\beta) = (1, 0)$) is twice the harmonic mean of the thresholds on two factor networks. Similarly, the epidemic threshold on a DPMN ($(\alpha, \beta) = (0, 1)$) is simply the product of the thresholds on two factor networks.

A particularly interesting application of this result is the prediction of epidemic thresholds on random networks generated by a stochastic block model with equal community size. Let $P$ be an $r
\times r$ edge probability matrix, and assume each community is made of exactly $m$ nodes (thus the total number of nodes is $rm$). Then, the ensemble average of adjacency matrices of networks generated by this model is given by $P \operatorname*{\otimes}R$, an adjacency matrix of a nonsimple DPMN, where $R$ is an $m \times m$ all-one square matrix. A typical assumption in matrix perturbation theory suggests that the spectrum of a specific binary adjacency matrix generated from this stochastic block model behaves similarly with the spectrum of this ensemble average. Therefore, we can estimate the epidemic threshold on a random network generated with this model by calculating the largest eigenvalue of $P \operatorname*{\otimes}R$. Since $R$ has $m$ as the single largest eigenvalue (while all other eigenvalues are 0s), this estimation 