Magnitude Characterization Using Complex Networks in Central Chile

Studies using complex networks are applied to many systems, like traffic, social networks, internet and earth science. In this work we make an analysis using complex networks applied to magnitude of seismicity in the central zone of Chile, we use the preferential attachment in order to construct a seismic network using local magnitudes and the hypocenters of a seismic data set in central Chile. In order to work with a complete catalogue in magnitude, the data associated with the linear part of the Gutenberg-Richter law, with magnitudes greater than 2.7, were taken. We then make a grid in space, so that each seismic event falls into a certain cell, depending on the location of its hypocenter. Now the network is constructed: the first node corresponds to the cell where the first seismic event occurs. The node has an associated number which is the magnitude of the event which occured in it, and a probability is assigned to the node. The probability is a nonlinear mapping of the magnitude (a Gaussian function was taken), so that nodes with lower magnitude events are more likely to be attached to. Each time a new node is added to the network, it is attached to the previous node which has the larger probability; the link is directed from the previous node to the new node. In this way, a directed network is constructed, with a ``preferential attachment''-like growth model, using the magnitudes as the parameter to determine the probability of attachment to future nodes. Several events could occur in the same node. In this case, the probability is calculated using the average of the magnitudes of the events occuring in that node. Once the directed network is finished, the corresponding undirected network is constructed, by making all links symmetric, and eliminating the loops which may appear when two events occur in the same cell. The resulting directed network is found to be scale free (with very low values of the power-law distribution exponent), whereas the undirected one turns out to have small world behavior. These results are compared with a second, ficticious network, constructed in the same way, but where each successive node is chosen randomly in the grid, and its associated probability is also random, but taken from a Gaussian distribution. We find that this also generates a scale free network, but not a small world one. These results show an interesting behavior, another evidence of the complex organization of seismicity.