Introduction

Statistical features of seismicity have been investigated since the early years of seismology, focusing on topics such as the spatial and temporal distributions of earthquakes or their size distribution. Various statistical approaches have been developed and applied to model earthquake occurrence, applying methods as scaling laws, universality, fractal dimension, and renormalization group, to study the physics of the earthquake. In recent years the visibility graph method has been applied to the study of earthquake catalogs1, and other geophysical observables2. This method involves transforming an earthquake time series into a graph object, which is then described by the full set of relationships between all nodes (i.e. earthquakes) that form the graph. The application of this method to various earthquake catalogs, has revealed that the number of connections k of each node with the others (i.e. graph edges) is linearly related to the earthquake magnitude M. Thus, for a given earthquake, the greater its magnitude, the higher its connectivity degree k within the graph representation. Furthermore, the slope of this relationship is, in turn, proportionally related to the slope of the magnitude-frequency relationship, commonly known as the b-value of the Gutenberg-Richter law. Notable, this last relation has only very recently emerged as a topic of significant interest3,4,5. Detailed explanation of the visibility graph (VG) method and its application in statistical seismology are provided in the section “Method”. The literature indicates that the b-value can be influenced by a variety of physical processes and may also serve as precursor of major seismic events. In this context, the application of the graph analysis in the study of complex time series such as the earthquake catalogs, can provide insights on these physical processes.

This work aims to provide natural examples of seismic catalog that could test the idea of the universal relationship between the b-value and the k–M slope derived from the visibility graph conversion. Previous studies have not explored multiple catalogs at the same time. Analyzing multiple catalogs using exactly the same method reduce the bias in determining studied parameters, particularly in calculating the b-value, allowing for a more robust verification of its relationship with the \(k-M\) slope. In particular, we analyze six different earthquake catalogs characterized by broad ranges in terms of number of events and maximum magnitude, and each of them associated with different source mechanisms. We also provide a collection of values from the literature and discuss about the universal character of the relationship between the b-value and the k–M slope.

Data

In this work, six different seismic catalogs have been considered, selected to enclose a variety of different characteristics of seismicity. Specifically, we provide two examples for each of the following types of seismicity: (1) tectonic seismicity; (2) volcano-related seismicity; and (3) induced seismicity.

Tectonic seismicity is the result of the activation of a fault (or a fault system) under the control of the tectonic stress acting in the region. In these contexts, seismic sequences often follow the activation of fault systems after the occurrence of main shocks. These sequences typically involve very large crustal volumes (\(10^4\)–\(10^5\) \(km^3\)) and may persist from months or even years. We provide examples of both reverse and normal fault seismic sequences, as these exhibit different characteristics. In particular, earthquake size distributions may differ between extensional and compressional tectonic regimes6,7.

Volcanic earthquakes are defined as those earthquakes that occur at, or near, volcanoes. Volcano seismic signals can span over a wide frequency band, and the volcano seismicity can result from a variety of source mechanisms that can be linked to different magmatic processes8. Volcanic earthquakes typically occur in regions where the stress is concentrated, which may not always be strictly related to the location of magma. The spatio-temporal propagation of volcanic earthquakes may follow magma migration. In particular, the two volcanic earthquakes catalogs considered in this study represent: (1) seismicity preceding and accompanying a volcanic eruption, and (2) the seismicity occurring in a caldera experiencing a phase of unrest.

Induced seismicity results from stress perturbations linked to various types of anthropogenic activities9. Most of the induced seismicity is triggered by fluid movements that perturb the state of stress of the surrounding rocks sufficiently to cause earthquakes10. The induced seismicity catalogs selected for this study regard: (1) the exploitation of a gas field, and (2) the seismicity associated with the loading effect from a water reservoir.

These six catalogs have different features, including the number of events, the involved crustal volume, the magnitude of completeness, and the maximum magnitude. In particular, the maximum magnitude spans across a very wide range: from 2.6 to 7.3 (Fig. 1). The catalogs are described in detail in the following paragraphs and summarized in Table 1.

