Analysing the topology of seismicity in the Hellenic arc using complex networks

Abstract

Based on the theory of complex networks, we quantify for the first time correlations between seismic events occurring in the Hellenic arc and Trench system, which is the most seismogenic structure in the European–Mediterranean region. We examine how relatively strong events with their aftershock sequences trigger phase transitions of the underlying network topology of seismic activity from random to scale-free structures. In particular, we show that the network is characterized by a highly clustered spatial structure giving rise to heterogeneous networks exhibiting enhanced small-world attributes.

Appendix

Usually, the statistical properties of networks are studied in terms of the following basic quantitative measures (Watts and Strogatz 1998; Albert and Barabasi 2002; Newman 2003):

• (a) The average path length (APL). It is defined as the mean value of all the shortest paths between any two nodes, reading:

$$ \mathrm{APL}=\frac{{\displaystyle \sum {d}_{i\to j}}}{N\left(N-1\right)} $$

where d _i→j is the shortest path between i and j nodes and N is the size of the network. The average path length is a global property of a network indicating the average number of steps needed to reach any two nodes.

• (b) The global efficiency. In the case where a network is disconnected that is at least two nodes do not communicate, the APL is infinity. In order to overtake this problem, the global efficiency E is defined as

$$ E=\frac{{\displaystyle \sum \frac{1}{d_{i\to j}}}}{N\left(N-1\right)} $$

If two nodes are disconnected, then $\frac{1}{d_{i\to j}}=0$, i.e. the efficiency is zero. The inverse of global efficiency 1/E is the harmonic mean of the shortest paths, and it is similar to APL.

• (c) The clustering coefficient c _i . The clustering coefficient c _i of the node i is the number E _i of linked triangles it forms with its neighbours, divided by the number of all possible triangles that i node forms. In the case of directed networks, the c _i takes the form (Fagiolo 2007):

$$ {c}_i=\frac{{\left(A+{A}^T\right)}_{ii}^3}{2\left[{k}_{\mathrm{tot}}\left({k}_{\mathrm{tot}}-1\right)-2{\left({A}^2\right)}_{ii}\right]} $$

where A is the adjacency matrix of the network, k _tot is the summation of inward and outward degrees, i.e. k _tot = k _in + k _out and the parenthesis (⋅) _ii indicate the main diagonal of the ⋅ matrix. The ACC of the whole network is defined as the mean value of the clustering coefficients c _i .

• (d) The degree distribution P(k) which gives the fraction of nodes with exactly k edges connected to it. Characteristic examples of almost symmetric-around the mean value of the degree distributions are the Erdős–Rényi networks (Albert and Barabasi 2002; Barrat et al. 2008; Newman 2003); scale-free networks are characterized from power-law distributions with exponent γ (Barrat et al. 2008; Newman 2003). In the case of directed network, the degree distribution is replaced by the distribution of inward or outward degrees.

• (e) The entropy of the degree distribution is defined as

$$ H=-{\displaystyle \sum_{k=1}^{k_{\max }}P(k) \log P(k)} $$

which provides a measure for the heterogeneity of the network. Higher values of H indicate the existence of heterogeneous degree distribution (Costa et. al 2007): There are few nodes acting as hubs (i.e. having big degree). Examples of such distributions are the power laws (or scale-free) with heavy tails in their degree distribution.