Characterization of Soil CO₂ Flux from an Active Volcano Through Visibility Graph Analysis

Abstract

The comprehension of the complex dynamics of degassing is critical for volcano monitoring and assessing volcanic hazards. In this study, we apply visibility graph analysis (VGA) to a decadal, high-resolution time series of daily soil CO₂ flux recorded by a standardized monitoring network at Mt. Etna volcano (Italy). By mapping these time series into complex networks, we demonstrate that the connectivity degree distributions follow a power law described by the exponent $\gamma$, which reveals a self-similar behavior of gas emissions. We introduce the $\gamma$-deviation, namely the variation of the scaling exponent from its long-term site-specific baseline, as a novel proxy for degassing efficiency. The long-term baseline is interpreted as a site-specific measure of flux efficiency, while its variations are attributed to other factors, such as fluctuations in the sources or changes in the efficiency of fluids transport pathways. Our results identify a transition from a period of discordance across the monitoring sites (pre-2016) to a phase of network-wide concordance (after 2016). The striking correlation between topological $\gamma$-deviations and the established normalized network signal (${\Phi }_{norm}$) validates the methodology, suggesting that VGA is able to capture the same underlying magmatic drivers. This study establishes VGA as a robust and reliable tool for medium- and long-term monitoring, potentially capable of identifying the occurrence of large-scale magmatic processes and refining the characterization of fluid transport dynamics in active volcanic systems.

1. Introduction

The study of volcanic gases is a fundamental tool for the investigation of magmatic processes in active volcanic systems because of their potential evolution into eruptive phenomena. For this reason, the monitoring of gas emissions is today considered a fundamental part of monitoring programs for active volcanoes around the world [1,2,3,4]. Among the volatiles present in magma, carbon dioxide is the most abundant after water; it is relatively straightforward to sample, and is the most widespread species in volcanic soils. In addition, there is a substantial scientific literature that establishes a correlation between the variability of emission rates and the eruptive state of volcanoes [5,6]. This correlation is also evident in the connection with degassing volcano-tectonic structures [7,8,9], thus establishing it as a fundamental geochemical parameter that requires monitoring.

In this work, visibility graph analysis (VGA) is applied for the first time to explore soil CO₂ emission on an active volcano. Data come from a permanent monitoring network [10,11] installed on the slopes of the active Mount Etna volcano (Sicily, Italy). The advantage of exploiting this dataset is the possibility of analysing simultaneous measurements from a network that is homogeneous in terms of geological, volcanological, and environmental conditions, as well as the type of monitoring instrumentation [12].

In broad terms, the VGA method transforms a generic time series into a graph, so that the reciprocal “visibility” among the series’ values represents the connections among nodes [13]. The application of this method to the study of environmental and geophysical variables is quite recent [14]. Among the data analyzed with VGA, we list: pollutants like PM₁₀ and PM_2.5, ozone (O₃), NO₂, SO₂, temperature, wind speed, streamflow, the solar cycle [15,16,17,18,19,20,21,22,23,24,25,26], and numerous examples in earthquake catalogs [27,28,29,30,31,32]. Within this overview, the application to CO₂ emissions in an active volcanic environment represents a true novelty.

The main objective of this work is to check the suitability of the VG transformation for this type of data, and to verify whether certain features of the visibility networks could represent an effective tool for characterizing complex signals, such as soil CO₂ flux time series. Validating the applicability of this approach will allow for the exploration of CO₂ flux dynamics by assessing its temporal and spatial variations, as well as the relationship between soil gas emissions and active volcanic processes.

2. Data and Methods

The CO₂ time series were recorded by a monitoring network of 14 sites located at Mt. Etna volcano (

Table 1. Locations of the soil CO₂ monitoring sites at Mt. Etna volcano and results of the power-law fitting of the VGA-converted time series. Details are provided in the main text.

