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Active and passive faults detection by using the PageRank algorithm Amir H. Darooneh and Nastaran Lotfi EPL, 107 (2014) 49001
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August 2014 EPL, 107 (2014) 49001 doi: 10.1209/0295-5075/107/49001
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Active and passive faults detection by using the PageRank algorithm Amir H. Darooneh and Nastaran Lotfi Department of Physics, University of Zanjan - University Blvd., 45371-38791, Zanjan, Iran received 26 April 2014; accepted in final form 18 July 2014 published online 8 August 2014 PACS PACS PACS
91.30.Px – Earthquakes 91.30.-f – Seismology 02.10.Ox – Combinatorics; graph theory
Abstract – Here we try to find active and passive places for earthquakes in the geographical region of Iran. The approach of Abe and Suzuki is adopted for modeling the seismic history of Iran by a complex directed network. By using the PageRank algorithm, we assign to any places in the region an activity index. Then, we determine the most active and passive places. c EPLA, 2014 Copyright
Introduction. – Recently, scientists have been greatly interested in studying earthquakes within the context of complex systems theory [1,2]. The faults are the main components of a geological system whose motions are responsible for most of spatiotemporal phenomena in geological regions. When the activity of a fault occurs abruptly, it can produce an earthquake. Earthquakes have an influence on their surrounding faults by transferring the stresses to them and may trigger subsequent events [3–5]. Seismic waves are generated by the sudden releasing of the energy in an earthquake. They propagate through the Earth crust. The fluctuation of stresses associated with seismic waves may temporarily exceed the local failure threshold in the far-field faults and may lead to other earthquakes in the places that are not close to the original earthquake’s epicenter [4–7]. Many evidences support the fault-fault interaction because of the earthquake stress changes [8]. The nature of such interactions are more complex and the complex network method seems to be a good candidate for studying and understanding the geophysical phenomena in a system of faults belonging to an area. In this paper, we continue our study about Iran’s earthquakes network and try to find out information about the active and passive regions by calculating the centrality of nodes in the corresponding network. The active region is a place that most likely can trigger the next earthquakes in other places. Similarly the most probable place for bringing about a new earthquake by seismic activity in other places is called the passive region. The information about activity or passivity of places can be used for earthquake forecasting or as
an alternative way for preparing the seismic zoning map of an area. In the following section, we describe how an earthquakes network is constructed for a geological region and what its properties are. The third section is devoted to explaining the centrality concept particularly of the PageRank (CheiRank) which was used for attributing the passivity (activity) to the different areas. In the fourth section, we report our results obtained for the geological region of Iran and in the final section we summarize our work. Complexity: the earthquakes network. – The concept of an earthquake network has been recently introduced in order to reveal the complexity of seismicity, both qualitatively and quantitatively [9]. The process of constructing the network is an important step in order to quantify the complexity of the earthquake phenomena in the framework of the complex network theory. The network for a particular geographical region is simply constructed by dividing the region into small cubes or squares which are called cells. The cells play the role of the network vertices. Occurrence of successive events in two of the vertices can link them. Abe and Suzuki have studied the different properties of such networks in California, Japan and Iran in series of papers [9–14]. Their results reveal the scale-free and small-world structure of the earthquake networks [9]. They also discovered that the evolving networks of earthquakes have a scale-free network behavior. The hubs of a network demonstrate the main shocks [10]. Besides, they find that the distribution of periods corresponding to cycles on a network obeys a
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Amir H. Darooneh and Nastaran Lotfi power law [11] which is important for the prediction of the future earthquakes. In the next step, the hierarchical and mixing properties of the earthquake networks are examined. The hierarchical organization is shown by the clustering coefficient asymptotic power-law decay with respect to connectivity [12]. One of the most interesting results that belong to the dynamics of the clustering coefficient is that before the main shocks it remains stationary and has a sudden jump up at the main shocks. After this, it slowly decays following a power law to become stationary again [13,14]. By using finite data-size scaling, the universal behavior of the clustering coefficient in different regions is shown [15]. The following approach of Abe and Suzuki to demonstrate the role of the cell size on the features of the earthquakes network in Iran is carried out by introducing a new parameter named resolution that equals the inverse of the cell size [16]. The clustering coefficient, the characteristic