Table 1 Details about the six earthquake catalogs selected for this study.
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Fig. 1
figure 1

Bar plots comparing \(M_{max}\) and \(M_c\) (left), and the earthquake number (right) across the six catalogs. Refer to the Table 1 for the abbreviations in the labels.

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Reverse faulting sequence (RF)

As an example of reverse faulting seismicity, we selected the 1999 Chi-Chi earthquake sequence (Taiwan). The island of Taiwan exhibits complex seismotectonic features, primarily dominated by ongoing collision11. This region is characterized by very high seismic release; over the past 25 years, more than one hundred M>6.0 earthquakes were recorded by a well developed seismic network12,13. The \(M_w\) 7.6, 1999 Chi-Chi earthquake is the largest earthquake occurred in Taiwan in recent time and produced a surface rupture approximately 100 km in length14. Seismological, geodetic, and structural data indicate the activation of multiple fault segments, with a main reverse component and a secondary left-lateral component15. The main shock activated a sequence of aftershocks that persisted for several months16. Seismicity is widespread over an area of about 7,000 \(km^2\), with the majority of earthquakes occurring at depths between 4 and 17 km.

Normal faulting sequence (NF)

As an example of normal fault seismicity, we selected the 2016–2017 Apennines Mountains sequence (Central Italy). The Apennines Mountains form a 1,000 km long, active extensional belt extending along the Italian peninsula. Various portions of this belt activated in historic time, producing earthquakes up to M 717. The 2016–2017 sequence occurred in the central portions of Apennines, where a 60 km long fault system activated on August, \(24^{th}\) with a first \(M_w\) 6.0 mainshock. The sequence evolved over the following months, with the largest event occurring on October \(30^{th}\), a \(M_w\) 6.5 earthquake18. Both main events produced surface ruptures, and in particular, the second one caused a complex surface faulting that spanned about 30 km19. Seismological and geodetic data from dense local networks reveal the activation of a complex normal fault system, primarily exibithing an extensional component20. The sequence of aftershocks lasted for a very long period, affecting an area of about 2,000 \(km^2\). Most of the events are located within the depth interval from 3 to 15 km21.

Volcanic eruption (VE)

Volcanic eruptions are often preceded and accompanied by seismic activity, as was also the case with the Tajogaite volcano eruption (La Palma, Canary Islands, Spain). Signs of unrest (i.e. seismic swarms, ground deformation, geochemical anomalies) began years before the onset of the eruption22 which started on September 19, 202123. The eruption occurred in the southern part of the island, in correspondence of the well-known Cumbre Vieja rift zone, where many other eruptions occurred in the historic time24. Earthquake locations allowed to track the rapid magma ascent few days before the onset of the eruption; in particular earthquake become progressively shallower and migrated reaching the eruption site23,25. The eruption ended suddenly on December 13, 2022, being active for 85 days and producing a total amount of more than \(200 \times 10^6\) m\(^3\) of volcanic products24.

The considered seismic catalog begins ten days before the eruption, when a sharp increase in the seismicity was observed26. It covers the entire duration of the eruption and extend beyond it, when the seismic activity stabilized, after a gradual decreasing. Seismicity is widespread over an area of aproximately 100 km\(^2\), with depths basically in the interval between 7 and 16 km.

Unresting caldera seismicity (UC)

The term caldera refers to a type of complex, meta-stable, and potentially dangerous volcano. Calderas can experience periods of anomalous behaviour, called unrest, which may eventually lead to eruptions. However, not all unrest periods culminate with an eruption27. Among the geophysical parameters, seismicity is considered one of the most significant indicators of caldera unrests27.

As a case study, we consider the long-term evolving unrest which is taking place at Campi Flegrei caldera (Italy)28,29. Previous episodes of non-eruptive unrest were observed during 1950–1952, 1969–1972, and 1982–1984, each marked by ground uplift up to \(\sim 1.8\) m and moderate, shallow seismicity30. The current phase of unrest began between 2000 and 2005, when a series of observation, including gas composition, seismicity, and ground deformation, indicated the upward movement of magmatic fluids30 which interacted with the shallow hydrothermal system31.

