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In this work, we studied the correlation properties of seismic networks by analyzing the assortativity of worldwide and synthetic earthquake networks. We used data from the World Earthquake Catalog for the period from 2002 to 2016, considering earthquakes with magnitude thresholds 4.5 and 5.0. Shallow earthquakes (a depth of up to 70 km) and deep earthquakes (a depth greater than 70 km) were analyzed separately. Synthetic data were produced from simulations using a modified version of the Olami-Feder-Christensen model, which can reproduce several statistical characteristics of actual earthquakes. The study was carried out for two methodologies of connections between the network elements, where the correlation measures were calculated for all networks. The results for shallow earthquakes and synthetic data indicate: assortative correlation (locations with similar seismic activities tend to have a greater number of connections between them); mainshocks induce other mainshocks in both close and further away regions; the structure found has a type of “attracting dynamics”, where the places with a more intense seismic activity produce large numbers of connections in other locations around them. Deep earthquake networks are neutral and therefore do not have an explicit correlation type. Our findings agree with previous works for specific areas and contribute to better understand correlations between seismological regions.