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arxiv: 2605.29767 · v1 · pith:YAAFEXJ6new · submitted 2026-05-28 · 🌊 nlin.CD · physics.comp-ph· physics.data-an

Complex network topological and spectral determinants of extreme events

Christian Hechler , Timo Br\"ohl , Ulrike Feudel , Klaus Lehnertz This is my paper

Pith reviewed 2026-06-28 23:41 UTC · model grok-4.3

classification 🌊 nlin.CD physics.comp-phphysics.data-an
keywords extreme eventscomplex networkscoupling topologyalgebraic connectivityedge densitynetworked dynamical systemspower-law scalingcollective dynamics
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The pith

The coupling strength needed to produce extreme events in networks follows a power-law relationship with edge density and algebraic connectivity.

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

The paper examines how the structure of connections in networks influences when collective extreme events appear in different dynamical systems. It finds that the minimal coupling strength required for these events relates to network properties like how densely connected the nodes are and how well the network is connected overall through a power-law scaling. This scaling holds similarly across various systems and ways of generating extremes, pointing to the network's wiring as the main driver. A reader would care because it suggests that predictions about when extremes occur can be made from topology without detailed knowledge of the individual node dynamics or the precise trigger.

Core claim

The authors determine the coupling strength necessary to generate an extreme event in the collective dynamics of networked systems and observe a power-law-like relationship between this threshold and both the edge density and algebraic connectivity of the coupling topologies. This relationship is largely independent of the specific dynamical system and the mechanism generating the extreme events, suggesting it is primarily mediated by aspects of the coupling topology.

What carries the argument

The power-law relationship between the coupling threshold for extreme events and the topological property of edge density together with the spectral property of algebraic connectivity.

If this is right

  • The threshold for extreme events can be predicted from network topology alone across different systems.
  • Adjusting edge density or algebraic connectivity can control the onset of extreme events.
  • The finding applies regardless of whether extremes arise from the same or different mechanisms.
  • Topological features dominate over dynamical details in determining extreme event occurrence.

Where Pith is reading between the lines

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

  • Similar scaling might appear in real-world networks such as climate or financial systems if topology is the dominant factor.
  • Designing networks with specific connectivity could prevent or promote extremes in engineered systems.
  • Further tests on very large or heterogeneous networks could confirm if the power law persists beyond the simulated cases.

Load-bearing premise

The observed power-law relationship is driven by the topological and spectral properties of the network rather than by the details of the dynamical models or the limited size of the networks used in simulations.

What would settle it

Simulating the same systems on much larger networks or with different coupling mechanisms and checking whether the power-law scaling between coupling threshold and edge density still holds with the same exponent.

Figures

Figures reproduced from arXiv: 2605.29767 by Christian Hechler, Klaus Lehnertz, Timo Br\"ohl, Ulrike Feudel.

Figure 1
Figure 1. Figure 1: FIG. 1. Relation between edge density [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Left: exemplary time series of observables of the regarded systems S1 – S4 exhibiting rare, high-amplitude events, which we consider [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Relationship between coupling threshold [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Same as Fig. 3, but for coupling topologies based on small [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Power law exponents [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: shows how the coupling threshold kth varies for dif￾ferent realizations of the networked systems S1 – S4 of differ￾ent sizes N. FIG. 6. Estimated coupling threshold kth for the networked systems S1 – S4 with different sizes N and for edge densities ε = 0.2 (upper panel) and ε = 0.5 (bottom panel). Coupling topologies based on random (blue), small-world (pr = 0.25; red) and scale-free (green) networks. 1E. … view at source ↗
read the original abstract

We study the impact of the coupling topology on the ability of various networked dynamical systems to generate extreme events. By determining the coupling strength that is necessary to generate an extreme event in the collective dynamics of a given system, we observe a power-law-like relationship between this coupling threshold and both topological (edge density) and spectral (algebraic connectivity) properties of various coupling topologies. Interestingly, this relationship appears to be largely independent of both the investigated system and the underlying mechanism to generate extreme events. This may indicate that the observed relationship is primarily mediated by aspects of the coupling topology.