Site id	Name	Lat. (°N)	Lon. (°E)	Elev. (m a.s.l.)	$\mathit{\gamma }$	$\mathit{\gamma }$ S.E.	p-Value
1	3c	37.6086	15.0822	518	2.14	0.07	0.043
2	agro	37.5336	14.8989	122	1.68	0.07	0.028
3	albano1	37.7253	14.9422	1724	2.03	0.07	0.010
4	brunek	37.8081	15.0742	1418	2.28	0.07	0.693
5	fondachello	37.7706	15.2167	11	2.20	0.09	0.597
6	maletto	37.7936	14.8997	1192	1.81	0.06	0.207
7	msm1	37.8258	14.9836	1535	2.37	0.09	0.624
8	p78	37.6953	15.1433	324	2.56	0.10	0.011
9	parcoetna	37.6306	15.0231	836	2.15	0.08	0.491
10	passop	37.8669	15.0456	704	2.59	0.10	0.100
11	roccacampana2	37.8003	15.1369	736	1.44	0.07	0.025
12	sml1	37.6569	14.9208	878	2.15	0.09	0.881
13	sml2	37.6636	14.9050	855	2.31	0.08	0.026
14	sv1	37.6967	15.1353	378	2.33	0.09	0.488

). These were placed to achieve uniform coverage around the flanks of the volcano. The analyzed time series cover a period of almost nine years, ranging from February 2011 to December 2019.

Raw time series were pre-processed before being converted into networks for their subsequent characterization. The pre-processing followed the well-established approach of [12,33] which was specifically developed to filter and homogenize measurements of soil CO₂, and filled the data gaps in the Etna network. Details on the procedure can be found in the source papers; it is briefly described in the following list:

(i)

Reduction of flux series from hourly to daily measurements.

(ii)

Modeling the annual variations by means of running averages with different numbers of terms.

(iii)

Filtering daily flux series to remove seasonal signals.

(iv)

Normalization of each of the 14 CO₂ flux series in the range $\phi$[0, 1] as a preparatory step to combine them into a global, unique signal representative of the CO₂ flux which describes the degassing state of the volcano. After this step, the physical unit of flux (kg · m² · d⁻¹) is dropped, and the normalized flux unit ${\Phi }_{norm}$[0, 1] is used.

(v)

Filling gaps (about 6%) with a combination of a linearly weighted moving average part and a white noise part. The imputation procedure is specifically tuned to the dataset, so that the statistical properties of time series remain unaltered. The resulting 14 series are composed of 3251 daily flux data points each.

The time series were then converted into graphs using the method called “visibility graph analysis” (VGA). This method transforms a generic series into a graph on the basis of the relationships between the nodes of the series. The network resulting from the geometric transformation performed by the VGA method retains the properties of the original time series [27,34]. In particular, VGA is based on the reciprocal “visibility” [34] between two generic nodes (Figure 1). Using this representation, values in the time series can be described as nodes, and the connections between the various nodes represent their relationships. This technique has proven to be highly informative for studying many different environmental and geophysical observables [17,18,20,21,23,26,31,32,35,36].

Adopting the notation from [34], the visibility relationship is verified when all the intermediate observations ($t_c,y_c$) between two generic observations of a time series, ($t_a,y_a$) and ($t_b,y_b$), fulfil the following relationship:

$$y_c<y_b+(y_a-y_b){\displaystyle \frac{t_b-t_c}{t_b-t_a}}$$

(1)

Therefore, the connectivity degree k is defined as the number of links that each observation i holds with all the other observations of the time series (i.e., network nodes)—$k_i=\Sigma a_{ij}$ [14]. Since each observation has at least two connections with its adjacent observations, the minimum value the connectivity degree can assume is $k=2$ (except for the two tip-observations). A node with a large value of k indicates that the corresponding observation is linked to many others and therefore represents a hub within the network.

Modifications of this method have been adopted, such as horizontal visibility graphs, which only consider horizontal “lines-of-sight” within the time series [37], the implementation of thresholds [38], and the definition of directional (forward or backward) criteria for mapping the time series [28,39,40]. In this work, we use the general, undirected notation proposed by [34]. The construction of a visibility graph from a time series can involve significant computational cost, especially for large time series, since VGA transformation has quadratic time complexity [41,42].

