  Researchers have developed several theoretical methods for predicting
epidemic thresholds, including the mean-field like (MFL) method, the quenched
mean-field (QMF) method, and the dynamical message passing (DMP) method. When
these methods are applied to predict epidemic threshold they often produce
differing results and their relative levels of accuracy are still unknown. We
systematically analyze these two issues---relationships among differing results
and levels of accuracy---by studying the susceptible-infected-recovered (SIR)
model on uncorrelated configuration networks and a group of 56 real-world
networks. In uncorrelated configuration networks the MFL and DMP methods yield
identical predictions that are larger and more accurate than the prediction
generated by the QMF method. When compared to the 56 real-world networks, the
epidemic threshold obtained by the DMP method is closer to the actual epidemic
threshold because it incorporates full network topology information and some
dynamical correlations. We find that in some scenarios---such as networks with
positive degree-degree correlations, with an eigenvector localized on the high
$k$-core nodes, or with a high level of clustering---the epidemic threshold
predicted by the MFL method, which uses the degree distribution as the only
input parameter, performs better than the other two methods. We also find that
the performances of the three predictions are irregular versus modularity.
