Universalities of earthquake-network characteristics

The scale-free and small-world properties are studied in detail for the complex earthquake networks constructed from the seismic data sets taken from California (USA), Japan, Iran and Chile. It is found that, in all these geographical regions, both the exponent  of the power-law connectivity distribution and the clustering coefficient C take the universal invariant values ≈1 and C≈0.85, respectively, as the cell size, which is the scale of coarse graining needed for construction of network, becomes larger than a certain value. A possible physical interpretation is given to the emergence of such remarkable invariance.

Dynamics governing seismicity is yet largely unknown, and accordingly the relevant research areas are necessarily empirical, owing their developments to case studies. This seems to be true even in the processes of establishing two celebrated classical laws known as the Omori law for temporal pattern of aftershocks and the Gutenberg-Richter law for the relation between frequency and magnitude.
In the absence of detailed knowledge about underlying dynamics, it is of central importance to clarify the pattern of correlations in order to extract information on features of the dynamics as a time series. Here, our primary interest is in the event-event correlations between successive earthquakes. As reported in the literature, both the spatial distance [1] and time interval [2,3] between two successive events strongly deviate from Poissonian statistics, supporting rationality of the concept of the event-event correlations. Accordingly, the following working hypothesis may be framed: two successive events are indivisibly correlated at least at the statistical level, no matter how distant they are. In this context, it is also noted that an earthquake can in fact trigger the next one which is more than 1000 km away [4]. Thus, the correlation length can indeed be divergently large, indicating a similarity to critical phenomena.
Although seismicity is generically assumed to be a complex phenomenon, it is actually a nontrivial issue to characterize its complexity in a unified way. The network approach offers a powerful tool to analyzing kinematical and dynamical structures of complex systems in a holistic manner (see [5] as a general reading). In a series of recent works [6][7][8][9], we have introduced and developed the concept of earthquake network in order to reveal the complexity of seismicity. In this approach, the event-event correlations are replaced by edges connecting vertices (for the details of constructing an earthquake network, see Section 1). It was found that the networks constructed in some geographical regions are complex ones, which are scale-free [10] and small-world [11]. Close studies have shown [12] that they are hierarchically organized [13] and possess the property of assortative mixing [14]. It has also turned out [15] that time evolution of network characterizes main shock in a peculiar manner. In addition, it has been found [16] that the scaling relation holds for the exponents of the power-law connec-tivity distribution and network spectral density.
Thus, the network approach may offer a novel possibility to deeper physical understandings of seismicity. Since the approach is inherently empirical in the sense that it is based on analyses of real data, it is essential to examine how universal the properties observed are.
In this paper, first we summarize the recent discovery [17] about the universal behaviors of the characteristics of earthquake networks found through analysis of the data sets taken from California, Japan and Iran, as well as Chilean one included here anew. The earthquake networks in these four geographical regions are found to be of the scale-free and small-world type. We carefully study the dependencies of the exponent  of the power-law connectivity distribution and the clustering coefficient C on a single parameter that is the cell size of division needed for constructing a network (see Section 1). We show a remarkable fact that both  and C come to take the universal invariant values: respectively, as the cell size becomes larger than a certain value. Then, we shall present a possible physical explanation about the emergence of such invariance.

