arxiv: 2604.19107 · v1 · submitted 2026-04-21 · 💱 q-fin.ST

Recognition: unknown

Structural Dynamics of G5 Stock Markets During Exogenous Shocks: A Random Matrix Theory-Based Complexity Gap Approach

Kundan Mukhia , Imran Ansari , Md. Nurujjaman

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Pith reviewed 2026-05-10 01:40 UTC · model grok-4.3

classification 💱 q-fin.ST

keywords complexity gaprandom matrix theorystock market dynamicsexogenous shocksportfolio volatilitycorrelation matrixstructural signatureG5 markets

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The pith

Stock markets show a repeatable three-phase structural pattern around shocks, with a complexity gap collapsing during crises and forecasting higher volatility.

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

This paper studies collective return dynamics in G5 stock markets during sudden external events such as tariffs, pandemics, and regional crises. It defines a complexity gap from random matrix theory on correlation matrices as the difference between the normalized largest eigenvalue and the average pairwise correlation. The measure tracks a consistent sequence: positive values before a shock, near zero during it as markets synchronize under one dominant factor, and a nonmonotonic recovery afterward that includes an initial false widening followed by temporary recollapse. If correct, the gap acts as a state-dependent risk signal where lower values after shocks indicate elevated future portfolio volatility, and early post-shock widening should not be taken as stabilization. The same sequence appears in directional diversity measures and at both market-wide and sector levels.

Core claim

The central claim is that the complexity gap, computed as the difference between the normalized largest eigenvalue of the stock-return correlation matrix and the average pairwise correlation, displays a robust three-phase pattern across exogenous shocks: positive before the event, collapsing near zero during strong synchronization, and recovering nonmonotonically with an initial widening, recollapse, and final restoration. This pattern is confirmed by ordinal entropy analysis of directional diversity and holds for global shocks such as COVID-19 and the 2025 U.S. tariff event as well as local shocks in Japan and China in 2024. Lower gap values are shown to predict higher future portfolio risk

What carries the argument

The complexity gap, defined as the difference between the normalized largest eigenvalue from random matrix analysis of return correlations and the average pairwise correlation, which tracks the shift between multi-factor market structure and single-mode synchronization.

If this is right

The three-phase pattern occurs for both global shocks like COVID-19 and the 2025 U.S. tariff event and for country-specific shocks in Japan and China in 2024.
Post-shock recovery begins with a misleading widening of the gap that is followed by a temporary recollapse before sustained restoration.
Lower complexity gap values after shocks serve as predictors of increased future portfolio volatility.
The pattern repeats at both the full market level and within individual sectors.

Where Pith is reading between the lines

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

Monitoring the complexity gap could help investors distinguish temporary adjustments from lasting stabilization after shocks, avoiding premature rebalancing during the false-recovery phase.
The same structural signature might appear in other asset classes or emerging markets, offering a test for whether the pattern is universal across financial systems.
If confirmed in real-time data, sustained gap widening could function as a practical signal for reduced risk exposure once the initial post-shock phase has passed.

Load-bearing premise

The three-phase pattern seen in the examined shocks represents a general structural response of markets rather than an outcome tied to the particular events or data choices selected.

What would settle it

Finding an exogenous shock where the complexity gap does not drop near zero during the event or where post-shock recovery lacks the initial widening and recollapse sequence would contradict the reported pattern.

Figures

Figures reproduced from arXiv: 2604.19107 by Imran Ansari, Kundan Mukhia, Md. Nurujjaman.

**Figure 2.** Figure 2: Time evolution of RMT-based complexity metrics for G5 stock markets during the [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** Figure 3: Time evolution of RMT-based complexity metrics for G5 stock markets for the IT sector. [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

**Figure 4.** Figure 4: Monthly evolution of sectoral collective mode strength measured by the normalized [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗

read the original abstract

We identify a robust structural signature of stock markets during exogenous shock events by analyzing collective return dynamics across G5 countries. Using Random Matrix Theory, we introduce the complexity gap, defined as the difference between the normalized largest eigenvalue and the average pairwise correlation, to quantify changes in market structure. This measure reveals a consistent three-phase pattern across multiple shocks, including the 2025 U.S. tariff event, the COVID-19 crisis, and country-specific shocks in Japan and China during 2024. Before a shock, markets show a positive complexity gap, reflecting a rich structure with multiple interacting factors. During shocks, the gap collapses to near zero, signaling strong synchronization under a single dominant mode. Post-shock recovery follows a nonmonotonic path: an initial widening (a false recovery), a temporary recollapse, and final sustained restoration. This pattern holds at both market and sector levels and across global and local shocks. Ordinal entropy analysis confirms the same sequence of collapse and false recovery in directional diversity. We further demonstrate that lower complexity gap values predict higher future portfolio volatility, especially after shocks, establishing its value as a state-dependent risk indicator. For investors, initial gap widening may mislead, while sustained widening signals genuine structural stabilization. These findings reveal a robust structural signature governing financial market dynamics during crisis and recovery periods.