After converting the simulated catalogs into graphs, we evaluated the frequency distribution of the connectivity degree $P\left(k\right)$ to investigate the nature of CO₂ time series. In fact, if a time series shows scale-free fractal behavior, rather than periodic or random, then the tail region of the log–log plot of $P\left(k\right)$ can be fitted with a power-law relation in the form $P\left(k\right)\propto k^{-\gamma }$. The exponent $\gamma$ also represents the slope of the straight line in the linearized relation [34,43]. The degree distribution is one of the most common metrics used to evaluate complex networks (ref. [13] and reference therein). The approach we propose was used by several authors to investigate complex networks with the objective of characterizing the structure and the properties of the original time series [16,18,20,21,22,25,26,34,36,44,45].

The reliability of power-law distributions was tested following the approach proposed by [46]. This is based on a goodness-of-fit test which measures the deviation between the observed data and the hypothesized power-law model. This distance is expressed as a p-value, defined as the fraction of modeled deviations that are larger than the empirical deviation. For p-values above given thresholds (e.g., ≥0.05) the difference between observations and model can be attributed to stochastic fluctuations, and the power-law distribution represents a likely fit for the data [46]. However, as the authors warn, this test should not be considered as a decisive evidence to prove, or discard, whether the power law is the best distribution to describe the data.

3. Results

Results of the graph transformation of the normalized daily soil CO₂ time series are shown in Figure 2, where they are displayed together with the corresponding connectivity degree k after the graph conversion. It might be expected that higher values in the series would also have higher k values, but they often exhibit a wide range of degree (Figure 3). Therefore, higher values of CO₂ flux do not necessarily correspond to “hubs” in the VGA; rather, their values can span a broad range of k. This derives from the sequential order of observations within the time series [15,36] which reflects the time correlation of observations, such as, for instance, when two or more high values are located very close to each other (Figure 2). The patterns of the $\phi$-k plots are quite similar, even though peak k values vary from about 70 at the “passop” site to about 700 at the “rocacampana2” site (Figure 3).

The distributions of connectivity degrees P(k) are analyzed to investigate the nature of CO₂ flux time series (Figure 4). The greater the connectivity degree is, the lower the associated probability, because large degrees are exclusive to nodes with the highest values (i.e., hubs) due to their more probable high visibility. All time series have similar distributions, which, in a log–log plot, can be fairly approximated to a straight line. The regressions of k frequency distributions were calculated considering a minimum k threshold value because the power-law trend holds only for the tail region of the distribution. Such deviation from the trend is ascribable to edge effects (i.e., time series tips) or short-scale features within the time series. Regressions were performed for $k\ge 5$: this value minimizes the global residual error of regressions over the 14 time series. The calculated value of $\gamma$ represents a scale-free parameter that describes the characteristics of each power-law distribution; in particular, it defines the relative proportion of low-k and high-k nodes within the network. Because the distributions have decreasing trends, $\gamma$-values are negative; however, for the sake of simplicity, in the description of the results and their following discussion, we consider their absolute values $\left|\gamma \right|$. A high $\gamma$-value indicates a higher fraction of low-k nodes than large-k ones; conversely, a low $\gamma$-value indicates a lower fraction of small-k nodes than large-k ones (Figure 4).

Metrics used to evaluate the regression (residual standard error, coefficient of determination, and p-value in Figure 4) together concur in indicating reliable estimations of the $\gamma$-values. At the “albano” and “p78” sites, the p-value = 0.01, below the threshold for accepting the power-law distribution; however, the corresponding coefficients of determination (0.91 and 0.93) suggest very good correlation for $k\ge 5$. The breakdown of the power law of these series may suggest a transition in the underlying physical mechanism of degassing. The joint evaluation of the available metrics (Table 1 and Figure 5a) enables us to provide reliable estimations of the $\gamma$-value, which ranges between 1.44 (at the “roccacampana” site) and 2.59 (at the “passop” site), with the associated standard error (S.E.) in the range between 0.07 and 0.1.

The spatial distribution of $\gamma$-values does not show evidence of a clear spatial correlation (Figure 5b); rather, the arrangement appears uneven.

To further investigate the characteristics of the CO₂ flux time series and the dynamics of the volcanic system’s emissions, we examined both the temporal and spatial variations in $\gamma$-value.

3.1. Temporal Variability

The proposed methodology (VGA conversion and power-law fitting of the degree distribution) was also applied to analyze temporal variations in CO₂ signals at the monitoring sites. In particular, we considered 15 time windows (

Table 2. Details of the fifteen two-year time windows.

