Network theory, which is one of the largest branches of mathematics, was first proposed in 1736 with the publication of an article by Leonard Euler, which is closely related to algebra and matrix theory. In mathematician terms, a network is a set of interconnected points and lines. In fact, a network is a mathematical model for a discrete set whose members are interconnected in some way. Network members are defined as a set of nodes and the interaction between members by a set of edges. The members of a set in society can be human beings and the edge between them can be a friendship, and in a molecular set, atoms as members of the set and the edge between them can be chemical bonds (Babelian 2007)

Recent advances in mathematics, especially in its applications, have led to the dramatic expansion of network theory, so that network theory is now a very useful tool for research in various fields such as coding theory, statistical operations research, electrical networks, Computer science, chemistry, biology, social sciences, design of electrical circuits, geometric modification of streets to solve traffic problems, in urban planning and civil engineering and other fields. In some networks, such as the network of earthquakes, due to various unknown parameters of earthquakes, it is more complex to understand the behaviors of complex networks, methods derived from statistical mechanics and define statistical quantities such as degree distribution function, cluster coefficient and etc, useful to provide information about the general state of the network (Boccaletti et al. 2006).

To examine these qualities an appropriate network must be set up. To build any network (earthquake network), firstly, must know the nodes and edges that are the main parameters of each network. There are two main approaches to constructing an earthquake network, which are as follow:

In the first approach, the geographical area under study is divided into boxes of the same size. If earthquakes occur in these boxes, the earthquakes will act as a node. Edges are also defined in such a way that consecutive seismic events are connected by edges in terms of time. This network model is related to the model presented by Abe and Suzuki (Abe and Suzuki 2009, 2007, 2005, 2004; Abe et al. 2011). They showed that the network of earthquakes is scaleless and has the structure of a small world. Also, the effect of box size on the characteristics of Iran seismic networks has also been studied and network characteristics such as degree distribution function, cluster coefficient and characteristic length have been obtained and the study of seismic active points in Iran has been obtained (Darooneh et al. 2014).

In the second approach, seismic events are considered as nodes and two nodes are connected when they are interconnected by the relationships that are true for an earthquake event. Telesca and Lovallo used phenomenal algorithms to make connections between earthquakes, which This algorithm was proposed as a tool for time series analysis (Lacasa et al. 2008).

Baiesi et al. (2004) to express the relationship between earthquakes, presented a quantitative relationship that included the quantities of time interval and spatial distance between two earthquakes, as well as the magnitude of the first earthquake.

Davidson et al. (2003) used the recursive seismic method to construct the earthquake network, which is the method used in this study to construct the earthquake network. This method uses the concept of waiting time to return the occurrence of earthquakes in space and time. Two earthquakes, *A* and *B*, are considered to have occurred in time A before earthquake *B*. Earthquake *B* is a return from Earthquake *A* provided that no other earthquakes have occurred spatially closer to Earthquake *B* after Earthquake *A*.

In fact, \(\stackrel{-}{AB}\)should be the lowest value in terms of distance, in the time interval [\({t}_{A}-{t}_{B}\)] that time between the event *A* to *B*. Each return is denoted by the interval \(L=\stackrel{-}{AB}\) and the time interval \({t}_{A}-{t}_{B}\). In other words, the space window is the center of the first event, and each return to this event is at the closest spatial distance to the other events.

The distribution of *L* intervals between return events for different thresholds of magnitude m is calculated using the centrifugal coordinates of the earthquake catalog and using Eq. 1 presented by Baiesi et al. (2004).

$${L}_{ij}={R}_{0} Arc cos[{sin}\left({\theta }_{i}\right){sin}\left({\theta }_{j}\right)+{cos}\left({\theta }_{i}\right){cos}\left({\theta }_{j}\right)cos({\phi }_{i}-{\phi }_{j})$$

1

In relation to index *i* and *j*, respectively, represent the first and second earthquakes, and *θ* and *φ* are the longitude and latitude of the epicenter of the earthquake, respectively, and *R**0*, the radius of the earth.

Earthquake networks are constructed in such a way that seismic events represent nodes, and to create an edge between nodes in the earthquake catalog, define the temporal and spatial return of earthquakes, and each return from the next event to the previous event is represented by an edge between a pair of nodes in the temporal and spatial series of earthquakes.

Separate events have different input and output edges that indicate their relationship to other events, and thus nodes have different degrees. The output edges of each node determine the structure of the returns in its neighbors. In other words, the output edges of each node are the input edges of its neighbors.