We analyze the seismicity at the Campi Flegrei caldera for a period of about 6.5 years, starting from September 2017 when an increase of the rate of occurrence of volcano-tectonic earthquake was observed32. The seismicity covers an area of about 50 km\(^2\), with the majority of the earthquakes occurring at depths shallower than 3 km.

Gas field induced seismicity (GF)

Injection or extraction of fluids from underground reservoirs can trigger local seismicity33. In the Groningen gas field (The Netherlands), the exploitation of a natural gas reservoir is recognized as the cause of small-magnitude, shallow-source, induced earthquakes. The first events occurred in 1991, long after the gas production had begun in 196334. A devoted seismic monitoring network was developed since 1995, and progressively extended, to closely monitor the evolution of seismicity35. Since 2003, the earthquake rate has increased, with the most significant event occurring in 2012 (\(M_L\) 3.6) which also caused damages36. The mechanism leading to seismicity is believed to be the ground compaction (or, more precisely, the compaction rate) of the reservoir consequently to gas extraction36,37.

The seismic catalog counts almost 1,500 events that occurred over a period of about 30 years (Table 1). The seismicity is widespread over an area of about 1,800 km\(^2\) and it is located at a depth of about 3 km, which coincides with the depth of the exploited sandstone reservoir (2,600–3,150 m depth)34,38.

Water reservoir induced seismicity (WR)

The filling of water reservoirs is proved to be a cause of induced seismicity39. The invoked mechanism is thought to involve perturbations of the stress regime, resulting from an increase of the vertical stress due to the load of the reservoir, coupled with a decrease of effective stress due to pore pressure40. These two effects can arise with different timing: while the former can be relatively rapid, the diffusion of water at depth is more gradual, potentially taking several seasons or filling cycles. Moreover the temporal distribution of seismicity may be also controlled by seasonal variations in water level, both in terms of frequency and amplitude41. As a case study, we analyzed the seismicity affecting the area surrounding the ”Pertusillo” artificial lake (Southern Apennines, Italy). This area is affected by low-magnitude, swarm-like seismicity, which has been correlated with seasonal water level variations in the reservoir42,43. In particular, the seismicity rate shows an annual periodicity corresponding to the loading/unloading reservoir cycle of the resevoir44. The proposed mechanism includes the effect of the loading/unloading of the water column which oscillations are in the order of 10–15 m45, as well as the effect of poro-elastic deformation and pore-pressure diffusion in the saturated rock, similarly controlled by the seasonal cycle46,47.

The catalog counts more than 5,000 events and was compiled after a 13-months-long seismic experiment, which involved the deployment of a dense monitoring network of 46 seismic stations. Most of the earthquakes are located within the depth interval between 2 and 5 km, and the seismicity extends over an area of approximately 60 km\(^2\).

Results

The six catalogs previously introduced were analyzed using the visibility graph method. Figures 2, 3, 4, 5, 6 and 7 display time series distributions of the earthquake magnitude (panel a) and the respective frequency-magnitude distributions (panel b), where the \(M_c\) and b values are reported. The \(M_c\) is used to truncate the catalogs and then perform the visibility graph analysis on the complete catalogs. Therefore, the connectivity degree (k) is calculated only for those events with magnitudes equal or greater than \(M_c\). For each of the six catalogs, we verified if \(M_c\) and b values from previous works were reported in the literature. The aim is to check whether the calculated values (Tables 1 and 2) are comparable to those previously published.

