Definition of the Subject
Complexity is an emergent collective property, which is hardly understood by the traditional approach in natural science based onreductionism. Correlation between elements in a complex system is strong, no matter how largely they are separated both spatially and temporally,therefore it is essential to treat such a system in a holistic manner, in general.
Although it is generally assumed that seismicity is an example of complex phenomena, it is actually nontrivial to see how and in what sense it iscomplex. This point may also be related to the question of primary importance of why it is so difficult to predict earthquakes.
Development of the theory of complex networks turns out to offer a peculiar perspective on this point. Construction of a complexearthquake network proposed here consists of mapping seismic data to a growing stochastic graph. This graph, or network, turns out to exhibita number of remarkable behaviors both physically and...
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Abbreviations
- Network or graph:
-
A network (or a graph) [28] consists of vertices (or nodes) and edges (or links) connecting them. In general, a network contains loops (i. e., edges with both ends attached to the same vertices) and multiple edges (i. e., edges more than one that connect two different vertices). If edges have their directions, such a network is called directed. A simple graph is a network, in which loops are removed and each multiple edge is replaced by a single edge. In a stochastic network, each connection is inherently probabilistic. A classical random graph is a simple example, in which each two vertices are connected by an edge with probability p and unconnected with probability \( 1-p\enskip (0<p<1) \).
- Connectivity distribution or degree distribution:
-
The connectivity distribution (or the degree distribution), \( { P(k) } \), is the probability of finding vertices with k edges in a stochastic network. In a directed network, the number of incoming/outgoing edges is called the in‐degree/out‐degree. Connectivity of a classical random graph obeys the Poissonian distribution in the limit of the large number of vertice [11,14,20], \( { P(k)=\text{e}^{-\lambda}\lambda^{k}/k! } \) (λ: a positive parameter, \( { k=0,1,2,\ldots } \)), whereas a scale-free network [11,12,14,20] has a power-law shape, \( { P(k)\sim k^{-\gamma} } \) (γ: a positive exponent), for large k.
- Preferential attachment rule:
-
This is a concept relevant to a growing network, in which the number of vertices increases. Preferential attachment [11,12,14,20] implies that a newly created vertex tends to link to pre‐existing vertices with the probability \( { \Pi(k_{i})=k_{i}/\sum\nolimits_j {k_{j}} } \), where \( { k_i } \) stands for the connectivity of the ith vertex. That is, the larger the connectivity of a vertex is, the higher the probability of getting linked to a new vertex is.
- Clustering coefficient :
-
The clustering coefficient [27] is a quantity characterizing an undirected simple graph. It quantifies the adjacency of two neighboring vertices of a given vertex, i. e., the tendency of two neighboring vertices of a given vertex to be connected to each other. Mathematically, it is defined as follows. Assume the ith vertex to have \( { k_i } \) neighboring vertices. There can exist at most \( { k_i(k_{i}-1)/2 } \) edges between the neighbors. Define \( { c_i } \) as the ratio
$$ c_i=\frac{\parbox{50mm}{\center actual number of edges between the neighbors of the $i$th vertex}}{k_i(k_i -1)/2}\:. $$(1)Then, the clustering coefficient is given by the average of this quantity over the network:
$$ C=\frac{1}{N}\sum\limits_{i=1}^N c_i\:, $$(2)where N is the total number of vertices contained in the network. The value of the clustering coefficient of a random graph, \( { C_\text{random} } \), is much smaller than unity, whereas a small‐world network has a large value of C which is much larger than \( { C_{\text{random}} } \).
- Hierarchical organization :
-
Many complex networks are structurally modular, that is, they are composed of groups of vertices that are highly interconnected to each other but weakly connected to outside groups. This hierarchical structure [22] can conveniently be characterized by the clustering coefficient at each value of connectivity, \( { c(k) } \), which is defined by
$$ c(k)=\frac{1}{N P_{\text{SG}}(k)}\sum\limits_{i=1}^N c_i \delta_{k_i,k}\:, $$(3)where \( { c_i } \) is given by (1), N the total number of vertices, and \( { P_{\text{SG}}(k) } \) the connectivity distribution of an undirected simple graph. Its average is the clustering coefficient in (2): \( { C=\sum\nolimits_{k}{c(k)P_{\text{SG}}}(k) } \). A network is said to be hierarchically organized if \( { c\,(k) } \) varies with respect to k, typically due to a power law, \( { c(k)\sim k^{-\beta} } \), with a positive exponent β.
- Assortative mixing and disassortative mixing:
-
Consider the conditional probability, \( { P(k^\prime\vert k) } \), of finding a vertex with connectivity \( { k^\prime } \) linked to a given vertex with connectivity k. Then, the nearest‐neighbor average connectivity of vertices with connectivity k is defined by [20,21,26]
$$ \bar{k}_{nn}(k)=\sum\limits_{k^\prime}{k^\prime P(k^\prime\vert k)}\:. $$(4)If \( { \bar {k}_{nn}(k) } \) increases/decreases with respect to k, mixing is termed assortative/disassortative. A simple model of growth with preferential attachment is known to possess no mixing. That is, \( { \bar{k}_{nn}(k) } \) does not depend on k.
The above‐mentioned linking tendency can be quantified by the correlation coefficient [17] defined as follows. Let \( { e_{kl}(=e_{lk}) } \) be the joint probability distribution for an edge to link with a vertex with connectivity k at one end and a vertex with connectivity l at the other. Calculate its marginal, \( { q_k=\sum\nolimits_l {e_{kl}} } \). Then, the correlation coefficient is given by
$$ r=\frac{1}{\sigma_{q}^{2}}\sum\limits_{k,l} {kl\left(e_{kl}-q_{k}q_l\right)}\:, $$(5)where \( { \sigma_{q}^{2}=\sum\nolimits_{k}{k^2}q_k -(\sum\nolimits_k {k}q_k)^2 } \) stands for the variance of \( { q_k } \). \( { r\in [-1,1] } \), and if r is positive/negative, mixing is assortative/disassortative [17,20].
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Abe, S., Suzuki, N. (2009). Earthquake Networks , Complex. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_153
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