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We investigate the dynamical behavior of aftershocks in earthquake networks, and the earthquake network calculated from a time series is constructed by contemplating cell resolution and temporal causality. We attempt to connect an earthquake network using relationship between one main earthquake and its aftershocks from seismic data of California. We mainly examine some topological properties of the earthquake such as the degree distribution, the characteristic path length, the clustering coefficient, and the global efficiency. Our result cannot presently determine the universal scaling exponents in statistical quantities, but the topological properties may be inferred to advance and improve by implementing the method and its technique of networks. Particularly, it may be dealt with a network issue of convenience and of importance in the case how large networks construct in time to proceed on earthquake systems.

Scientists have treated emerging problems in order to cover the basic concept and the important principle or clearly describe the scientific phenomenon. In several drifts of the complex system, they have pursued and settled many scientific phenomena [

Particularly, the remarkable potential to calculate and analyze the dynamical behavior of complex systems has gradually been an increasing trend in new fields of research in the natural, engineering, medical, social sciences over the last two decades. In the network theory, small-world and scale-free network models [

Abe and Suzuki have analyzed the spatio-temporal properties of seismicity from the viewpoint of the Tsallis entropy under appropriate constraints [

The aftershocks represent many smaller earthquakes that it occurs after a large earthquake. Our method is to construct the network between one main earthquake and its aftershocks, different from the Abe and Suzuki method [

In this section, we mainly consider the theoretical background of the several global network metrics. First of all, there are some important ingredients of complex networks, and these are different from ingredients of random networks. The mean degree 〈 k 〉 is defined as

〈 k 〉 = 1 N ∑ m = 1 N k m , (1)

where k_{m} is the degree of a node m, and N is the number of total degrees. The random network is that it constructed by randomizing the earthquake network under fixed links and nodes. From our method for constructing network, a newly creating node of the growing earthquake network is linked with preferential attachment probability. A network generated with this rule characterized by power-law connectivity distribution. The degree distribution that is the probability distribution function as a function of degrees k is represented in terms of

P ( k ) ~ k − γ , (2)

where γ is the degree exponent. Theoretically, the scaling exponent γ of the scale free network [_{i} for a node i is defined as the fraction of links that exists among its nearest-neighbor nodes to the maximum number of possible links among them. The clustering coefficient of a node with degrees k follows the scaling law

C ( k ) ~ k − β , (3)

where the scaling exponent β is a hierarchy coefficient. The network with the mentioned feature of Equations (2) and (3) is called the scale free network.

The characteristic path length is defined the statistical quantity that the sum of all the shortest path length between two nodes is divided by all links of nodes. We introduce the characteristic path length L given as

L = 2 N ( N − 1 ) ∑ m = 1 N − 1 ∑ m = i + 1 N L m n , (4)

where L_{mn} is the shortest path length between nodes m and n [_{m} for a node m is defined as the fraction of links that exist among its nearest neighbor nodes to the maximum number of possible links among them. The global clustering coefficient C_{g} is defined as the transitivity ratio that is the fraction of the closed triangles over the whole triangles.

The global efficiency is defined as the average of inverses of the global distance for all nodes [

E g = 2 N ( N − 1 ) ∑ i = 1 N − 1 ∑ j = i + 1 N 1 L i j . (5)

When we construct a network for the neighbors of node m, the local efficiency E_{lc} can be calculated to be the average value of the efficiencies of node as E l c = 1 / N ∑ m = 1 N E m , where E_{m} is the subgraph efficiency of the neighbors of the m-th node.

From Equations (1)-(5), we mainly calculate and analyze the topological measures in the complex network in next section. These methods and techniques are able to be treated in the study of the diverse earthquake models. Further result of other phase metrics will appear in a future publication.

The aftershocks have a trend occurring during definitive time intervals after one main earthquake occurs. If an aftershock is larger than the main shock, then the aftershock is considered as the role of the main shock and the previous main shock is designated as a foreshock. Aftershocks are formed as the crust around the displaced fault plane adjusts to the effects of the main shock. Hence, we suggest our method to construct complex network using the property of aftershocks. An earthquake network is constructed by segmenting the whole region into three-dimensional cubic cells and making a link between consecutive events. Each cubic cell is regarded as node of a network, and the network constructed in that manner is basically directed, but we transform it into an undirected one because we focus on the topology of the network.