For the RF sequence, the b and Mc values (0.82 and 2.2, respectively) are consistent with those calculated by other authors for the same seismic sequence. In particular, Ref.48 report \(b=0.97\pm 0.1\), but for a period extending 5 years before the sequence started, which is much longer than the time interval considered in this study (Fig. 2), Ref.16 calculate \(b=0.84\) considering a longer sequence of aftershocks, while49 calculate \(b=0.804\pm 0.004\) for the period 1996–2003. All three studies agree on \(M_c=2\) for the period of the Chi–Chi earthquake sequence. For the NF sequence, the calculated Mc and b values are consistent with the ones obtained by other authors, considering that the spatial and temporal windows used for event selection may differ. In detail, the b and Mc values suggested by50,51,52 are similar, if not identical, to the values calculated in this study (1.02 and 1.5 respectively). Reference53 analyze not only the whole sequence but also cluster different seismogenic active sub-volumes, finding again comparable Mc and b values. The estimated b value for the VE catalog is 1.12, which is comparable to the one calculated by26 for the clusters they analyse during the eruptive period. The temporal trend of Mc shows wide fluctuation during the eruption, primarily controlled by the amplitude of background volcanic tremor. It increases from values lower than 1.0 in the pre- and post-eruptive periods to values greater than 2.0, as reported by to26, while according to54, Mc reaches values up to 2.8. These estimations are consistent with the calculated \(Mc=2.7\) for this catalog (Table 1), which is relatively high and implicate a clear reduction of the catalog later transformed with the visibility method (Fig. 4c). Reference32,55 analyzed the temporal and spatial distribution of b across the area of Campi Flegrei caldera (UC catalog) in the period 2014–2020 and 2005–2023, respectively. They found values in the range between 0.8–1.4 and 0.7–1.2, respectively. The calculated value of 0.78 in this work is, considering all differences, coherent with those reported in the literature. The \(Mc=1.0\) calculated for UC catalog aligns with the \(Mc=0.9\) reported by55. The GF catalog has recently been studied by other authors37,56,57, all of whom report similar results in terms of Mc values. However, the b value calculated in this work (0.86) is slightly lower compared to the values reported in the previously mentioned studies (0.95, 1.0, 0.94, respectively). The seismicity in the area of Pertusillo lake (WR catalog) was investigated by43,44,46. Their catalogs differ from the one considered in this work in terms of spatial and temporal extensions, which likely explains why their b (1.4 and 1.14) and Mc (1.2 and 1.1) values are not consistent with the ones calculated here (\(b=1.73\) and \(Mc=0.4\), respectively). Reference42 also investigated the identical WR catalog and reported \(Mc=0.4\) as well.

In summary, considering the variety of methods employed for the calculation of b and Mc which can lead to slightly different results even for the same analysed catalogs, we can confirm that the methods we applies provide reliable estimations for both the values of b and Mc. These estimations are therefore suitable to cut the catalogs and perform the visibility graph transformations. This, in turn, enables to calculate the relationship between the connectivity degree k and the earthquake magnitude M. The ultimate objective is to determine, for each of the six different catalogs, the pairs of values of b and k–M slope. None of these six catalogs has been previously analyzed with the visibility graph method. We stress that they have been selected to include the broadest collection in terms of source mechanism, maximum magnitude, and number of events.

Results of the graph transformation are presented as time series of the connectivity degree k (panel c in Figs. 2, 3, 4, 5 and 6 and 7) and scatter distributions between k and earthquake magnitude M (panel d in Figs. 2, 3, 4, 5 and 6 and 7). These results are displayed as heat maps to show also the frequency of occurrence of observations. The grid dimension is set to 0.1 M \(\times\) 2 k, and it is identical for the six case studies for easier comparisons. For each scatter-plot, the linear regression and the k–M slope are also reported. The slope value is primarily controlled by the contribution of small-magnitude earthquakes, which are more abundant than the larger ones. The larger magnitude earthquakes tend to exhibit a larger dispersion of the connectivity degree k, due to the sequential order of the observations.

To further remark the differences between the catalogs (Fig. 1), the six k–M distributions are displayed overlapped on the same plot (Fig. 8); each catalog occupies a distinct region within the k–M space, with some partial overlaps. This graphical comparison also illustrates the different arrangement of the linear regression lines: k–M slopes varies from 7.7 (UC catalog) to 17.86 (WR catalog) (Fig. 8 and Table 2). The slope of the k–M slope relation can be considered an indicator of earthquake productivity of a given seismic area3. Table 2 summarizes the values of b and k–M slope for the six catalogs, including their relative uncertainties, as well as their ratio.

