can be used to describe a wider variety of complex networks. In the following sections, we illustrate the use of GPMNs through three application examples.

Application I: Predicting Epidemic Thresholds {#sec:app1}
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GPMNs can be used to predict epidemic thresholds (i.e., critical ratio of infection and recovery rates) of epidemic processes on networks. It is known that the epidemic threshold is given by the reciprocal of the largest eigenvalue of the adjacency matrix of the network [@chakrabarti2008epidemic]. If the network can be modeled as a GGPMN, the largest eigenvalue of its adjacency matrix can be obtained analytically as $$\lambda_{\max}^{G \operatorname*{\boxtimes}_{\alpha, \beta} H} = \max_{i,j} \left( \alpha \lambda^G_i + \alpha \lambda^H_j + \beta \lambda^G_i \lambda^H_j \right) . \label{eq:max-eigval}$$ In most cases, the adjacency spectrum of a large simple graph is characterized by a dense area around the origin accompanied by a small number of substantially larger (positive) outliers [@farkas2001spectra]. Therefore, if $\alpha \ge 0$ and $\beta \ge
0$, Eq. (\[eq:max-eigval\]) is simplified to $$\lambda_{\max}^{G \operatorname*{\boxtimes}_{\alpha, \beta} H} = \alpha \lambda^G_{\max} + \alpha \lambda^H_{\max} + \beta \lambda^G_{\max} \lambda^H_{\max} . \label{eq:lambdamax-formula}$$ Because the epidemic threshold is the reciprocal of the largest eigenvalue, this result indicates that the epidemic threshold on a CPMN ($(\alpha, 