Variation dynamics of the complex topology of a seismicity network

Abstract

Reduction in the scaling exponent of the frequency–magnitude power law of regional seismic activity as a precursor to sizable earthquakes has been widely documented and debated. Recently, postulation based on a modified sand-pile model has been proposed as a plausible explanation. The model demonstrates systematic variations in the frequency–size scaling exponent of avalanches through the introduction of varying degrees of randomness to the conventional regular, nearest-neighbor network connection. In this study, we examined a connection network of successive events in the Taiwan seismicity, in an attempt to shed lights on the behavior of the actual earthquake network. The revealed nature of connection is indeed quite different from the nearest-neighbor interaction usually presumed in most conventional seismicity modeling. However, monthly variations in the statistics of the connection degree, the connection time and the connection distance that reflect important transition dynamics of the regional seismicity network, are inconsistent with the postulation based on the modified sand-pile model that attributes the scaling exponent variation to the varying degree of long range connections.

Introduction

The Gutenberg–Richter (G–R) law [1] is one of the most well known empirical rules characterizing the frequency–magnitude distribution of regional seismic activity. Although the ubiquity of power-law distribution extends to other complex systems (natural as well as artificial; e.g., Refs. [2], [3]), the scaling exponent of the G–R law has been of particular interest in seismotectonic studies. Seismologists have been very persistent in reserving the term “$b$-value” for this particular scaling exponent. Spatial variations in the G–R $b$-value have been confirmed to be related to the local deformation regimes [4], whereas temporal variations acting as precursors to sizable earthquakes have been proposed (e.g., Refs. [5], [6], [7], [8]) amid controversy (e.g., Ref. [9]).

Agent based models (ABM) such as the sand-pile model [10] and the classical slider-block model [11] have been proposed for the study of earthquakes. These conceptual models are characterized by nearest-neighbor interactions and are capable of generating the ubiquitous power-law frequency–size events distribution. Chen et al. [12] proposed a Long Range Connected Sand-pile (LRCS) model that modifies the nearest-neighbor interactions by gradually increasing a slight rewiring probability of random Long Range Connections (LRCs). A simple experiment is reproduced here to elucidate the effect. Briefly, grains of sand are thrown, one at a time, onto the board of a two-dimensional grid of 50 by 50 cells of uniform size. Once the total amount of the accumulated sands within a particular cell exceeds the threshold of 4, the sand is redistributed to its 4 nearest-neighbors. Each of the cells receiving grains from their neighbors is then checked, and the redistribution is continued away if any of the neighbors exceeds the threshold too. For each throw of a new grain, the redistribution proceeds until none of the peripheral cells in the cascading event receive grains exceeding the threshold. The total number of cells involved in the redistribution process initiated by a single throw is defined as the avalanche size. Because the grains redistributed outside the boundary of the square grid are permanently lost, it is interesting to note that the total number of grains retained within the grid increases quickly in the beginning but remains at a quasi-static level with only small fluctuations, once the system has reached and stays at a critical state (Fig. 1a). At that critical state, the frequency–size distribution of the avalanche events are scale invariant (Fig. 1b).

A scheme of rewiring the nearest-neighbor connections that is analogous to a two-dimensional version of the Watts and Strogatz’s small-world configuration [13], is then adopted. That is, for any particular cell, each individual connection that originally links two nearest-neighbors, is given a probability ($0<p<1$), such that the connection might be broken and rewired from the considered cell to any randomly chosen cell (besides itself) within the network (Fig. 2). In other words, whereas the original network topology without rewiring, $p=0$, is strictly characterized by connections confined within the nearest neighborhood, an increased rewiring probability, $p>0$ results in a higher degree of randomly distributed LRCs being implemented within the network (Fig. 2). The connections become completely random for $p=1$, but the topology is not the same as the classical random network [14] since the average connections for each vertex will be retained around 4. With the rewired network topology, the redistribution of grains during a cascading avalanche event follows the modified paths instead of the fixed, nearest-neighbor interactions. A weighted least squares scheme [15] is then adopted that puts different weights on data points of different magnitude to reweight the unbalanced data quantity during regression for the scaling exponent of each experiment. In fact, all power law fittings in this study follow the same strategy. Also, a consistent algebraic notation that follows the seismologists’ convention is adopted to denote the log–log expression of power laws by two parameters, that is, the $a$-value for the intercept on the log scale and the $b$-value for the slope. They are also the proportional constant and the scaling exponent of the power law. To avoid confusion with the original G–R $a$-value and $b$-value, subscripts are inserted to mark the different power laws. For example, the $b_{Sand}$-value would be specifically related to the frequency–size distribution of numerical sand avalanches in the LRCS model. The interesting discovery is the systematic increase in the scaling exponent, $b_{Sand}$-value, as well as the decrease in the sustained quasi-static number of on-board grains (Fig. 1f) along with the increase of the rewiring probability, $p$. Noticeably, there is an obvious relative reduction in the number of moderate to sizable events for topologies with higher $b_{Sand}$-values that are associated with more LRCs (compare Fig. 1c and d and see also the cumulative distribution in Fig. 1e). Based on the LRCS model, Chen et al. [12] hypothesized that sizable earthquakes might disrupt a wide area and provide a higher probability of connecting local cracks. This leads to an increased number of LRCs, and thus a higher $b_{Sand}$-value. In a sense, an increase in the number of LRCs helps to increase the probability of small avalanches (as implied from Fig. 1c to e). It is when the healing of the crust becomes effective along with the slowly driven tectonic loading that gradually closes these LRCs, thereby reduces the $b_{Sand}-$value and manifests as the reported precursor. In addition to offering a hypothetical physical analogy for the temporal variations in the G–R $b$-value, it is also conceivable that accordingly, regions within the compressive regime will have less opportunity to establish LRCs, and thus manifest a lower $b$-value, whereas it will be the opposite for the tensional regime. This is, in fact, consistent with a number of recently established statistics of observed regional seismicity [4].

The ABM highlights insights of the macroscopic characteristics of the collective behavior of the system. However, it is unclear whether the predictive behavior of the system would actually apply to the regional seismicity. Even though the network perspective has already been an indispensable tool for the study of complex systems, postulations based strictly on conceptual models would not be very useful unless it can be validated by field observations. The problem with the seismicity network, however, is that there is currently no consensus on what exactly constitutes the triggering network. This is certainly very different from the studies of the network topology of the World Wide Web or regional power grids where internal connections are without ambiguity. An empirical approach examining the event–event correlations simply treats connections between successive earthquakes as viable edges connecting active vertices, and then studies the network woven by these accumulated edges [16], [17]. One strong reason to adopt the similar implementation in this study is that this network of successive earthquakes has been shown to reveal characteristics of the small-world network [18], just like the essence of the LRCS model that is distinctive from the original Bak–Tang–Wiesenfeld (BTW) sand-pile model [19]. However, unlike Abe and Suzuki’s work [16], [17] that builds a complex earthquake network by connecting the complete catalog, we envision that the seismogenic crust is repeatedly and continually operated upon by competing ruptures and healing mechanisms, such that signatures of temporal variations in the network topology manifest. The goal of analyzing the actual seismic activity by inspecting the network of successive earthquakes is thus to compare the variation dynamics to the implied postulation of the reference LRCS model. That is, do we observe ubiquitous reduction and increasing of LRCs before and after the occurrence of major earthquakes?