Dynamical evolution of clustering in complex network of earthquakes

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

(b)

FIG.1

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

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FIG. 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

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FIG.3(a)

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

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FIG.3(b)

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

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FIG.3(c)

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

major event kmukmukm cell sizeM0(u102)

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109.8448.00.6181.830.1221.66D2.24.00.330.400.550.59Joshua TreeEarthquakeLanders Earthquake Hector Mine Earthquake

TABLE I

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

Dynamical evolution of clustering in complex

network of earthquakes

Sumiyoshi Abe1,2 and Norikazu Suzuki3

1Institute of Physics, University of Tsukuba, Ibaraki 305-8571, Japan

2Institut Supérieur des Matériaux et Mécaniques Avancés,

44 F. A. Bartholdi, 72000 Le Mans, France

3College of Science and Technology, Nihon University, Chiba 274-8501, JapanAbstractThe network approach plays a distinguished role in contemporary scienceof complex systems/phenomena. Such an approach has been introduced into seismologyin a recent work [S. Abe and N. Suzuki, Europhys. Lett. 65, 581 (2004)]. Here, wediscuss the dynamical property of the earthquake network constructed in California andreport the discovery that the values of the clustering coefficient remain stationary beforemain shocks, suddenly jump up at the main shocks, and then slowly decay following apower law to become stationary again. Thus, the network approach is found tocharacterize main shocks in a peculiar manner.

PACS number(s):89.75.Da, 91.30.–f, 05.65.+b

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

Looking at seismic data from the physics viewpoint, it may be of interest to recognizethat it is essentially a field-theoretical system. It consists of the series of a set of valuesof occurrence time, hypocenter, and magnitude of each earthquake. In other words,seismic moment (its logarithm being magnitude) as a field strength is defined on eachdiscrete spacetime point. However, unlike ordinary field dynamics in physics, both thefield strength and spacetime points are inherently random. In spite of such apparentcomplicatedness, known empirical laws are rather simple. There are in fact twocelebrated classical examples. One is the Gutenberg-Richter law [1] for the relationshipbetween frequency and seismic moment. The other is the Omori law [2] for thetemporal decay of frequency of aftershocks. Both of them are power laws, indicatingcomplexity/criticality of seismicity.

Instantaneous release of huge energy by a main shock can be thought of as a“quenching” process. The disorder of a complex landscape of the stress distribution atfaults in the relevant area is then reorganized by it. Accordingly, a swarm of aftershocksmay follow. This process constitutes nonstationary parts of a seismic time series, and,due to the power-law nature of the Omori law, “relaxation” to a stationary state is veryslow. In a recent work [3], it has been found that there are striking similarities betweenthe aftershock phenomenon and glassy dynamics, including aging and scaling.

In the previous works [4,5], we have studied the spatio-temporal complexity ofseismicity and found that both the spatial distance and time interval between twosuccessive earthquakes obey specific but remarkably simple statistical laws. Thoseresults indicate that successive events are indivisibly correlated, no matter how large

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

their spatial separation is. In fact, there is an investigation [6], which points out that anearthquake can be triggered by a foregoing one, which is more than 1000 km away. Thisimplies that the seismic correlation length may be enormously large, exhibiting a strongsimilarity to phase transition and critical phenomena. Accordingly, it is inappropriate toput spatial windows in analysis of seismicity, in general.

To characterize complexity of event-event correlation in seismicity, we have recentlyproposed the network approach [7-10], in which seismic data is mapped to a growingrandom graph. This graph, termed the earthquake network, is constructed as follows. Ageographical region under consideration is divided into a lot of small cubic cells. A cellis regarded as a vertex of a network if earthquakes with any values of magnitudeoccurred therein. Two successive events define an edge between two vertices. If theyoccur in the same cell, a loop is attached to that vertex. The edges efficiently representevent-event correlation mentioned above. The network thus constructed representsdynamical information of seismicity in a peculiar manner. (Another procedure ofconstructing an earthquake network, which is more complicated than the present oneintroducing seven parameters including the spatial distance, time interval, magnitude,and so on, is considered for example in Ref. [11].) Several comments on thisconstruction are in order. Firstly, it contains a single parameter, the cell size, whichdetermines a scale of coarse graining. Once the cell size is fixed, the earthquakenetwork is unambiguously defined. Since there are no a priori operational rules todetermine the cell size, it is of importance to examine the dependence of the property ofearthquake network on it. Secondly, the earthquake network is a directed graph in its

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 http://www.77cn.com.cn/. 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 stationary before the main shocks, suddenly jumps up at themoments of the main shocks, and then gradually decays.

