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Rigidity spectra and onset geometry of the two largest Forbush decreases of solar cycle 25 from visibility-graph curvature

This study applies nodal Forman-Ricci curvature of natural visibility graphs to neutron monitor data, revealing that this geometric network diagnostic provides a statistically robust, rigidity-resolved descriptor of Forbush decrease onset geometry and morphology, outperforming traditional degree-based metrics and linking curvature extremes to spectral hardness and amplitude.

Original authors: D. Sierra-Porta

Published 2026-07-16

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CC BY 4.0

Original authors: D. Sierra-Porta

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the Earth is wrapped in a giant, invisible magnetic bubble, a protective shield that keeps us safe from the constant stream of tiny, high-speed particles raining down from deep space. These particles are called cosmic rays, and they are like a steady, invisible drizzle hitting our planet. Usually, this drizzle is steady and predictable. But sometimes, the Sun gets angry. It hurls massive clouds of magnetized gas and shockwaves toward Earth, like a cosmic fire hose. When these solar storms hit our magnetic bubble, they can push the cosmic rays away, causing a sudden, sharp drop in the "drizzle" hitting our detectors. Scientists call these drops "Forbush decreases."

To understand these events, scientists have traditionally looked at how deep the drop is and how long it takes to recover. But there's a new way to look at the data, using a branch of math called "network science." Think of a time series (like a graph of cosmic ray counts over time) not just as a line, but as a social network. In this network, every data point is a person, and two people are "friends" (connected by a line) if they can "see" each other without any other data points blocking their view. This is called a "visibility graph." The paper uses a specific mathematical tool called "Forman–Ricci curvature" to measure the shape of this network. In simple terms, this curvature tells us how "hub-like" or "tree-like" the connections are. A flat, boring line looks like a straight path, but a sudden, steep drop creates a structure where a few points become super-connected hubs, and the math detects this as a sharp change in the network's "curvature."

This paper takes a fresh look at two of the biggest solar storms of the last few years: the "Gannon storm" from May 2024 and a massive event in June 2025. Instead of just measuring how low the cosmic ray counts fell, the authors used this "network curvature" tool to see exactly what the shape of the drop looked like. They found that when the solar storm hits, the curvature of the data network doesn't just wiggle; it collapses. It plunges into a deep, negative minimum that acts like a perfect fingerprint of the storm's arrival.

The researchers tested this method using data from six different neutron monitor stations around the world, which act like cosmic ray rain gauges. They discovered that at the exact moment the solar shockwave hits, the "curvature" of the data drops dramatically—by a factor of 2.5 to 8 times deeper than normal. This drop happens before the cosmic ray counts reach their lowest point, making it a very early warning sign. The team was very careful to prove this wasn't just a fluke. They ran a special statistical test that kept the natural "rhythm" of the data intact, and the result was rock-solid: the chance that this drop happened by random luck is less than 1 in 500.

The paper also compared the two storms. The 2024 Gannon storm was a fast, single punch, and the curvature drop was a sharp, quick spike. The 2025 storm was a slower, two-step event driven by multiple clouds of solar gas, and its curvature drop was deeper and lasted longer. Interestingly, the deeper the storm was, the harder the "curvature" dropped, following a specific mathematical rule. The authors suggest that this method is a powerful new way to describe the "shape" of these solar storms, offering a compact, mathematically precise way to see how the storm's structure changes as it moves through space. It's not a magic detector for every tiny storm, but for the big, dramatic ones, it provides a clear, geometric picture of the chaos happening between the Sun and Earth.

Technical Summary: Rigidity Spectra and Onset Geometry of the Two Largest Forbush Decreases of Solar Cycle 25 from Visibility-Graph Curvature

Problem Statement
Forbush decreases (FDs) are rapid depressions in galactic cosmic-ray flux driven by interplanetary coronal mass ejections (ICMEs) and their associated shocks. While the detection of large FDs is well-established via the worldwide neutron-monitor network, the scientific challenge lies in characterizing the magnitude, rigidity dependence (spectrum), and detailed time profiles of individual events. Existing methods often rely on global fitting or statistical averaging, which may obscure local, time-resolved geometric features of the decrease. The authors propose that the "amplitude geometry" of a time series—specifically how extreme drops create specific visibility structures—contains diagnostic information about the onset and morphology of FDs that is not captured by standard degree-based network metrics alone.

Methodology
The study applies a geometric network diagnostic, the nodal Forman–Ricci (FR) curvature, to Natural Visibility Graphs (NVGs) constructed from neutron monitor data.

