# Similar Seismic Activities Analysis by Using Complex Networks Approach

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## Abstract

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## 1. Introduction

## 2. Data and Network Construction

- a
- Southern California Seismic Network (SCSN). California belongs to the Pacific Rim seismic zone, where earthquakes occur frequently. The center provides seismic data for the California region for more than 80 years from 1932 to the present, with more than 500,000 public seismic records. With the continuous improvement of data acquisition and measurement methods, SCSN has more than 400 detection stations in different places.
- b
- Japan University Network Earthquake Catalog (JUNEC). Japan is located at the junction of the Asia–Europe plate and the Pacific plate, belongs to the Pacific Rim seismic zone. The crustal movement is active, so Japan is a country with frequent earthquakes. JUNEC provides seismic data for the geographical region of Japan from 1985 to 1998.
- c
- Advanced National Seismic System (ANSS). ANSS includes a national Backbone network, the National Earthquake Information Center (NEIC), the National Strong Motion Project, and 15 regional seismic networks operated by USGS and its cooperative organizations. This system provides a huge amount of worldwide seismic data, close to 3 million records.

## 3. Method

#### 3.1. Basic Network Properties

- a
- Degree distribution is the probability distribution of the degrees k over the whole network. Scale-free property of earthquake networks with $P\left(k\right)\sim {k}^{-\lambda}$ has been studied by many studies [9,14,16]. The exponent $\lambda $ is proved to approach a fixed value, remaining invariant, as the cell size becomes larger than a certain value [35].
- b
- Strength of the ${i}^{th}$ node is defined as ${s}_{i}={\sum}_{j}{w}_{ij}$, where ${w}_{ij}$ is the weight of a connection between two different nodes as mentioned before. Here we use average strength $S=\left({\sum}_{i=1}^{n}{s}_{i}\right)/n$ to describe the average influencing degree among nodes in the network. It has been observed that the probability distributions of nodal strengths follow power law decay forms. Chakraborty et al. used strengths to analyze the rich-club feature in the weighted earthquake network. However, this feature is not found in unweighted network [36].
- c
- Clustering coefficient is the ratio of existing edges connecting a node’s neighbors to each other to the maximum possible number of such edges. The weighted clustering coefficient [37] is defined as ${C}^{w}\left(i\right)=\left(\frac{1}{{s}_{i}({k}_{i}-1)}\right){\sum}_{j,h}(({w}_{ij}+{w}_{ih})/2){a}_{ij}{a}_{ih}{a}_{jh}$ where ${a}_{ij}$(s) are the elements of the adjacency matrix. The average clustering coefficient is defined by $C=\left({\sum}_{i=1}^{n}{C}^{w}\left(i\right)\right)/n$. The clustering coefficient has been investigated in many studies of earthquake network [10,14,19]. It implies the degree of clustering of the network. Abe et al. found that before main shocks, the values of the clustering coefficient is stable but suddenly jump up when the main shocks occur, and then slowly decay following a power law to become stable again [19].
- d
- Average path length l is defined as the average number of steps along the shortest paths for all possible pairs of network nodes. Let $l({v}_{1},{v}_{2})$, where ${v}_{1},{v}_{2}\in V$, denotes the shortest distance between ${v}_{1}$ and ${v}_{2}$. Then, the average path length L is $L=\frac{1}{n(n-1)}{\sum}_{i\ne j}l({v}_{1},{v}_{2})$. Many researchers have found that earthquake networks have a very short average path length leading to the concept of a small world where one node is connected to another through a very short path [10,14]. Short average path length in earthquake network may implies quick transfer of energy [10].
- e
- Coreness. K-core decomposition analysis is used to define the location of the node. This process assigns an integer index or coreness ${k}_{s}$ to each node, representing its position according to successive layers (k shells) in the network. The coreness index ${k}_{s}$ is a very robust metric, and in the case of incomplete information, the node is not significantly influenced. The max value of coreness K represents the number of layers of the network. He et al. [33] found different values of K in different regions. They also found that the highest core (or layer) is generally related to a deep hierarchy of the earthquake network structure.
- f
- Entropy is a metric of uncertainty for a network. The weighted entropy is defined as $H=-{\sum}_{i=1}^{n}p\left({s}_{i}\right)logp\left({s}_{i}\right)$, where $p\left({s}_{i}\right)$ is the probability distribution of ${s}_{i}$. The network entropy is an important indicator describing the heterogeneity of network. Lin et al. [21] focused on the dynamic evolution of the structure entropy of the earthquake network and captured the changes of network structure entropy before and after the main shocks.