Table 2 Results of the statistical analyses of the six earthquake catalogs; the standard errors (SE) for b and k–M slope are also reported.
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Fig. 2
figure 2

Analysis of the reverse faulting sequence (RF). (a) Magnitude-time distribution of the seismicity; the blue line represents the cumulative number of earthquake. (b) Frequency-magnitude distribution; grey dots refer to the cumulative distribution; \(M_c\) and b values are also reported. (c) Temporal distribution of the connectivity degree of the visibility graph analysis (k) relative to the complete catalog. (d) Scatter distribution of connectivity degree (k) versus earthquake magnitude (M) and linear regressions (black line); the value of the k–M slope is also reported.

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Fig. 3
figure 3

Analysis of the normal faulting sequence (NF). (a) Magnitude-time distribution of the seismicity; the blue line represents the cumulative number of earthquake. (b) Frequency-magnitude distribution; grey dots refer to the cumulative distribution; \(M_c\) and b values are also reported. (c) Temporal distribution of the connectivity degree of the visibility graph analysis (k) relative to the complete catalog. (d) Scatter distribution of connectivity degree (k) versus earthquake magnitude (M), along with the linear regressions (black line); the value of the k–M slope is also reported.

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Fig. 4
figure 4

Analysis of the volcanic eruption sequence (VE). (a) Magnitude-time distribution of the seismicity; the blue line represents the cumulative number of earthquakes. (b) Frequency-magnitude distribution; grey dots refer to the cumulative distribution; \(M_c\) and b values are also reported. (c) Temporal distribution of the connectivity degree of the visibility graph analysis (k) relative to the complete catalog. (d) Scatter distribution of connectivity degree (k) versus earthquake magnitude (M), along with the linear regressions (black line); the value of the k–M slope is also reported.

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Fig. 5
figure 5

Analysis of the unresting caldera catalog (UC). (a) Magnitude-time distribution of the seismicity; the blue line represents the cumulative number of earthquake. (b) Frequency-magnitude distribution; grey dots refer to the cumulative distribution; \(M_c\) and b values are also reported. (c) Temporal distribution of the connectivity degree of the visibility graph analysis (k) relative to the complete catalog. (d) Scatter distribution of connectivity degree (k) versus earthquake magnitude (M), along with the linear regressions (black line); the value of the k–M slope is also reported.

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Fig. 6
figure 6

Analysis of the gas field induced seismicity catalog (GF). (a) Magnitude-time distribution of the seismicity; the blue line represents the cumulative number of earthquake. (b) Frequency-magnitude distribution; grey dots refer to the cumulative distribution; \(M_c\) and b values are also reported. (c) Temporal distribution of the connectivity degree of the visibility graph analysis (k) relative to the complete catalog. (d) Scatter distribution of connectivity degree (k) versus earthquake magnitude (M), along with the linear regressions (black line); the value of the k–M slope is also reported.

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Fig. 7
figure 7

Analysis of the water reservoir induced seismicity catalog (WR). (a) Magnitude-time distribution of the seismicity; the blue line represents the cumulative number of earthquake. (b) Frequency-magnitude distribution; grey dots refer to the cumulative distribution; \(M_c\) and b values are also reported. (c) Temporal distribution of the connectivity degree of the visibility graph analysis (k) relative to the complete catalog. (d) Scatter distribution of connectivity degree (k) versus earthquake magnitude (M), along with the linear regressions (black line); the value of the k–M slope is also reported.

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Discussion

The relationship between the k–M slope and the b value has become a popular topic in recent years. Studies on both on natural and simulated dataset suggest that this relationship might hold global validity, similar to the well-established like the G–R relationship4,5,58,59,60,61,62,63. In particular, the VG method considers not only the magnitude of the events in a time series but also their sequential order. As can be observed on the slope of the curves of cumulative earthquake number (blue lines in Figs. 2a, 3, 4, 5, 6 and 7a), the slope of the curves is variable. For a given an earthquake catalog, a different temporal distribution of the events (i.e. a shuffled series) would results in a different graph network and therefore in a different k–M relation, even though the parameters of the G-R law would be unchanged. Thus, the statistical properties of a seismic catalog are analyzed more extensively, extending beyond the simple G-R relationship3.