Data Source: The analysis utilizes 30-minute count rates from six NM64 neutron monitor stations (CALM, BKSN, NEWK, YKTK, OULU, TERA) spanning cutoff rigidities ( $R_c$ ) from 0 to 7 GV. The dataset covers two specific events in Solar Cycle 25: the Gannon superstorm (May 10–11, 2024) and the June 1, 2025 event.
Graph Construction: Each time series is mapped to an NVG where nodes represent samples and edges connect pairs that "see" each other (i.e., no intermediate sample lies above the line connecting them).
Curvature Metric: The study computes the nodal Forman–Ricci curvature, $F(u)$ , defined as the mean of edge curvatures incident to a node. Crucially, this metric combines the node's own degree ( $k_u$ ) with the mean degree of its neighbors ( $\langle k \rangle_{N(u)}$ ), effectively encoding two-hop structural information.
Statistical Framework: To address the strong autocorrelation inherent in time-series data, the authors employ a placement (block-permutation) test. This preserves the exact autocorrelation structure of the pre-event interval to generate a null distribution, avoiding the overestimation of significance common in naive two-sample tests.
Controls and Robustness: The study includes controls using Horizontal Visibility Graphs (HVG), which are invariant to monotone amplitude transformations, to isolate the role of amplitude geometry. Robustness is further tested via edge weighting (temporal gaps) and sliding-window graph construction (6–24 h).
Spectral Analysis: FD rigidity spectra are derived by folding station amplitudes through the Dorman coupling function to fit a power-law model $\delta J/J \propto R^{-\gamma}$ .

Key Results

Curvature Collapse at Onset: Nodal FR curvature exhibits a sharp, coherent minimum at the onset of both FDs. The curvature values drop from quiet-time medians of $F \approx -10$ to $-14$ to extremes of $-66$ to $-164$ during the onset phase.
Statistical Significance: A placement test confirms the onset signature is statistically robust across all 24 station–phase combinations ( $p \lesssim 2 \times 10^{-3}$ , the test floor), with displacement values $|z|$ ranging from 4.7 to 23.7.
Superiority over Degree Sequence: The nodal curvature nearly doubles the effect size (Cliff's $\delta$ ) compared to using the NVG degree sequence alone (mean $\delta = 0.89$ vs. $0.54$). This confirms that the two-hop structure (neighbor degrees) encodes essential information about the decrease's geometry that the degree sequence misses.
Specificity to Amplitude Geometry: The signal is specific to the amplitude geometry of the decrease. The HVG control, which retains only ordinal structure, shows significantly reduced effect sizes, confirming the signal arises from the depth of the drop rather than just the order of samples.
Event Morphology and Timing:
- Gannon (May 2024): A single-step, shock-dominated event with a rapid main phase (~6.5 hours). The curvature response is a single, deep, short-lived spike.
- June 2025: A compound, two-step event driven by interacting CMEs. The curvature response is deeper (reaching $F \approx -164$ ) and more extended, reflecting the prolonged gradient phase.
- In both events, the curvature minimum precedes the count-rate minimum, responding to the steep gradient of the main phase rather than the depression floor.
Rigidity Spectra and Scaling:
- The Gannon event has a spectral index $\gamma = 0.80$ ( $A_{10} = 11.2\%$ ).
- The June 2025 event is spectrally harder with $\gamma = 0.50$ ( $A_{10} = 18.8\%$ ), indicating modulation extending to higher rigidities.
- A power-law relation is established between the curvature extreme and FD amplitude across all station-event pairs: $|F_{min}| \propto A^{0.57}$ (Spearman $\rho = 0.75$ ).

Significance and Claims
The paper claims to present the first application of nodal Forman–Ricci curvature to Forbush decreases. The authors position this method not as a detection tool for small events (which are often below the noise floor of 30-minute data), but as a compact, rigidity-resolved descriptor for characterizing the morphology of significant decreases.

The primary contribution is the demonstration that local graph curvature provides a time-local, single-station diagnostic that:

Is statistically robust against autocorrelation.
Encodes physical information about the transition from fluctuation-dominated to gradient-dominated transport (the "curvature collapse").
Distinguishes between shock/sheath-dominated and ejecta-dominated event morphologies.
Offers a quantitative link between the geometric structure of the time series and the physical rigidity spectrum of the cosmic-ray modulation.

The authors conclude that visibility-graph curvature is a promising addition to the FD analysis toolbox, specifically for characterizing event structure anchored to interplanetary context, rather than for blind detection. Future work suggested by the authors includes calibrating the amplitude-curvature relation on larger catalogs, extending the method to muon detectors, and testing precursory sensitivity against loss-cone methods.

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