To clarify the property of the slow decay in more detail, we present Fig. 3, in whichshorter-time analysis with 24 hours is performed by examining two different cell sizes,5km×5km×5km and 10km×10km×10km. As can clearly be appreciated, the“cumulative” clustering coefficient,

C(≤n)=

M=1∑nCM,(4)

obeys a definite law, where CM stands for the clustering coefficient of the networkconstructed in the interval 24×(M 1)~24×M [hours] after the moment of the mainshock at M=0, and n=(hours)/24. Indeed, it is well represented by the followingpower law:

CM~1,(1+M/M0)α(5)where α and M0 are positive constants, and their values are given in Table I.

In conclusion, we have found that the clustering coefficient of the evolvingearthquake network remains stationary before a main shock, suddenly jumps up at themain shock, and then slowly decays to become stationary again following the power-law relaxation. In this way, the clustering coefficient is shown to successfully

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

characterize main shocks. We would like to emphasize that the power-law decay after amain shock described in Eq. (5) might remind one of the Omori law, but actually theyare not directly related to each other. This is because, in the definition of the clusteringcoefficient, loops are removed and multiple edges are replaced by single edges, that is, anumber of aftershocks are excluded in the analysis.

One of the authors (S. A.) would like to thank Carmen P. C. Prado for discussions.

[1]B. Gutenberg and C. F. Richter, Seismicity of the Earth and Associated

Phenomenon, 2nd edition (Princeton University Press, Princeton, 1954).

[2]F. Omori, J. Coll. Sci. Imper. Univ. Tokyo 7, 111 (1894);

T. Utsu, Geophys. Mag. 30, 521 (1961).

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[8]S. Abe and N. Suzuki, Physica A 332, 533 (2004).S. Abe and N. Suzuki, J. Geophys. Res. 108 (B2), 2113 (2003).S. Abe and N. Suzuki, Physica A 350, 588 (2005).D. W. Steeples and D. D. Steeples, Bull. Seismol. Soc. Am. 86, 921 (1996).S. Abe and N. Suzuki, Europhys. Lett. 65, 581 (2004).S. Abe and N. Suzuki, Physica A 337, 357 (2004); S. Abe and N. Suzuki,

Nonlin. Processes Geophys. 13, 145 (2006).

[9]

[10]S. Abe and N. Suzuki, Phys. Rev. E 74, 026113 (2006).S. Abe and N. Suzuki, Eur. Phys. J. B 44, 115 (2005).

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

[11]

[12]M. Baiesi and M. Paczuski, Nonlin. Processes Geophys. 12, 1 (2005).R. Pastor-Satorras and A. Vespignani, Evolution and Structure of the Internet:

A Statistical Physics Approach (Cambridge University Press, Cambridge, 2004).

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[14]

[15]T. P. Peixoto and C. P. C. Prado, Phys. Rev. E 74, 016126 (2006).D. J. Watts and S. H. Strogatz, Nature (London) 393, 440 (1998).R. J. Wilson, Introduction to Graph Theory, 4th edition

(Prentice Hall, London, 1996).

[16]A. Vázquez, R. Pastor-Satorras, and A. Vespignani,

Phys. Rev. E 65, 066130 (2002).

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

Figure and Table Captions

FIG. 1Schematic descriptions of an earthquake network. (a) A full directed network.

The vertices with high values of connectivity, A, B, and C, correspond to mainshocks. (b) The simple undirected graph reduced from the full network in (a).FIG. 2Evolution of the (dimensionless) clustering coefficient during each 240 hours.

The origins are adjusted to the moments of the main shocks, that is,

(a) the Joshua Tree Earthquake, (b) the Landers Earthquake, and

(c) the Hector Mine Earthquake.

FIG. 3Evolution of the (dimensionless) cumulative clustering coefficient defined in

Eq. (4) during each 24 hours. The solid curves are due to the model in Eq. (4)with the form in Eq. (5). (a, b, c-1) and (a, b, c-2) are the results for the cell

sizes, 5km×5km×5km and 10km×10km×10km, respectively,

for (a) the Joshua Tree Earthquake, (b) the Landers Earthquake, and

(c) the Hector Mine Earthquake. The values of the parameters in Eqs. (5) aregiven in Table I.

TABLE IThe values of the parameters in Eq. (5) used in Fig. 3.

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