Window id	Start Date (yyyy-mm-dd)	End Date (yyyy-mm-dd)	N. of Days
1	2011-02-06	2013-02-05	730
2	2011-08-05	2013-08-04	730
3	2012-02-01	2014-01-31	730
4	2012-07-30	2014-07-30	730
5	2013-01-26	2015-01-26	730
6	2013-07-25	2015-07-25	730
7	2014-01-21	2016-01-21	730
8	2014-07-20	2016-07-19	730
9	2015-01-16	2017-01-15	730
10	2015-07-15	2017-07-14	730
11	2016-01-11	2018-01-10	730
12	2016-07-09	2018-07-09	730
13	2017-01-05	2019-01-05	730
14	2017-07-04	2019-07-04	730
15	2017-12-31	2019-12-31	730

), each with a duration of two years and a 75% overlap (i.e., 6-month shift). With this windowing arrangement, we expected a certain degree of smoothing in the obtained values. In this way, we aimed to eliminate short-term transients attributable to local site effects and enhance long-period variations, which are more likely related to changes in the signal source shared among the monitoring sites or to major changes in the permeability of the system [12,33]. The ranges of the obtained values (

Table 3. Results of power-law fitting slope $\gamma$ of the degree distribution for the 15 time windows (w) at the 14 monitoring sites; $\gamma$-values relative to the entire flux series are reported for comparison.

Site id	$\mathit{\gamma }$ Series	w1	w2	w3	w4	w5	w6	w7	w8	w9	w10	w11	w12	w13	w14	w15
1	2.14	1.91	2.01	2.06	2.03	2.07	2.06	1.99	1.95	2.05	1.86	1.76	1.71	1.70	1.76	1.88
2	1.68	1.60	1.61	1.61	1.59	1.55	1.35	1.32	1.29	1.34	1.36	1.51	1.45	1.45	1.39	1.45
3	2.03	1.99	1.95	1.90	1.78	1.89	1.90	2.02	2.12	2.02	1.85	1.72	1.60	1.52	1.75	1.66
4	2.28	2.02	1.85	1.76	1.92	1.82	2.02	1.83	1.91	1.66	1.72	1.70	1.92	1.88	1.92	1.93
5	2.20	1.91	1.93	1.86	1.86	1.87	1.75	1.84	1.82	1.82	1.86	1.87	1.81	1.87	1.84	1.84
6	1.81	1.42	1.34	1.35	1.47	1.46	1.51	1.66	1.77	1.86	1.84	1.91	1.55	1.58	1.71	1.69
7	2.37	2.09	1.89	1.88	1.89	2.07	2.03	2.11	2.06	2.15	2.15	2.14	1.91	1.99	2.01	2.14
8	2.56	2.44	2.50	2.49	2.45	2.57	2.63	2.44	2.38	2.30	2.27	2.29	2.11	2.21	1.98	2.18
9	2.15	1.90	2.03	2.02	1.91	1.94	1.75	1.97	2.04	2.24	1.91	1.92	1.46	1.64	1.58	1.54
10	2.59	2.06	2.08	2.05	2.03	2.29	2.36	2.41	2.43	2.39	2.32	2.16	2.18	2.25	2.15	2.10
11	1.44	1.37	1.46	1.73	2.16	2.01	1.72	1.36	1.38	1.29	1.31	1.47	1.51	1.37	1.15	1.33
12	2.15	1.81	1.66	1.66	1.98	1.95	1.95	2.19	2.01	1.90	1.84	1.92	1.82	1.84	1.80	1.90
13	2.31	2.09	2.08	1.96	2.08	1.78	1.77	1.90	1.91	2.14	2.18	1.86	1.90	1.95	2.03	2.02
14	2.33	2.10	2.37	2.38	2.32	2.45	2.23	2.25	2.22	2.05	2.08	2.09	2.25	2.25	1.91	1.87