The b-value is believed to be linearly proportional to the k–M slope. The first to recognize the possible correlation between b and the slope of the k–M relationship was3 who investigated five seismic regions in the Mexican subduction zone. Since then, other authors have provided further evidence supporting this assumption.

In general, the k–M slope and b may present a certain variability both spatially and temporally. The two parameters of this relationship describe the scaling properties of a seismic catalog and, as such, reflect possible changes in the underlying seismic processes. The variability of b in a given area can occur at different scales64, and it is typically associated to changes in the effective state of stress of the crust such as those triggered by the occurrence of large earthquakes. The long-standing observation that b sometimes decreases before large earthquakes, was analyzed employing the natural time domain (i.e. indexing the sequence of events), demonstrating that the fluctuation of the order parameter of seismicity exhibit an increase upon approaching the mainshock64,65. For an overview on the possible factors influencing spatial and temporal changes of the b value, refer to50,66. Similarly, the variability of the k–M slope has been associated with the occurrence of the largest shocks within a seismic sequence60,67. This suggests that significant seismic events can influence the overall scaling properties of the seismicity, and reflected, in turn, in changes of the k–M relationship.

In this work, we calculate k–M slope and b values averaged over the entire catalogs (i.e. given areas and given time spans as described in the Data section), without considering changes at shorter temporal or spatial scales. For the six catalog considered, b varies from 0.78 (UC) to 1.73 (WR), while k–M slope varies from 7.70 (UC) to 17.86 (WR). Their ratio is relatively constant, averaging around 10.4 (Table 2), except for the UC catalog. where it is slightly lower (9.9), and the VE catalog, where it is higher (12.5). Reference5 analyzing various types of synthetic catalogs, suggest this ratio might be related to the number of events composing the catalog. However, we do not find evidence of such relationship, despite the broad variation in the number of events across the catalogs (Table 1).

The six pairs of values, along with with their uncertainties, are plotted together, and a linear fit is performed using the least square regression method. The resulting fit shows a strong correlation, with a coefficient of determination of \(R^2=0.93\) (light blue dashed line in Fig. 9). Our results are also combined with data retrieved from the literature, concerning the analysis of both synthetic and natural seismic catalogs analyzed using the VG method (Table 3). Regression lines proposed in other studies are also displayed in Fig. 9, alongside a new fitting line that includes all the available data (blue dashed line in Fig. 9) which equations is: \(b=0.07415* k {-}M_{slope}+0.17617\). The regression lines are quite similar, with slight variations in their slopes. They are indistinguishable in the ranges of approximately \(0.7<b<1.2\) (Fig. 9).

Fig. 8
figure 8

Comparison of the scatter distributions of connectivity degree (k) and earthquake magnitude (M) for all the six catalogs analyzed in this work. The linear regression lines are displayed, and the corresponding k–M slope values are reported.

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Fig. 9
figure 9

The correlation between k–M slope and the b value: data from this work (colored dots) and data from other studies (grey dots) (values in Table 3). The regression lines obtained in this study are reported and compared with those retrieved from the literature.

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Conclusions

This study performs the visibility graph analysis on six earthquake catalogs with the main objective of examine the relationship between k–M slope and b, and compare the results with the existing literature. The conclusions of this paper can be summarized as follows:

  • The application of the visibility graph (VG) method to the study of seismicity is quite recent. We analyzed seismic catalogs spanning across a wide range of scales, in terms of number of events, maximum magnitude, earthquake frequency rate, and types of seismicity.

  • For each catalog we calculated the relationship between the connectivity degree k and the earthquake magnitude M. This relationship is proportional, with the slope showing a strong correlation with the b parameter of the G-R law. Our results are consistent with results derived from other studies that applied the VG method.

