefully with system size. Our analyses show that M-information is resilient to noise, indexes critical behaviour in artificial neuronal populations, and reflects states of consciousness and task performance in real-world macaque and mouse neuroimaging data. Furthermore, M-information can be incorporated into existing information decomposition frameworks to reveal a comprehensive taxonomy of information dynamics. Taken together, these results help us unravel collective computation in large complex systems.

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.

Referee Report

2 major / 1 minor

Summary. The paper introduces M-information as a novel measure quantifying higher-order integration of information in complex dynamical systems from multivariate time series. It claims this measure admits an exact representation as a convex optimization problem, for which a robust scalable algorithm is derived. Empirical analyses demonstrate resilience to noise, indexing of critical behavior in artificial neuronal populations, and correlations with consciousness states and task performance in macaque and mouse neuroimaging data; the measure is further positioned for integration into information decomposition frameworks.

Significance. If the core equivalence between the defined M-information and its convex optimization solution holds without bias or hidden relaxations, the work would provide a valuable scalable tool for higher-order information analysis in large systems, addressing a key limitation of existing techniques in information theory and complex systems. The reported applications to neuroimaging data suggest potential for advancing studies of collective computation, though this hinges on the fidelity of the numerical estimator.

major comments (2)
  1. [Abstract] Abstract and central derivation: the claim that M-information 'can be calculated via a convex optimisation problem' whose solution equals the original higher-order quantity is load-bearing for all results, yet the available text provides no explicit objective function, constraints, or proof of exact recovery. Without this, it is impossible to rule out systematic bias or approximations that would affect the reported noise resilience and neuroimaging correlations.
  2. [Results] The empirical claims (resilience to noise, critical behavior indexing, consciousness/task correlations) rest on the optimizer outputs faithfully representing the intended interdependencies; any implicit assumptions about the joint distribution or auxiliary variables in the convex program could introduce under- or over-estimation, undermining the interpretation of the macaque and mouse data results.
minor comments (1)
  1. [Abstract] The abstract would benefit from a concise mathematical definition of M-information before stating its computational representation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive feedback, which highlights important aspects of clarity and validation in our work. We address each major comment below and have revised the manuscript to strengthen the presentation of the central results.

read point-by-point responses

  1. Referee: [Abstract] Abstract and central derivation: the claim that M-information 'can be calculated via a convex optimisation problem' whose solution equals the original higher-order quantity is load-bearing for all results, yet the available text provides no explicit objective function, constraints, or proof of exact recovery. Without this, it is impossible to rule out systematic bias or approximations that would affect the reported noise resilience and neuroimaging correlations.

    Authors: We agree that the explicit formulation of the convex program is essential and should be more prominent. In the revised manuscript we have added a dedicated subsection to the Methods that states the objective function (minimization of a convex functional equivalent to a regularized mutual information subject to marginal constraints encoding higher-order dependencies), the full set of linear constraints, and a self-contained proof that the optimal value recovers the defined M-information exactly for discrete finite alphabets with no hidden relaxations or bias. This addition directly addresses concerns about systematic error in the subsequent analyses. revision: yes

  2. Referee: [Results] The empirical claims (resilience to noise, critical behavior indexing, consciousness/task correlations) rest on the optimizer outputs faithfully representing the intended interdependencies; any implicit assumptions about the joint distribution or auxiliary variables in the convex program could introduce under- or over-estimation, undermining the interpretation of the macaque and mouse data results.

    Authors: We share the concern that estimator fidelity must be demonstrated. The revised manuscript now includes a new validation subsection that compares optimizer outputs against exact enumeration on small systems (N≤6), confirming recovery to machine precision. We have also added explicit discussion of the stationarity assumption on the empirical joint distribution and the role of auxiliary variables, together with sensitivity analyses showing that variations in these elements do not materially alter the reported noise resilience, criticality signatures, or correlations with consciousness and task performance. These checks support the interpretations while acknowledging the underlying modeling choices. revision: yes

Circularity Check

0 steps flagged

M-information definition and convex optimization are presented as independent derivation steps

full rationale

The paper introduces M-information as a novel measure quantifying higher-order integration in dynamical systems, then separately shows it admits a convex optimization representation with a derived scalable algorithm. No equations or self-citations reduce the target quantity to a fitted parameter or prior self-result by construction. The central claims rest on the optimization faithfully implementing the defined measure rather than redefining it, and the provided abstract and skeptic notes give no explicit reduction of the form 'M-information equals the optimizer output by definition'. This is the common case of a self-contained computational derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

The central claim rests on the definition of M-information as a distinct higher-order quantity and on the claim that its computation reduces to a convex program whose solution yields the desired measure.

invented entities (1)
  • M-information no independent evidence
    purpose: Quantify higher-order integration of information beyond pairwise terms in dynamical systems
    New information-theoretic quantity introduced by the paper.

pith-pipeline@v0.9.0 · 5739 in / 1055 out tokens · 54214 ms · 2026-05-19T08:03:28.370449+00:00 · methodology

discussion (0)

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