) are comparable to those of the entire time series, with $\gamma$-values spanning from 1.15 to 2.63 (Figure 6 and Supplementary Material). At each site, variations are relatively limited; some of them appear to coincide with the occurrence of volcanic activity, although it is not possible to identify consistent trends across different sites (Figure 7a). Windowed $\gamma$-values are almost always lower (except at the “roccacampana” site) than the corresponding values calculated over the entire time series (Figure 6), suggesting that the latter act as upper thresholds for the windowed values. It is reasonable to assume that decadal values characterize the long-term flux conditions at each site, while relative variations are controlled by shorter-term dynamics. In this view, decadal $\gamma$-values can be used as reference values to compare $\gamma$ variations across different sites (Figure 7b). The $\gamma$-deviation values generally range in the interval between −0.5 and 0. A common pattern clearly emerges when computing the heat map of the $\gamma$-deviation (Figure 7c,d). Values remain consistent at most of the sites, except during the period of 2014–2016 when values are spread over a wider interval.

3.2. Spatial Variability

The windowed $\gamma$-values (Table 3) were also mapped to obtain progressive spatial distributions. In particular, Kriging interpolation was computed by modeling the omnidirectional semivariograms to infer the interpolation parameters. Kriging was preferred over other interpolation methods because it provides the error estimation, which is essential for assessing the reliability of the modeled spatial distributions. For all computed spatial interpolations, Kriging predictions and their respective standards errors are displayed in the Supplementary Material. Interpolations were performed on a 3 × 3 km grid covering a surface of approximately 2600 km²; this grid represents a trade-off between over-fitting and resolution. Interpolations were masked in the portions where errors were greatest and then smoothed using a moving-window resampling technique. The 14 monitoring sites are homogeneously distributed over the Etna area, thereby enabling reliable interpolation over a roughly circular region that corresponds to the entire volcanic edifice of Mt. Etna.

Spatial interpolations were calculated for the $\gamma$-values relative to the entire time series, and for the $\gamma$-deviations relative to the 15 time windows (Figure 8). Interpolations are displayed alongside the fault system affecting the eastern and southern flanks of Etna, which is recognized to interact with the flux dynamics as preferential emission zones [49,50,51]. Additionally, shallow seismicity is reported, which is often linked to the same volcanic and magmatic processes that control the soil CO₂ flux dynamics at the surface [50,52,53,54,55,56].

As mentioned in the previous paragraph, the spatial pattern of the $\gamma$-values for the entire times series is irregular and characterized by small-scale variations (Figure 5b and Figure 8), while interpolations of the windowed $\gamma$-deviation generally show more regular patterns. In windows 1 to 6, higher $\gamma$-deviation marks the northwestern and western flanks, while lower deviations are concentrated exactly on the opposite flanks; the range of values is similar across all these windows. Window 7 marks a transition to windows 8 to 11, where only the northwestern portion of the area exhibits low deviation values; deviation becomes progressively larger in the eastern flank. In this case, the range of values also remains consistent across the windows. Finally, windows 12 to 15 are characterized by a greater spatial variability in $\gamma$-deviation values and by a progressive increase in the number of earthquakes (Figure 8).

4. Discussion

VGA is a recognized tool for the characterization of complex signals. In this work, we applied this technique for the first time to the study of soil CO₂ flux time series, a key parameter in understanding magmatic processes on active volcanoes. In particular, we have robust evidence that VGA-converted soil CO₂ time series recorded at Mt. Etna generally follow a power-law distribution. Because the graphs inherit the underlying structure of their source time series [34,37], we can infer that the properties observed in the graph can be traced back directly to the original observations. The main implication of a power-law distribution is that the observed quantity is scale-invariant, meaning the data exhibit a self-similar structure, regardless of the scale. Such behavior is observed in several natural (and other) phenomena [46,58,59,60,61,62], specific to complex systems with non-linear, irregular dynamics that repeat across different scales. However, it is advisable that this novel application of VGA be supported by a more established method. For this reason, we processed the soil CO₂ time series, both the entire series and the windowed subsets, using fractal dimension (FD) analysis (see Supplementary Material). FD analysis captures the roughness (or smoothness) of a time series across scales by calculating the estimator D. We found that D-values are consistent with $\gamma$-values [63], supporting the idea that VGA is as robust as FD but provides unique topological insights. In fact, only the VGA suggests that the long-term $\gamma$-values represent an upper limit that can be considered as representative of the long-term behavior at each site (Figure S3).

