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Complexity synchronization of scaling exponents across coupled variables diagnoses cooperative performance in adaptive systems.

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2026-06-27 10:24 UTC pith:6T6A36T2

load-bearing objection This paper defines complexity synchronization as the correlation of sliding-window MDEA and DFA scaling exponents but supplies no results, statistics, or controls to show the measure is diagnostic rather than an artifact. the 3 major comments →

arxiv 2606.10948 v1 pith:6T6A36T2 submitted 2026-06-09 nlin.AO nlin.CD

Complexity synchronization as a diagnostic and control principle for adaptive systems

Korosh Mahmoodi , Scott E. Kerick , Piotr J. Franaszczuk , David L. Boothe , Paolo Grigolini , Bruce J. West This is my paper
classification nlin.AO nlin.CD
keywords complexity synchronizationadaptive systemsmulti-agent modelsscaling exponentscooperative performancetemporal complexitydiagnostic methodspredator-prey model
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proposes complexity synchronization as a way to measure how the temporal complexity of different parts of a system evolve together. By applying this to a model of selfish agents in a predator-prey setting with a dilemma payoff, it shows that this measure correlates with how well the system cooperates overall. Standard metrics only say if performance is good or bad, but this approach reveals the underlying coordination structure. If the idea holds, it could guide which parts to adjust when things go wrong in adaptive systems like teams or biological networks.

Core claim

Complexity synchronization is defined as the correlation between time-dependent scaling exponents obtained from modified diffusion entropy analysis and detrended fluctuation analysis applied in sliding windows to the outputs of interacting agents. In the high-interaction regime of the tested multi-agent model, the MDEA version of this measure increases as cooperative performance improves, while the DFA version identifies a different coordination mode based on persistence. This allows identification of functionally relevant subsystems that can be targeted for repair when performance declines.

What carries the argument

Complexity synchronization (CS), the correlation of time-varying scaling exponents that quantifies synchronization of evolving temporal complexity across coupled variables.

Load-bearing premise

The correlation of scaling exponents from the two analysis methods in sliding windows measures genuine synchronization of temporal complexity that is functionally tied to cooperative performance rather than depending on window size or fitting details.

What would settle it

Finding no increase in CS with performance in the high-interaction regime, or finding that adjusting subsystems identified by CS fails to change performance.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • MDEA-based CS increases with cooperative performance in high-interaction regimes.
  • DFA-based CS captures persistence-dominated coordination modes distinct from the MDEA version.
  • CS identifies functionally relevant subsystems within the adaptive system.
  • CS provides a basis for targeted repair interventions when performance fails.
  • CS serves as a general diagnostic framework for coordination in biological, social, and human-machine adaptive systems.

Where Pith is reading between the lines

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

  • Monitoring CS in other adaptive systems could flag coordination breakdowns before average performance metrics decline.
  • The separation of MDEA and DFA versions suggests that different complexity measures might be selected depending on the type of coordination being diagnosed.
  • If CS proves robust, it could support engineering control loops that adjust agent interactions to restore synchronization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 0 minor

Summary. The paper proposes complexity synchronization (CS) — defined as the correlation between time-dependent scaling exponents from sliding-window modified diffusion entropy analysis (MDEA) and detrended fluctuation analysis (DFA) — as a diagnostic and control principle for adaptive systems. It tests the idea in a multi-agent predator-prey model with prisoner's-dilemma payoffs and claims that, in the high-interaction regime, MDEA-based CS increases with cooperative performance, DFA-based CS captures a distinct persistence mode, and CS can identify functionally relevant subsystems for targeted repair.

Significance. If the central claims are substantiated with quantitative evidence and appropriate controls, CS could supply a new, non-average metric for diagnosing internal coordination modes in adaptive systems and guiding interventions, with potential applicability across biological, social, and engineered domains beyond standard performance or payoff measures.

major comments (3)
  1. Abstract: the assertion that 'MDEA-based CS increases with cooperative performance' is stated without any quantitative results, error bars, statistical tests, parameter values, window sizes, or figures; the central empirical claim therefore lacks verifiable support from the presented text.
  2. Abstract (definition of CS): CS is defined directly as the correlation of scaling exponents obtained by applying MDEA and DFA to identical sliding-window segments of the same time series; because both estimators respond to local variance and trends, the observed correlation may be an artifact of the shared analysis pipeline rather than evidence of genuine complexity synchronization, and no surrogate tests, null models, or variation of window/overlap parameters are described to rule this out.
  3. Abstract (subsystem claim): the statement that 'CS can reveal functionally relevant subsystems' is made without any concrete procedure for identifying subsystems from the CS time series, any validation against known functional divisions in the agent model, or any demonstration that CS-based targeting improves repair outcomes over baseline methods.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive comments. We address each major point below and indicate planned revisions to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [—] Abstract: the assertion that 'MDEA-based CS increases with cooperative performance' is stated without any quantitative results, error bars, statistical tests, parameter values, window sizes, or figures; the central empirical claim therefore lacks verifiable support from the presented text.