Our network is introduced the method constructing in aftershock, while Abe-Suzuki network constructs in earthquakes for all the time series data consecutive earthquake events. Our method constructing the network is compared to that of Abe-Suzuki. Our procedure is as follows: 1) We segment the whole region into cells, each of which has the same size. 2) If the magnitude of second earthquake is smaller than the first one, we link two earthquakes. 3) If the magnitude of third earthquake is smaller than the second one, we also make a link between first earthquake and third one. In this manner, smaller earthquakes as the role of aftershock are linked with a main shock. Otherwise, if the magnitude of third earthquake is bigger than that of second earthquake, the third one becomes a main shock. 4) If two consecutive events belong to the same cubic cell, then their link is disregarded. 5) The number of links, if two directed links form between two cubic cells. 6) Hence, we regard the links made by all events belonging to the cubic cell with others in another cubic cell as links of a network, by considering each cubic cell as a node. Next, the method of Abe-Suzuki is as follows: 1) We segment the whole region into N-by-N-by-L cubic cells, each of which has the same size. 2) We link two earthquakes occurring consecutively. 3) If two consecutive events belong to the same cubic cell, their link is disregarded. 4) If two directed links form between two cubic cells, the number of links is counted as one. 5) By considering each cubic cell as a node, we regard the links made by all events belonging to the cubic cell with others in another cubic cell as links of a network.

We construct and analyze seismic data collected from California of USA. The data sources are USGS [^{3} to ((1˚/20) × (1˚/20) × 10 km^{3}.

In

cell width | N | k | L | C_{l} | E_{g} | ||||||
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OAN | ASN | OAN | ASN | OAN | ASN | OAN | ASN | OAN | ASN | ||

1˚/10 | 2,616 | 32863 | 50465 | 25.12 | 38.58 | 2.66 | 2.49 | 0.41 | 0.59 | 0.37 | 0.71 |

1˚/11 | 2,963 | 34501 | 53604 | 23.29 | 36.18 | 2.70 | 2.52 | 0.38 | 0.57 | 0.36 | 0.69 |

1˚/12 | 3,309 | 36735 | 57832 | 22.20 | 34.95 | 2.76 | 2.56 | 0.35 | 0.53 | 0.35 | 0.66 |

1˚/13 | 3,660 | 38557 | 61386 | 21.07 | 33.54 | 2.79 | 2.59 | 0.33 | 0.51 | 0.34 | 0.64 |

1˚/14 | 4,041 | 40156 | 64690 | 19.87 | 32.02 | 2.84 | 2.63 | 0.30 | 0.49 | 0.33 | 0.62 |

˚/15 | 4,350 | 41154 | 67578 | 19.66 | 31.07 | 2.86 | 2.66 | 0.29 | 0.47 | 0.33 | 0.60 |

1˚/16 | 4,691 | 43245 | 70811 | 18.44 | 30.19 | 2.91 | 2.68 | 0.27 | 0.45 | 0.33 | 0.57 |

1˚/17 | 5,063 | 44186 | 72824 | 17.45 | 28.77 | 2.94 | 2.70 | 0.25 | 0.44 | 0.32 | 0.56 |

1˚/18 | 5,408 | 45764 | 75902 | 16.92 | 28.07 | 2.99 | 2.74 | 0.23 | 0.42 | 0.31 | 0.54 |

1˚/19 | 5,783 | 46499 | 77580 | 16.08 | 26.83 | 3.01 | 2.76 | 0.21 | 0.39 | 0.31 | 0.51 |

1˚/20 | 6,127 | 47955 | 80398 | 15.65 | 26.24 | 3.04 | 2.79 | 0.20 | 0.38 | 0.30 | 0.49 |

plots the mean degree as a function of the cell width in our aftershock network and the ASN. We show that the mean degree of the earthquake network becomes relatively smaller than that of random network as the cell width approaches to small values.

In

We find that the scaling exponent in the degree distribution [^{3} and (1˚/10) × (1˚/10) × 10 km^{3} in the OAS (the ASN), respectively.

In this paper, we have calculated and analyzed the fundamental network metrics such as the mean degree, degree distribution, characteristic path length, average clustering coefficient, and global efficiency from seismic data of California. We have compared the OAS and the ASN to their random networks. Through other works, the seismicity has taken the features of complex network for the average clustering coefficient. Min et al. have found the values of average clustering coefficient between 0.85 and 0.90 in the cell widths between 60 km and 100 km [

We have novelly treated the network of aftershocks in the field of complex networks. In the future, we think that this method will extend and measure its topological metrics to other earthquake networks how to show a universal property in networks of other regions. We conclude from the results of the calculation that our aftershock network is a scale free network and has the hierarchical structure. Particularly, our method is able to perceive one way to construct the aftershock network, significantly different from the constructing method of the ASN.

The results of this investigation may provide useful and effective information for prediction of scaling behaviors under the impacts of earthquake network changes in other earthquake regions. Our findings support that a recent network approach to earthquake analysis is very useful and reliable in three-dimensional cells. In order to argue our suggestion, a further work about the calculation of network constructions in other nations is needed. It is anticipated that the formalism of our analysis can be extended to both discrimination and the characterization of various aftershocks and earthquakes.

This work was supported by a Research Grant of Pukyong National University (2017).

Baek, W.-H., Kim, K., Chang, K.-H., Seo, S.-K., Lee, J.-H. and Lee, D.-I. (2018) On the Dynamical Analysis in Aftershock Networks. Open Journal of Earthquake Research, 7, 28-38. https://doi.org/10.4236/ojer.2018.71002