Having established the degree distributions of the CO₂ flux, we observe that the exponent $\gamma$ differs significantly among the various monitoring sites (Figure 5a). The $\gamma$-value describes the proportion of nodes with large and small k-values: a higher proportion of small k-values with respect to higher k-values results in a higher $\gamma$-value, and vice versa.

It is important to note that the CO₂ time series were recorded employing standardized uniform instrumentation (all of which was designed by INGV) and were pre-processed and filtered to remove high-frequency noise and seasonal signals. Furthermore, the monitoring stations are almost evenly distributed across the flanks of the volcano. Consequently, since the non-volcanic components of the signal were removed, leaving only the “meaningful” residual component [12,64,65], the observed differences in $\gamma$-values must result from some other factors worth investigating.

Understanding the variability in $\gamma$-values is of significant interest, as it may suggest a link to the factors controlling soil gas emissions. For this reason, we investigated the temporal and spatial variations of the $\gamma$ exponent in more detail. The same degree distribution behavior persists when the signals are windowed; however, these windowed $\gamma$-values exhibit fluctuations and generally remain lower than the long-term $\gamma$ at each monitoring site (Figure 6). Differences in $\gamma$-value between sites can be attributed to local conditions, as frequently observed in soil gas emission studies [3,12,65,66]; thus, long-term $\gamma$-values may be considered site-specific measures of the flux regime. The reduction in windowed $\gamma$-value is not attributable to the shorter length of the VGA-converted series, which remain scale-invariant, but rather to change occurring in the flux regime over the time. As reported by [33], at a yearly time scale, the processes controlling the dynamics of soil CO₂ flux, and in particular its extreme values, differ from those operating over longer periods. Given that decadal values act as a baseline, temporal variations are expressed as $\gamma$-deviations.

When combined, the temporal fluctuations of $\gamma$ across the various monitoring sites show a certain concordance; namely, they assume similar values, which is particularly marked after 2016 (Figure 7c). The spatial arrangement of $\gamma$-values suggests the occurrence of periods characterized by smooth, large-scale variations that can be persistent over time, interspersed with periods of higher variability occurring over shorter time scales (Figure 8). The major changes in the spatial patterns (windows 1–6, 8–11, and 12–15) also correspond to significant changes in the $\gamma$-deviation trend: a dispersed signal (before 2016), a consistent increasing signal (2016–2018), and a consistent decreasing signal (after 2018) (Figure 7c). The homogeneous behavior on the western and northwestern flanks, in contrast to the more articulated behavior on the eastern flank, is well recognized. This difference results from the active fault system affecting the eastern and southern flanks [67,68] which regulates the release of underground fluids [33,50,51]. Interestingly, we also observe a strong correspondence between these variations and the trend of the entire-network signal ${\Phi }_{norm}$ (Figure 7d).

The methodology developed by [12] to integrate time series data from the Etna soil CO₂ network into a unified signal is well-established in the literature. This approach characterizes the state of the volcanic system on a broad scale and has provided a robust interpretive framework for assessing volcanic activity over the last ten years [33,54,55]. Therefore, the similar trends observed between these two metrics (time–space $\gamma$-deviation and ${\Phi }_{norm}$) can be interpreted as a strong indication that they provide measurements of the same physical quantity: the shared CO₂ source that simultaneously affects the majority of the sites.

Temporal variations in $\gamma$-deviation and changes in its spatial pattern can be interpreted together. When $\gamma$-deviations across all sites are discordant (windows 1–6), we observe smooth, large-scale spatial variations with lower deviations on the E and S flanks, the areas primarily influenced by the volcano’s structural system. Therefore, this discordance likely arises from a low-activity regime of the shared CO₂ source (Figure 7d and Figure 9), rather than significant permeability changes along the transport pathways to the surface, as also suggested by other authors [33].