lengths, the mean degree of neighbors and q-exponent for the degree distribution have a power law distribution behavior as a function of resolution. It is also explained that all network features are highly dependent on one another for a resolution greater than 10 which results in the point that just one topological feature is sufficient for describing the earthquake network. Recently, we showed that the q-exponential appears as an appropriate function for fitting the degree distribution of earthquake networks [17]. It is also discovered that the q-exponent, as a function of resolution, has a peak representing a threshold for the validity of our previous assertion. Centrality: the important nodes. – The centrality of a vertex can be measured via different ways such as degree, betweenness, closeness and PageRank to determine its relative importance within a graph. The degree is the simplest way only for measuring the centrality of a vertex by using the local structure around the vertices. In an undirected network, the degree is the number of links a vertex has. In a directed network, a vertex can have a different number of outgoing and incoming links. As a result, the degree is divided into out-degree and in-degree, respectively [18]. In a connected graph, the distance between all pairs of vertices is defined by the length of their shortest paths. The farness of a vertex is calculated as the sum of its distances to all other vertices, and its closeness is defined as the inverse of the farness [19]. Therefore, the more central a vertex is, the lower its total distance to all other vertices is. Closeness can be considered as a measure of how fast it will take to spread information from one vertex to all other vertices successively [20]. Betweenness centrality quantifies the number of times a vertex acts as a bridge along the shortest path between two other vertices. In this definition, vertices with a high betweenness are those that have a high probability to occur on a randomly chosen shortest path between two randomly chosen vertices [21].
PageRank is an algorithm used in the Google search engine for ranking the web pages [22]. It can be explained through the random walker model. A walker starts from a random vertex, and then proceeds along one of the outlinks that is selected in a random fashion. The PageRank of a specific vertex is the asymptotic probability that the walker meets the vertex. The random walker may be faced with difficulty when reaching a vertex that has no outlinks. To resolve this difficulty, we allowed him to jump randomly with a constant probability to other vertexes. The following equation describes such walking procedure: ri =
rj d + (1 − d) , N kjout
(1)
j∈Bi
where ri is the PageRank of i-th vertex, N is the total number of vertices, Bi is the set of vertices that are adjacent to the i-th vertex, kjout is the out-degree of vertex j. The parameter d is the probability of jumping to any vertex. It is usually set to 0.15. The CheiRank is constructed for a directed network with the inverted directions of links [23]. It is similar to the PageRank but it ranks the network vertices in the average to a number of outgoing links. Results: the active an passive faults in Iran. – The seismic data we use here are those taken from http://irsc.ut.ac.ir. The period and the geographical region covered are as follows: between 1 January, 1996 and 16 October, 2005, 24N-44N latitude and 40E-64E longitude. The number of events is recorded in this period and region is about 14420. In order to construct the two-diminutions network, we divide the geographical region under our consideration into small square cells with size of 0.25. This cell size indicates a resolution of 8 and the length of a cell is 27.9 km. It is shown that after a specific amount of resolution, the network’s characteristics are modeled by a power law function in terms of resolution which means that they depend on each other [16,17]. A cell is regarded as a vertex of a network if the earthquake with a magnitude greater than a threshold value occurred in it. A link between two vertices is defined when two successive earthquakes occurred in them. In the directed earthquake network, any link demonstrates a causal relationship between its starting and ending vertices. In other words, the seismic event in the starting place triggers the ending event. The links within a cell itself are removed due to the fact that they are not so informative in the context of the network theory. In such networks, the CheiRank shows how likely the occurrence of a seismic event in a place may cause the other places to quake. Hence it measures the activity of the places. In the same fashion, the PageRank shows how likely seismicity in other places may have an influence on the occurrence of an earthquake in a specific place. Thus, it quantifies the passivity of the places in a geological area.
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Active and passive faults detection by using the PageRank algorithm
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 1: (Color online) The results for CheiRank scores or activity (upper plots) and PageRank scores or passivity (lower plots) for different magnitude thresholds are shown. From left to right, plots correspond to the values of thresholds, 2, 3 and 4, respectively. The dark color shows the larger activity (passivity) values and the light color is used for less active places.
Table 1: The first five active and passive faults for different threshold values for magnitude.