#### 3.2. Network Similarity

## 4. Results

#### 4.1. Hierarchical Clustering Result

#### 4.2. Wavelet Analysis

#### 4.3. Period Analysis of Seismicity

## 5. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The topological structure of earthquake network constructed from the catalog of the Southern California Seismic Network (SCSN) covering the region 30${}^{\circ}$ N–39${}^{\circ}$ N latitude and 111${}^{\circ}$ W–124${}^{\circ}$ W longitude in the period between 1 January 1992 and 31 December 2011. The total number of events is 395274.

**Figure 2.**Dendrogram representing the hierarchical clustering of the different earthquake networks in California over the past decades.

**Figure 3.**Wavelet power spectrum of time series for (

**a**) the world, (

**b**) California and (

**c**) Japan. The red region represents the wavelet power with a bigger value while the blue region represents the wavelet power with a smaller value. We can see the specific time span for certain time scale.

**Figure 4.**Average wavelet variance of time series for (

**a**) the world, (

**b**) California and (

**c**) Japan. The right part of the dashed line indicate confidence level of 90%.

**Figure 5.**Wavelet power spectrum of time series for California with $w=5$. The red region represents to the wavelet power with bigger value while blue region represents the wavelet power with smaller value. We can see the specific time span for certain time scale.

**Figure 6.**Average wavelet variance of time series for California with $w=5$. The right part of the dashed line indicate confidence level of 90%. Its corresponding period is said to be with high confidence.

**Table 1.**The information of different earthquake catalogs. The abbreviation for the name of earthquake catalogs, the time spans for the collecting data, and the URL of the corresponding catalogs are shown in this table.

Name | Period | Site URL |
---|---|---|

SCSN | 1932 to the present | scedc.caltech.edu/ |

JUNEC | 1985–1998 | kea.eri.u-tokyo.ac.jp |

ANSS | 1863 to the present | www.quake.geo.berkeley.edu/anss/ |

Name | Symbol | Implication |
---|---|---|

Exponent of power law | $\lambda $ | The heterogeneity of degree distribution |

Strength | S | Influencing degree among nodes |

Clustering coefficient | C | The degree of clustering of earthquake network |

Average path length | L | Efficiency of transfering energy |

Max coreness | K | Hierachy of earthquake network |

Entropy | H | The heterogeneity of nodal strength |

**Table 3.**The time, magnitude, latitude and longitude of the great shocks in California in the year 1992, 1999 and 2010.

Time | Magnitude | Latitude | Longitude |
---|---|---|---|

1992/06/28 | 7.3 | 34.200 | −116.437 |

1999/10/16 | 7.1 | 34.603 | −116.265 |

2010/04/04 | 7.2 | 32.286 | −115.295 |

**Table 4.**The summary of possible periods in different regions. The mark “√” represents the possible period with 90% confidence rate.

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**MDPI and ACS Style**

He, X.; Wang, L.; Liu, Z.; Liu, Y.
Similar Seismic Activities Analysis by Using Complex Networks Approach. *Symmetry* **2020**, *12*, 778.
https://doi.org/10.3390/sym12050778

**AMA Style**

He X, Wang L, Liu Z, Liu Y.
Similar Seismic Activities Analysis by Using Complex Networks Approach. *Symmetry*. 2020; 12(5):778.
https://doi.org/10.3390/sym12050778

**Chicago/Turabian Style**

He, Xuan, Luyang Wang, Zheng Liu, and Yiwen Liu.
2020. "Similar Seismic Activities Analysis by Using Complex Networks Approach" *Symmetry* 12, no. 5: 778.
https://doi.org/10.3390/sym12050778