When $\gamma$-deviations across all sites converge into a concordant signal, we observe both decreasing and increasing trends. During the decreasing $\gamma$-deviation phase (windows 8–11), the $\gamma$-value at the single site generally decreases; consequently, the degree distributions become relatively less “clustered”, leading to a higher expected maximum flux value (Figure 9). Simultaneously, the spatial pattern is again characterized by smooth, large-scale variations, but with higher absolute deviation values on the E and S flanks. In this setting, the increased input from the shared CO₂ source facilitates the upward migration of fluids and consequent gas discharge, resulting in anomalous CO₂ flux behavior marked by extreme peaks and significant variability. The spatial pattern thus reflects the inherent efficiency of the structural pathway system (i.e., large-scale permeability), with higher flux concentrations on the E and S flanks [54,56]. In the increasing phase (windows 12–15), $\gamma$-values at single sites generally increase, and degree distributions become more “clustered”. Simultaneously, spatial patterns exhibit high short-scale variability, and the number of earthquakes clearly increases (Figure 9). Under these conditions, the system likely remains susceptible to fluid release; however, during periods of low volcanic degassing, as inferred from CO₂ emission data, the overall effectiveness of this process appears tp be reduced (Figure 9).

In summary, regardless of the structural control of the efficiency of fluid emission pathways, the larger the negative $\gamma$-deviation from its long-term value, the more efficient the flux regime at a given site.

5. Conclusions

In this study, we employed visibility graph analysis (VGA) to investigate the structure and the spatio-temporal dynamics of a nine-year, high-resolution soil CO₂ flux dataset recorded by a standardized monitoring network at the Mt. Etna active volcano. Because the topology of a VG inherits the features of its associated time series, it represents a tool for extracting potential additional information. The identified power-law trends in the connectivity degree distribution highlight self-similar behavior, suggesting that the mechanism that controls surface emissions is a complex physical system influenced by many different factors acting at different scales. The occurrence of these scaling properties allow us to exclude purely periodic effects or random noise as the primary controllers of flux dynamics. The main conclusions of this work can be summarized as follows:

• The scaling exponent of the long-term power-law distribution (i.e., $\gamma$) is a characteristic feature of each monitoring site, representing a stable, long-term threshold. Notably, while temporal fluctuations are observed within two-year windows, these windowed values remain essentially lower than the decadal $\gamma$-value. We conclude that the long-term degassing behavior at each site is primarily controlled by local conditions such as the permeability structure at the site (i.e., site-specific $\gamma$-value), while temporal variations are driven by other factors, such as fluctuations in the sources or episodic changes in the efficiency of fluid transport pathways.
• Temporal changes in soil CO₂ are not best described by absolute $\gamma$-value, but rather by variation from the corresponding long-term reference value (i.e., $\gamma$-deviation). In this context, $\gamma$-deviation can be interpreted as a site-specific measure of flux efficiency, as lower $\gamma$-values indicate less “clustered” time series (Figure 9). We identify a period during which $\gamma$-deviations are uncorrelated, followed by a period where these values are consistent, showing both negative and positive trends, across the majority of monitoring sites. The emergence of common patters in $\gamma$-deviation lends support to the hypothesis of a shared mechanism, identified as variability in the volcanic CO₂ source, which is capable of exerting an effect on all flanks of the volcano simultaneously. These findings not only reinforce the validity of integrating multi-site data into a unified systemic degassing signal (i.e., ${\Phi }_{norm}$ by [12]), but also establish that coupling ${\Phi }_{norm}$ with the $\gamma$-value substantially refines the characterization of soil CO₂ dynamics.
• Even though spatial variations exhibit complex distributions around the volcano’s flanks, regular arrangements emerge, which contrast with periods characterized by short-scale, irregular variations. By synthesizing the spatial distributions of $\gamma$-deviation, volcanic activity, the pattern of the active lineaments, and seismicity, we propose a comprehensive interpretative model. This model is consistent with conclusions drawn by other authors utilizing the same dataset, indicating that the methodology proposed in this work provides coherent and robust results.

In conclusion, this study demonstrates that VGA, specifically through the investigation of degree distribution, is a robust and suitable method for characterizing the complexity of soil CO₂ emission and exploring its underlying dynamics. Future works could be focused at investigate different time scales, both longer and shorter, as well periods characterized by specific volcanic activity. Moreover, the proposed method could potentially be applied to other volcanoes where data availability supports the characterization of temporal and/or spatial variations.