Latitude
Passivity Longitude
Fault
Latitude
Activity Longitude
Fault
30.75 30.50 36.25 40.00 35.50
56.50 56.75 51.50 50.00 48.75
Zahedan Zahedan Kandovan No fault South Parandak
30.50 30.75 37.00 36.25 40.00
56.75 56.50 54.50 51.50 50.00
Zahedan Zahedan Khazar Kandovan No fault
35.50 36.25 30.50 39.75 36.25
49.00 51.50 56.75 49.75 51.25
South Parandak Kandovan Darivan No fault Kandovan
35.50 30.50 36.25 39.75 35.75
49.00 56.75 51.50 49.75 49.00
South Parandak Darivan Kandovan No fault South Parandak
35.50 35.50 36.25 35.75 35.75
49.00 49.25 52.00 53.00 49.50
South Parandak South Parandak Kandovan South Parandak South Parandak
35.50 35.50 36.25 35.50 35.75
49.00 49.25 52.00 49.50 53.00
South Parandak South Parandak Kandovan South Parandak Astaneh
Magnitude > 4
Magnitude > 3
Magnitude > 2
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Amir H. Darooneh and Nastaran Lotfi We construct three types of networks for the seis- REFERENCES mic history in Iran. In the first type, we consider the [1] Bak P. and Tang C., J. Geophys. Res., 94 (1989) 15635. earthquakes with magnitude larger than 2; the second and [2] Cowie P. A., Vanneste C. and Sornette D., J. third types contain data for the earthquakes with magniGeophys. Res., 98 (1993) 21809. tude larger than 3 and 4, respectively. This means that we [3] King G. C. P., Stein R. S. and Lin J., Bull. Seismol. assume threshold magnitudes for the remote earthquake Soc. Am., 84 (1994) 935. triggering. [4] Belardinelli M. E., Bizzarri A. and Cocco M., J. The results of the CheiRank and the PageRank for difGeophys. Res., 108 (2003) 2135. ferent values of the threshold magnitude are plotted in [5] Freed A. M., Annu. Rev. Earth Planet. Sci., 33 (2005) fig. 1. The upper plots are related to the results of the 335. CheiRank which demonstrate the activity of different cells [6] Hill D. P., Bull. Seismol. Soc. Am., 98 (2008) 66. in Iran. The lower plots show the results of the PageRank [7] Hill D. P. and Prejean S. G., in Treatise which quantify the passivity in different cells in Iran. The on Geophysics: Earthquake Seismology, edited by dark color shows the large value of activity (passivity) Kanamori H., Vol. 4 (Elsevier, Amsterdam) 2009, p. 258. and the light color is used for the small value of activ[8] King G. C. P., in Treatise on Geophysics: Earthquake Seismology, edited by Kanamori H., Vol. 4 (Elsevier, ity (passivity). The activity (passivity) values could be Amsterdam) 2009, p. 225. used as prior probability distribution for earthquake fore[9] Abe S. and Suzuki N., Physica A, 337 (2004) 357. casting or raw material for the seismic risk assessment in [10] Abe S. and Suzuki N., Europhys. Lett., 65 (2004) 581. the earthquake engineering. If it is possible, to any cell we [11] Abe S. and Suzuki N., Eur. Phys. J. B, 44 (2005) 115. can associate a fault that crosses it. As is seen in fig. 1, [12] Abe S. and Suzuki N., Phys. Rev. E, 74 (2006) 026113. the central region of Iran is most likely to have geological [13] Abe S. and Suzuki N., Eur. Phys. J. B, 59 (2007) 93. activity. Since the differences between activity and passiv[14] Abe S. and Suzuki N., Braz. J. Phys., 39 (2009) 428. ity of cells are not clear in fig. 1, hereupon we report the [15] Abe S., Pasat’en D. and Suzuki N., Physica A, 390 first five active and passive cells and their corresponding (2011) 1343. faults in table 1. South Parandak and Kandovan are two [16] Lotfi N. and Darooneh A. H., Eur. Phys. J. B, 85 faults that are most active and also passive for any value (2012) 1. of the magnitude threshold. [17] Lotfi N. and Darooneh A. H., Physica A, 392 (2013) Conclusion. – Here, we presented a method for finding the active and passive places (faults) in a geographical area based on the PageRank algorithm. The method was examined on the seismic data of Iran. We reported our results for activity (passivity) of places as plots similar to the seismic zoning maps. We also obtained that the central part of Iran is most likely to have geological activity. Such results could be used for earthquake forecasting or the seismic risk assessment in the earthquake engineering.
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