    Authors: The abstract is intended as a concise summary of the central findings. The quantitative results, including error bars, statistical tests, specific parameter values, window sizes, and supporting figures, appear in the Results section of the full manuscript. To make the abstract more self-contained and directly responsive to this concern, we will revise it to include key quantitative metrics and explicit references to the relevant figures and parameter settings. revision: yes

  2. Referee: [—] Abstract (definition of CS): CS is defined directly as the correlation of scaling exponents obtained by applying MDEA and DFA to identical sliding-window segments of the same time series; because both estimators respond to local variance and trends, the observed correlation may be an artifact of the shared analysis pipeline rather than evidence of genuine complexity synchronization, and no surrogate tests, null models, or variation of window/overlap parameters are described to rule this out.

    Authors: This concern about possible methodological artifacts is well taken. The manuscript defines CS via the correlation of the two scaling exponents computed on identical windows. While MDEA and DFA target distinct aspects of scaling behavior, we agree that explicit controls are needed. In the revised manuscript we will add surrogate tests, null-model comparisons, and sensitivity analyses that vary window length and overlap to demonstrate that the reported correlations are not artifacts of the shared pipeline. revision: yes

  3. Referee: [—] Abstract (subsystem claim): the statement that 'CS can reveal functionally relevant subsystems' is made without any concrete procedure for identifying subsystems from the CS time series, any validation against known functional divisions in the agent model, or any demonstration that CS-based targeting improves repair outcomes over baseline methods.

    Authors: We acknowledge that the abstract statement would benefit from greater specificity. The full manuscript outlines the procedure for extracting subsystems from CS time series, validates the identified subsystems against the known functional structure of the predator-prey model, and compares CS-guided repair against baseline interventions. We will revise the abstract to briefly describe the identification procedure and the validation approach, and we will ensure the main text makes the comparative performance of CS-based targeting explicit. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected; empirical test of defined measure.

full rationale

The paper explicitly defines CS as the correlation of time-dependent scaling exponents from sliding-window MDEA and DFA applied to the same trajectories, then reports an empirical observation that MDEA-based CS increases with cooperative performance in the high-interaction regime of the multi-agent model. This is an observational claim tested in simulation rather than a derivation that reduces by construction to its inputs via self-definition, fitted-parameter renaming, or self-citation chains. No equations, uniqueness theorems, or ansatzes are presented that would force the diagnostic utility from the definition alone. The central result remains an independent empirical finding whose validity can be assessed externally.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 1 invented entities

The claim rests on the domain assumption that scaling exponents from DFA and MDEA capture temporal complexity, plus the fitting procedures that produce those exponents; no independent evidence for the new construct CS is supplied.

free parameters (2)
  • scaling exponents (MDEA and DFA)
    Obtained by fitting to sliding-window time series; central to the definition of CS.
  • window size and overlap for sliding analysis
    Chosen to produce time-dependent exponents; not specified in abstract.
axioms (1)
  • domain assumption Temporal complexity of a time series is adequately quantified by the scaling exponent obtained from DFA or MDEA
    Invoked when the authors equate the scaling exponents with complexity measures.
invented entities (1)
  • Complexity synchronization (CS) no independent evidence
    purpose: Diagnostic and control principle for adaptive systems
    Newly introduced construct whose validity is asserted via the model results.

pith-pipeline@v0.9.1-grok · 5756 in / 1382 out tokens · 35509 ms · 2026-06-27T10:24:21.853186+00:00 · methodology

0 comments
read the original abstract

Adaptive systems can exhibit similar levels of performance while relying on fundamentally different internal modes of coordination. Standard metrics such as average cooperation or payoff indicate whether a system succeeds, but do not reveal how coordination is organized across interacting components or which adaptive variables should be targeted when performance fails. Here we propose complexity synchronization (CS), the synchronization of evolving temporal complexity across coupled variables, as a diagnostic and intervention guiding principle for adaptive systems. We test this idea in an adaptive multi agent system composed of Selfish Algorithm agents interacting in a reduced Predator Prey model with a Prisoners Dilemma like payoff structure. Temporal complexity is quantified using sliding window modified diffusion entropy analysis (MDEA) and detrended fluctuation analysis (DFA). CS is defined as the correlation between the resulting time dependent scaling exponents. In the high-interaction regime, MDEA-based CS increases with cooperative performance, whereas DFA based CS captures a distinct persistence dominated coordination mode. Our results show that CS can reveal functionally relevant subsystems and provide a principled basis for targeted repair. More broadly, CS offers a general diagnostic and engineering framework for understanding and controlling coordination in biological, social, human machine, and other adaptive systems.

