Dynamical evolution of clustering in complex network of eart(2)

The network approach plays a distinguished role in contemporary science of complex systems/phenomena. Such an approach has been introduced into seismology in a recent work [S. Abe and N. Suzuki, Europhys. Lett. 65, 581 (2004)]. Here, we discuss the dynamic

nature. Directedness does not bring any difficulties to statistical analysis of connectivity(degree, the number of edges attached to the vertex under consideration) since, byconstruction, in-degree and out-degree [12] are identical for each vertex with possibleexceptions for the first and the last ones in the analysis: that is, the in-degree and out-degree do not have to be distinguished each other in the analysis of connectivity.However, directedness becomes essential when the path length (i.e., the number ofedges between a pair of connected vertices) and the period (meaning after how manysubsequent earthquakes the event returns to the initial vertex) are considered. Finally,directedness has to be ignored and the path length should be defined as the smallestvalue among the possible numbers of edges connecting the pair of vertices, when thesmall-world nature of the earthquake network is investigated. There, loops have to beremoved and multiple edges be replaced by single edges. That is, a full directedearthquake network is reduced to a corresponding simple undirected graph (see Fig. 1for the schematic description).

The earthquake network and its reduced simple graph constructed in this way arefound to be scale-free [7] and of the small world [8], exhibit hierarchical organizationand assortative mixing [9], and possess the power-law period distributions [10]. A mainreason why the earthquake network is heterogeneous is due to the empirical fact thataftershocks associated with a main shock tend to return to the locus of the main shock,geographically, and therefore the vertices of main shocks play roles of hubs of thenetwork.

The network approach has been used to examine self-organized-criticality models in

The network approach plays a distinguished role in contemporary science of complex systems/phenomena. Such an approach has been introduced into seismology in a recent work [S. Abe and N. Suzuki, Europhys. Lett. 65, 581 (2004)]. Here, we discuss the dynamic

the literature [13] if they can reproduce these notable features.

Here, we report a successful application of the dynamical network approach toseismicity. We find through careful analysis that the clustering coefficient exhibits asalient dynamical behavior: it is stationary before a main shock, jumps up at the mainshock, and then slowly decays as a power law to become stationary again. We ascertainthis behavior for some main shocks occurred in 1990’s in California. Thus, thedynamical network approach characterizes a main shock in a peculiar manner.

There are several known quantities that can structurally characterize a complexnetwork. Among them, in particular, we here consider the clustering coefficientintroduced in Ref. [14]. This quantity is defined for a simple graph, in which there areno loops and multiple edges contained. A simple graph is conveniently described by theadjacency matrix [15], A=(aij) (i,j=1,2, ,N with N being the number of verticescontained in the graph). aii=0, and aij=1(0) if the ith and jth vertices are connected(unconnected) by an edge. The clustering coefficient, C, is then given by

1C=N∑c,i

i=1N(1)where

ci=2ei

ki(ki 1)(2)

with

The network approach plays a distinguished role in contemporary science of complex systems/phenomena. Such an approach has been introduced into seismology in a recent work [S. Abe and N. Suzuki, Europhys. Lett. 65, 581 (2004)]. Here, we discuss the dynamic

ei=(A3)ii(3)and ki the value of connectivity (i.e., the degree) of the ith vertex. This quantity has thefollowing meaning. Suppose that the ith vertex has ki neighboring vertices. At most,ki(ki 1)/2 edges can exist between them. ci is the ratio of the actual number ofedges of the ith vertex and its neighbors to this maximum value. Thus, it quantifies thedegree of adjacency between two vertices neighboring the ith vertex. C is its averageover the whole graph. In the earthquake network, ci quantifies how strongly twoaftershocks associated with a main shock (as the ith vertex) are correlated.

Now, we address the question as to how the clustering coefficient changes in time asthe earthquake network dynamically evolves. For this purpose, we have studied thecatalog of earthquakes in California, which is available at URL /. In particular, we have focused our attention to three major shocksoccurred in 1990’s: (a) the Joshua Tree Earthquake (M6.1) at 04:50:23.20 on April 23,1992, 33 57.60'N latitude, 116 19.02'W longitude, 12.33 km in depth, (b) the LandersEarthquake (M7.3) at 11:57:34.13 on June 28, 1992, 34 12.00'N latitude, 116 26.22'Wlongitude, 0.97 km in depth, and (c) the Hector Mine Earthquake (M7.1) at 09:46:44.13on October 16, 1999, 34 35.64'N latitude, 116 16.26'W longitude, 0.02 km in depth. Wehave taken the intervals of the seismic time series containing these events, divided theintervals into many segments, and constructed the earthquake network of each segment.Then, we have calculated the value of the clustering coefficient of each network. In thisway, dynamical evolution of clustering has been explored.

In Fig. 2, we present the results on evolution of the clustering coefficient in the case

The network approach plays a distinguished role in contemporary science of complex systems/phenomena. Such an approach has been introduced into seismology in a recent work [S. Abe and N. Suzuki, Europhys. Lett. 65, 581 (2004)]. Here, we discuss the dynamic

when the length of the segments is fixed to be 240 hours long. Here, the cell size5km×5km×5km is examined. A remarkable behavior can be appreciated: theclustering coefficient stays st …… 此处隐藏：5906字，全部文档内容请下载后查看。喜欢就下载吧 ……