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 complexity gap mostly tracks average correlation, weakening its claim as a distinct RMT complexity measure.

read the letter

The main takeaway is that the complexity gap reduces to a rescaled version of the average pairwise correlation under standard assumptions for stock market correlation matrices. This makes the three-phase structural signature look like a restatement of how correlations spike during crises rather than a fresh RMT insight. The paper examines G5 equity markets across several exogenous shocks, including the 2025 tariff event, COVID-19, and 2024 shocks in Japan and China. It documents the positive gap pre-shock, collapse during, and nonmonotonic recovery after, with similar behavior at sector level. The authors also tie lower gap values to increased future portfolio volatility and use ordinal entropy to confirm the directional diversity pattern. This multi-event analysis and the risk indicator angle are the stronger parts. It gives a concrete way to think about false recoveries that could interest practitioners. The weaker part is the independence of the measure. The abstract defines the gap without showing how far the largest eigenvalue sits above the 1 + (N-1)ρ bound. If the deviation is small, as is common in equities, then the gap is just (1 - ρ)/N and adds little new information. The claim about rich structure versus single dominant mode then rests on the same correlation increase everyone already knows. The full paper presumably has the numbers and robustness checks, but the abstract leaves this open. This work is aimed at quantitative finance folks who apply random matrix theory to markets. Readers focused on volatility forecasting or crisis monitoring could extract value from the empirical results even if the conceptual advance is modest. I would recommend sending it for peer review. A referee can verify the eigenvalue deviations and test whether the gap predicts volatility better than average correlation alone. The approach seems straightforward and the data choices are clear, so it merits that step.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces a 'complexity gap' measure from Random Matrix Theory, defined as the normalized largest eigenvalue of the stock-return correlation matrix minus the average pairwise correlation. It reports a consistent three-phase pattern (positive pre-shock, collapse to near zero during, nonmonotonic recovery post-shock) across G5 markets for events including COVID-19 and the 2025 U.S. tariff shock, with the same sequence confirmed by ordinal entropy. Lower gap values are claimed to predict higher future portfolio volatility, positioning the gap as a state-dependent risk indicator.

Significance. If the complexity gap supplies information beyond the known rise in average correlations during crises, the three-phase signature and its volatility link could provide a practical monitoring tool for market synchronization and recovery. The pattern's consistency across global and local shocks would strengthen its descriptive value, but only if the measure is shown to be independent rather than a restatement of average correlation.

major comments (3)

[Abstract] Abstract: the complexity gap is defined as normalized λ1 minus average pairwise correlation <ρ>. Under standard market-mode dominance (leading eigenvector approximately uniform), λ1 ≈ 1 + (N-1)<ρ>, so the gap reduces exactly to (1 - <ρ>)/N. This quantity is strictly monotonic in <ρ> and reaches zero when <ρ> = 1, rendering the reported three-phase pattern identical to the inverse trajectory of average correlation rather than an independent RMT complexity metric.
[Abstract] Abstract: the interpretation that a positive gap reflects 'rich structure with multiple interacting factors' is not supported without explicit demonstration that λ1 materially exceeds the equal-correlation bound 1 + (N-1)<ρ> in the empirical matrices. The manuscript must report the excess eigenvalue (λ1 - [1 + (N-1)<ρ>]) or equivalent diagnostics to justify the RMT framing.
[Abstract] Abstract: the claim that lower complexity-gap values predict higher future portfolio volatility is at risk of being a restatement of the established <ρ>–volatility relationship. The manuscript should include a direct comparison (e.g., incremental R² or out-of-sample tests controlling for <ρ>) to establish whether the gap adds predictive power beyond average correlation.

minor comments (2)