Figures

Figures reproduced from arXiv: 2606.10948 by Bruce J. West, David L. Boothe, Korosh Mahmoodi, Paolo Grigolini, Piotr J. Franaszczuk, Scott E. Kerick.

Figure 1
Figure 1. Figure 1: Conceptual framework for complexity synchronization (CS) in interacting systems. Mutual adaptive interactions generate temporal dynamics that can be characterized through complementary scaling measures. For two interacting variables i and j (e.g., adaptive thresholds from two agents), detrended fluctuation analysis (DFA) quantifies persistence and long-range memory, producing scaling time series Hi(t) and … view at source ↗
Figure 2
Figure 2. Figure 2: Adaptive decision structure of the two interacting Selfish Algorithm (the two predator agents of the model). Each agent (S1 and S2) is defined by three thresholds governing information sharing (I), trust (T), and payoff sharing (P), with values in [0,1] that determine action probabilities through stochastic sampling. The red bars indicate the current threshold values, which are continuously updated via pay… view at source ↗
Figure 3
Figure 3. Figure 3: Representative snapshot of the spatial interaction dynamics in the Predator–Prey environment. Agents move on a two-dimensional domain with periodic boundary conditions, where the Prey (red) and Predators (blue) interact within finite sensing radii rS (circles). Arrows indicate instantaneous heading directions, and interactions are determined by local geometric proximity and predicted motion. This spatial f… view at source ↗
Figure 4
Figure 4. Figure 4: Representative short time series of the three adaptive threshold variables, information sharing I(t), trust T(t), and payoff sharing P(t), over 2000 trials (out of 106 ). The two colored traces correspond to the two Predator agents. These threshold dynamics form the raw signals from which local scaling exponents are extracted using MDEA and DFA. Accordingly, s = 0 corresponds to the damaged system, and s =… view at source ↗
Figure 5
Figure 5. Figure 5: Performance validation of the Predator–Prey model. Left panel: Cooperation rate Cr and payoff rate Pr (at t = 106 ) are shown as functions of the sensing radius ratio R for simulations with learning and without learning. Right panel: time evolution of the same quantities for R = 0.45. Scaling estimation from adaptive threshold dynamics The first step in the complexity analysis is to estimate local scaling … view at source ↗
Figure 6
Figure 6. Figure 6: Example of local scaling estimation from the adaptive threshold dynamics. Left panels: scaling-exponent time series obtained from sliding windows using MDEA (top) and DFA (bottom). Right panels: the corresponding scaling plots for a selected window, marked by circles, in the left panels. In the top-right panel, the entropy S(l) is plotted against log(l), where l is the length of the moving window in the MD… view at source ↗
Figure 7
Figure 7. Figure 7: Relationship between cooperation rate Cr and MDEA-based CS. Columns correspond to representative threshold pairs (I1–I2, I2–T1, and I1–P2), and rows correspond to stripe sizes used in MDEA (Str = 0.1,0.01,0.001). Each marker represents a simulation at a different sensing ratio R ∈ [0.25,0.45], with grayscale indicating the R value (color bar). Filled squares denote statistically significant CS values (p < … view at source ↗
Figure 8
Figure 8. Figure 8: Relationship between cooperation rate Cr and DFA-based CS. Each panel corresponds to a representative threshold pair (I1–I2, I2–T1, and I1–P2), with scaling exponents estimated using detrended fluctuation analysis (DFA). Each marker represents a simulation at a different sensing ratio R ∈ [0.25,0.45], with grayscale indicating the R value (color bar). Filled squares denote statistically significant CS valu… view at source ↗
Figure 9
Figure 9. Figure 9: Renewal aging test for the representative threshold I1. Survival functions Ψ(τ), aged survival functions, and shuffled-aged controls are shown for two sensing-radius regimes (R = 0.25 and R = 0.45) and three stripe sizes. The first half of the simulation was discarded, and one post-discard window of length 104 samples was analyzed. Events were defined as crossings from one stripe to another, and τ denotes … view at source ↗
Figure 10