  • All the data combined together were used to propose a robust linear relation linking the k–M slope with b. Our study confirms that this scale relationship has a universal character and can serve as a valuable took for verifying b when studying the properties of seismic catalogs. Furthermore, since the VG results are influenced by the sequential order of the events in a catalog (which is consistent with the introduction of the concept of natural time by68 ), the proposed method offers a more comprehensive understanding with respect of the simple \(b-\)value, and represents its valuable integration.

  • We emphasize the capability of this method; further studies on additional earthquake catalogs could offer more insights into the relationship, while applying the method at shorter temporal and spatial scales could reveal potential and meaningful variations within a single catalog.

Table 3 Values of b and k–M slope retrieved from the literature. Type: N=natural catalog; S=synthetic catalog.
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Method

The method called “visibility graph” (VG) converts a generic time series into a graph. In this method, the relationship which defines the connections among two nodes is determined by the reciprocal “visibility” between the series’ values73. Using this representation, values of the time series can be described as nodes, and the connections between the various nodes represent their relationships. Following the notation from73, the visibility relationship between two generic values of a time series (\(t_a, y_a\)) and (\(t_b, y_b\)) holds, and therefore two nodes are reciprocally visible, if for all the intermediate observations (\(t_c, y_c\)) is verified the following relationship:

$$\begin{aligned} y_c<y_b+(y_A-y_b)\frac{t_b-t_c}{t_b-t_a} \end{aligned}$$
(1)

The connectivity degree k is defined as the number of links that each observation i holds with the rest of the network: \(k_i=\Sigma a_{ij}\)2. Each observation has at least two connections with its adjacent observations, so the minimum value for the connectivity degree assumes is \(k=2\) (except only for the two tips-observations). A large value of k indicates that the corresponding observation is linked to many others, and therefore it represents a hub within the network. The network resulting from the geometric transformation performed by the visibility graph method holds the properties possessed by the original time series1,73.

In this work, we use the general, undirected notation proposed by73, which was later applied by1, for the first description and the analysis of seismic sequences. This technique has proven to be highly informative for studying seismicity. Many earthquake catalogs (both synthetic and real instrumental catalogs) as well as seismic sequences have been investigated in recent years using this approach3,4,58,59,60,62,63,67,69,70,71,74. Various algorithms have been proposed for the construction of the visibility graph from a time series, which can be extremely time-consuming75,76, especially for large time series, because visibility graph transformations have quadratic time complexity. In this work, we adopt the code developed by77, implemented for the R platform.

This work, after the mapping of seismic catalogs as visibility graphs, aims to investigate the relationship between the connectivity degree k and the earthquake magnitude M. The simple k–Magnitude slope, calculated as the slope of the line fitting the k–M plot provides insights into earthquake productivity and appears to be strongly related to the parameters of the Gutenberg-Richter law, as suggested by several studies3,4,5,61. In particular, the correlation between the k–M slope and the slope of the Gutenberg-Richter relationship, b, has been examined for each of the proposed case studies.

The Gutenberg–Richter (G–R) law describes the earthquake frequency magnitude distribution as a power-law function:

$$\begin{aligned} N(\ge M)=10^{a-bM} \end{aligned}$$
(2)

where \(N(\ge M)\) is the number of earthquakes larger, or equal, than magnitude M; a and b are two constants. The estimation of the minimum magnitude of completeness (\(M_c\)) represents a preliminary step for the assessment of the constants in the Eq. (2). Among the various methods for \(M_c\) calculation (c.f.66,78), we adopted the one proposed by79. This method is based on the calculation of goodness of fit between the cumulative number of observed earthquakes and those predicted by the G-R model. Specifically, \(M_c\) is defined as the lowest magnitude at which a fixed threshold of the normalized absolute difference is first met. This value was fixed at 0.95, and lowered to 0.90 if the threshold was not reached. The goodness of fit method fixes the systematic underestimation of \(M_c\) resulting from other methods. It is considered a robust technique providing reliable estimations78 and it is widely adopted in the study of synthetic and natural catalogs. The complete catalog (\(M>M_c\)) is used for the estimation of the b-value of the G–R model (2) and to perform the visibility graph transformation described earlier. In particular, b was estimated using the maximum likelihood estimation approach proposed by80.