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

3 major / 2 minor

Summary. The manuscript reports numerical observations across several networked oscillator models that the critical coupling strength required to produce extreme events in the collective dynamics exhibits a power-law-like dependence on the edge density and algebraic connectivity of the coupling graph. The authors conclude that this scaling appears largely independent of both the local dynamical system and the specific mechanism generating the extremes, implying that the relationship is primarily mediated by topological and spectral properties of the network.

Significance. A robust, mechanism-independent relationship between extreme-event thresholds and purely topological quantities would be a notable result for the study of collective extremes in complex networks, offering a potential topology-based diagnostic that does not require detailed knowledge of the node dynamics. The present evidence consists entirely of direct numerical measurements on finite networks; no analytic derivation, scaling collapse, or cross-model statistical test is supplied to substantiate the claimed universality.

major comments (3)
  1. [Abstract] Abstract and main results section: the claim that the observed power-law relationship is 'largely independent of both the investigated system and the underlying mechanism' is stated without quantitative support (e.g., variation of fitted exponents across models, number of independent realizations, or error bars on the thresholds). This directly affects the central claim of topology mediation.
  2. [Numerical results] Results on finite-N networks: all reported thresholds are obtained on networks of fixed, modest size. No finite-size scaling analysis or extrapolation to the thermodynamic limit is presented, leaving open the possibility that the apparent power-law is a finite-size artifact rather than a topological invariant.
  3. [Methods and models] Model selection: only a modest set of oscillator models is examined. No analytic argument or additional test is given to show that the same scaling would hold for qualitatively different local dynamics (e.g., non-oscillatory or excitable systems) whose instability mechanisms differ structurally from those tested.
minor comments (2)
  1. Notation for algebraic connectivity and edge density should be defined explicitly at first use and used consistently.
  2. Figure captions should state the number of realizations and the fitting procedure used to extract the reported power-law exponents.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment below and indicate where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract and main results section: the claim that the observed power-law relationship is 'largely independent of both the investigated system and the underlying mechanism' is stated without quantitative support (e.g., variation of fitted exponents across models, number of independent realizations, or error bars on the thresholds). This directly affects the central claim of topology mediation.

    Authors: We will revise the abstract and results sections to include quantitative support for the independence claim. This will consist of the range of fitted exponents across models and mechanisms, the number of independent realizations per threshold, and error bars on the reported thresholds. revision: yes

  2. Referee: [Numerical results] Results on finite-N networks: all reported thresholds are obtained on networks of fixed, modest size. No finite-size scaling analysis or extrapolation to the thermodynamic limit is presented, leaving open the possibility that the apparent power-law is a finite-size artifact rather than a topological invariant.

    Authors: We agree that finite-size effects merit explicit examination. In the revision we will add simulations across a range of network sizes and include a finite-size scaling discussion to assess robustness of the observed power-law. revision: yes

  3. Referee: [Methods and models] Model selection: only a modest set of oscillator models is examined. No analytic argument or additional test is given to show that the same scaling would hold for qualitatively different local dynamics (e.g., non-oscillatory or excitable systems) whose instability mechanisms differ structurally from those tested.

    Authors: The manuscript demonstrates the scaling across several oscillator models with distinct extreme-event mechanisms. While no analytic derivation of universality is provided, the numerical consistency across the tested systems underpins the topology-mediated claim. Extending the analysis to non-oscillatory or excitable systems lies beyond the present scope. revision: no

Circularity Check

0 steps flagged

No derivation offered; empirical measurements only

full rationale

The paper reports numerical experiments that measure a coupling threshold for extreme events in several oscillator networks and plot this threshold against edge density and algebraic connectivity, observing a power-law-like trend. No equations, ansatz, fitted parameters, or self-citations are used to derive or predict the observed relationship; the central claim rests entirely on direct simulation output across multiple models. Because no load-bearing step reduces by construction to the paper's own inputs, the analysis contains no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated. The work is presented as an empirical observation from simulations.

pith-pipeline@v0.9.1-grok · 5628 in / 1039 out tokens · 22495 ms · 2026-06-28T23:41:12.230103+00:00 · methodology

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