Figure 10. Figure 10: Autocorrelation of inter-event intervals extracted from the representative threshold I1. The first half of the simulation was discarded, and τ sequences were extracted from 10 windows of length 104 samples with 75% overlap for each sensing-radius and stripe-size condition. Autocorrelation was computed on the inter-event intervals τ, not on the raw threshold signal. Curves show the mean across windows, and… view at source ↗
Figure 11
Figure 11. Figure 11: CS, computed using modified diffusion entropy analysis (MDEA), and cooperation rate (Cr) as functions of learning strength k of the SharePay thresholds. The top row corresponds to R = 0.25 and the bottom row to R = 0.45. Columns represent stripe sizes of 0.1, 0.01, and 0.001, respectively. Colored curves denote CS between selected threshold pairs (I,T,P) involving the payoff sharing thresholds, while gray… view at source ↗
Figure 12
Figure 12. Figure 12: CS, computed using detrended fluctuation analysis (DFA), and cooperation rate (Cr) as functions of learning strength k of the SharePay thresholds. Panel (a) corresponds to R = 0.25 and panel (b) to R = 0.45. Colored curves represent CS between selected threshold pairs involving the payoff sharing thresholds, while gray curves denote other pairwise combinations. The dashed green curve indicates Cr . Shaded… view at source ↗
Figure 13
Figure 13. Figure 13: Complexity-synchronization (CS) networks under baseline, perturbation, and rescue conditions (MDEA, R = 0.45, ENS = 10). Graph representations of pairwise CS among the six adaptive thresholds (I1,T1,P1,I2,T2,P2), computed using modified diffusion entropy analysis (MDEA). Nodes correspond to threshold variables, and edges represent CS links. Edge thickness and grayscale intensity encode CS strength, dashed… view at source ↗
Figure 14
Figure 14. Figure 14: Cooperation recovery under different rescue strategies (R = 0.45, ENS = 10). Cooperation rate, Cr , is shown as a function of rescue strength, s, for three intervention strategies: random/local rescue, equal-budget global rescue, and CS-guided targeted intervention. Curves show the mean across ensembles, and shaded bands indicate ± SD. Horizontal dashed lines denote the baseline and perturbed (damage) lev… view at source ↗
Figure 15
Figure 15. Figure 15: Search efficiency of CS-guided and random intervention strategies (R = 0.45). The best achieved cooperation rate, Cr , is plotted as a function of the number of tested threshold pairs (out of 15 possible pairs). The CS-guided strategy evaluates threshold pairs ranked according to the strength of their baseline complexity-synchronization (CS) links, whereas the random strategy samples threshold pairs unifo… view at source ↗
Figure 16
Figure 16. Figure 16: Supplementary Fig. S4. DFA-based complexity-synchronization (CS) networks under baseline, perturbation, and rescue conditions (R = 0.25, ENS = 10). Graph representations of pairwise CS among the six adaptive thresholds (I1,T1,P1,I2,T2,P2), computed using detrended fluctuation analysis (DFA). Nodes correspond to threshold variables, and edges represent CS links. Edge thickness and grayscale intensity encod… view at source ↗
Figure 17
Figure 17. Figure 17: Supplementary Fig. S5. Cooperation recovery in the low-interaction regime (R = 0.25, ENS = 10) using DFA-based complexity-synchronization (CS). The cooperation rate, Cr , is shown as a function of rescue strength, s, for three intervention strategies: random/local rescue, equal-budget global rescue, and CS-guided targeted intervention. Curves show the mean across ensembles, and shaded bands indicate ± SD.… view at source ↗
Figure 18
Figure 18. Figure 18: Supplementary Fig. S6. Ordinary Pearson correlations as a function of sensory radius. The ensemble-averaged cooperation rate (Cr , black) and ordinary Pearson correlations (colored curves) are shown as functions of the sensory-radius parameter R. Colored curves represent the ordinary Pearson correlation computed for each of the 15 unique pairs among the six adaptive variables (I1, I2, T1, T2, P1, and P2),… view at source ↗
Figure 19
Figure 19. Figure 19: Supplementary Fig. S7. Schematics for the aging experiment. Events ti are extracted using the stripe-crossing method, and the sequence is aged by an aging time ta. Adapted from Ref. 14 with permission. on log–log axes. The red dashed line in [PITH_FULL_IMAGE:figures/full_fig_p032_19.png] view at source ↗

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Reference graph

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