Construction of earthquake network
The method of constructing an earthquake network proposed in [6] is as follows. A geographical region under consideration is divided into cubic cells. A cell is regarded as a vertex of a network if earthquakes with any values of magnitude (above a certain detection threshold) occurred therein. Two successive events define an edge between two vertices, but if they occur in the same cell, then a tadpole (i.e. a self-loop) is attached to that vertex. These edges and tadpoles represent the event-event correlations (recall the working hypothesis mentioned earlier).
This simple procedure enables one to map a given seismic time series to a growing stochastic network, which is an earthquake network that we have been referring to.
Several comments on the construction are in order. Firstly, it contains a single parameter: the cell size L, which is the scale of coarse graining. That is, all earthquakes occurred in a given cell are identified and represented by the relevant vertex. Once a set of cells is fixed, then an earthquake network is unambiguously defined. Secondly, an earthquake network is a directed one in its nature. However, directedness does not bring any difficulties to statistical analysis of connectivity (or degree, i.e. the number of edges attached to the vertex under consideration) needed for examining the scale-free property, since by construction the in-degree and out-degree are identical for each vertex ex-cept the initial and final vertices in analysis, so they need not be distinguished. Therefore, vertices except the initial and final ones have the even-number values of connectivity. Thirdly, a full directed earthquake network should be reduced to a simple undirected network, when its small-world property is examined (see Figure 1). That is, tadpoles are removed and each multiple edge is replaced by a single edge.
To practically set up cells and identify a cell for each earthquake, here we employ the following procedure. Let  0 and  max be the minimal and maximal values of latitude covered by a data set, respectively. Similarly, let   and  max be the minimal and maximal values of longitude. Define  av as the sum of the values of latitude of all the events divided by the number of events contained in the analysis. The hypocenter of the ith event is denoted by respectively. The north-south distance between ( 0 ,  0 ) and  Closing this section, we wish to emphasize the following point ascertained by our examinations. Although numerical values of characteristics of a network generically depend on how cells are set up, gross properties of a network do not change. We have constructed the earthquake networks form these data sets following the procedure explained in Section 1 and analyzed their properties. In particular, we focus our attention to its scale-free and small-world properties.
First, let us look at scale-freeness of earthquake network. In Figure 2, we present plots of the connectivity distribution (or, degree distribution) P(k), which gives the probability of finding a vertex with k edges, for those four geographical regions. As can be seen there, it decays as a power law: showing that the full earthquake networks are in fact scale-free. The scale-free nature indicates that there are quite a few hubs with large values of connectivity, which make the network heterogeneous. We have taken a close look at the vertices of some main shocks in each data set and found that they correspond to hubs. An important point is that, aftershocks following each main shock tend to return to the locus of the main shock. This empirical fact explains why main shocks make their vertices hubs of a network.
Second, we discuss small-worldness of the reduced simple earthquake. Here, the following two characteristic quantities are of primary relevance. One is the clustering coefficient and the other is the average path length [11], both of which are explained below.
The clustering coefficient C of a simple network with N vertices is defined by 1 1 .
c i appearing on the right-hand side is given by c i  (number of edges of the ith vertex and its neighbors)/[k i (k i 1)/2] with k i being the connectivity of the ith vertex. Equivalently, it is calculated also as follows. Let A be the symmetric adjacency matrix of the reduced simple network. Its element (A) ij is 1 (0), if the ith and jth vertices are linked (unlinked). Then, c i is also written as follows: c i tells about the tendency that two neighboring vertices of the ith vertex are linked together. By definition, C takes a value between 0 and 1. An important point is that C of a small-world network is much larger than that of the corresponding random network (i.e. classical Erdös-Rényi random graph) given by where <k> is the average value of connectivity of the random network. That is, The other important characteristic quantity is the average path length l , which quantifies that, for a pair of vertices, how many steps the shortest path linking them contains. A small-world network has a small value for it.
In Table 1, we present the results about these two characteristic quantities. It shows that the reduced simple earthquake networks in California, Japan, Iran and Chile are in fact of the small-world type.

Disappearance of cell-size dependence and universalities of network characteristics
Quantities characterizing a network are usually dimensionless, as the connectivity distribution, clustering coefficient and average path length are. On the other hand, in the case of earthquake network, their numerical values depend on the cell size L, in general. This issue is discussed in this section.
First of all, it is reasonable to make the cell size dimensionless. We rescale L using the dimension of a geographical region under consideration. Let L LAT , L LON and L DEP be the dimensions of the whole region in the directions of latitude, The cell size employed is 20 km × 20 km × 20 km. The total numbers of vertices are as in Figure 2. For comparison, the values of corresponding random networks, C random , are also presented. longitude and depth, respectively. From these, we construct the following dimensionless quantity: The The above result may be interpreted as follows. According to the method of network construction, there are two competitive factors. As the cell size increases, vertices get merged, yielding vertices with larger values of connectivity, while at the same time geographically neighboring vertices are absorbed each other, loosing the roles of hubs. The former decreases the value of , whereas the latter increases it. Disappearance of the cell-size dependence of  may be due to the balance between these two competitive effects.
Finally, let us discuss the cell-size dependence of the clustering coefficient C. In Figure 4, we present its plots with respect to l 3 for the earthquake networks in those four geographical regions. Again, quite remarkably, C also takes the universal invariant value C≈0.85 as the cell size becomes larger than a certain value.
This result may be explained as follows. As the cell size increases, the number of vertices decreases, and the network approaches a complete graph (i.e. a fully linked network) having the maximum value of the clustering coefficient (C=1), but at the same time cells swallow triangular loops (recall the A 3 -nature of C in eq. (5)) attached to them as they become larger. The former effect increases the value of C, whereas the latter decreases it. Disappearance of the cellsize dependence of C may be due to the balance between these two competitive mechanisms.

Concluding remarks
We have studied the scale-free and small-world properties of the earthquake networks in California, Japan, Iran and Chile. We have observed that these networks are of the scale-free and small-world type and their cell-size dependence qualitatively coincides with each other. We are firmly convinced that the discovered invariant values in eqs. (1) and (2) are universal and intrinsic in the seismicity of the Earth.
An additional remark is about incompleteness of a seismic data set. One might wonder if the incompleteness leads to change of the present results. Regarding this point, one should recall an important feature of a complex network: that is, it has a high degree of tolerance against "random attacks", i.e. random removals of vertices [18]. In fact, earthquake networks do not have a centrality with a small value of connectivity. Since the incompleteness of a data set is not biased (i.e. not due to "intelligent attacks"), we can confidently assume the robustness of the results presented here.