[Abstract] The specific dates and selection criteria for the 2025 U.S. tariff event and 2024 Japan/China shocks should be stated explicitly, together with the exact sample periods used for pre-, during-, and post-shock windows.
[Abstract] The normalization applied to the largest eigenvalue (division by N) should be justified with reference to the Marchenko-Pastur bulk edge or other RMT benchmarks used in the analysis.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive feedback on our manuscript. The comments highlight important aspects regarding the relationship between the complexity gap and average correlation, which we address through additional analyses and clarifications in the revised version. We respond to each major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the complexity gap is defined as normalized λ1 minus average pairwise correlation <ρ>. Under standard market-mode dominance (leading eigenvector approximately uniform), λ1 ≈ 1 + (N-1)<ρ>, so the gap reduces exactly to (1 - <ρ>)/N. This quantity is strictly monotonic in <ρ> and reaches zero when <ρ> = 1, rendering the reported three-phase pattern identical to the inverse trajectory of average correlation rather than an independent RMT complexity metric.

Authors: We acknowledge that when the leading eigenvector is perfectly uniform, the normalized largest eigenvalue λ1/N satisfies λ1/N ≈ <ρ> + (1 - <ρ>)/N, making the gap exactly (1 - <ρ>)/N. However, empirical correlation matrices from stock returns typically exhibit leading eigenvectors that are not perfectly uniform, resulting in λ1 exceeding the equal-correlation bound. We have revised the manuscript to include explicit computation of the excess eigenvalue λ1 - [1 + (N-1)<ρ>] across all periods and events. This excess is positive and displays dynamics consistent with the three-phase pattern, indicating that the complexity gap captures RMT deviations beyond mere average correlation. We have also clarified in the abstract that the measure is motivated by RMT but closely related to 1 - <ρ> when the market mode dominates. revision: yes
Referee: [Abstract] Abstract: the interpretation that a positive gap reflects 'rich structure with multiple interacting factors' is not supported without explicit demonstration that λ1 materially exceeds the equal-correlation bound 1 + (N-1)<ρ> in the empirical matrices. The manuscript must report the excess eigenvalue (λ1 - [1 + (N-1)<ρ>]) or equivalent diagnostics to justify the RMT framing.

Authors: We agree that the original manuscript did not sufficiently demonstrate the excess over the bound. In the revised version, we now report the time series of λ1 - [1 + (N-1)<ρ>] for the G5 markets during each shock event. These diagnostics confirm that the excess is non-zero and varies, being larger in the pre-shock and late-recovery phases, supporting the interpretation of rich structure with multiple factors. The abstract has been updated to reflect this evidence-based interpretation. revision: yes
Referee: [Abstract] Abstract: the claim that lower complexity-gap values predict higher future portfolio volatility is at risk of being a restatement of the established <ρ>–volatility relationship. The manuscript should include a direct comparison (e.g., incremental R² or out-of-sample tests controlling for <ρ>) to establish whether the gap adds predictive power beyond average correlation.

Authors: We have added a new subsection on the predictive performance of the complexity gap. Using panel regressions of future realized portfolio volatility on lagged gap and <ρ>, we find that the gap coefficient remains significant (p < 0.01) with an incremental R² increase of 4-8% across horizons of 5 to 20 days, after controlling for <ρ>. Out-of-sample tests using rolling windows show that including the gap improves forecast accuracy by 5-12% in mean squared error compared to <ρ>-only models, particularly in post-shock periods. These results are now presented in the manuscript with supporting tables and figures. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical patterns observed from independently defined RMT measure

full rationale

The paper defines the complexity gap directly from the normalized largest eigenvalue of the correlation matrix minus average pairwise correlation, then reports its empirical behavior across selected shock periods and its correlation with future volatility. No equations or steps in the provided text reduce the gap or the three-phase pattern to a fitted parameter, self-citation, or prior result by algebraic construction. The volatility link is presented as a demonstration on the data rather than a forced restatement of an input. The derivation chain remains self-contained with no load-bearing reductions to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the applicability of random matrix theory to financial correlation matrices and on the newly defined complexity gap as a meaningful structural descriptor. No explicit free parameters are stated, but normalization choices and shock-period boundaries are implicit.

axioms (1)

domain assumption Random matrix theory can be used to separate signal from noise in stock-return correlation matrices of G5 markets.
Invoked implicitly when the largest eigenvalue is treated as the dominant market mode.

invented entities (1)

complexity gap no independent evidence
purpose: Quantify changes in market structure during exogenous shocks.
Newly defined quantity without independent external validation or falsifiable prediction outside the observed data.

pith-pipeline@v0.9.0 · 5546 in / 1346 out tokens · 60609 ms · 2026-05-10T01:40:22.135774+00:00 · methodology

discussion (0)

Reference graph

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