becomes as simple as $$\begin{aligned}
\lambda_{\max}^{P \operatorname*{\otimes}R} = \lambda^P_{\max} m . \label{eq:threshold-estimation}\end{aligned}$$ This accomplishes a significant reduction of computational cost because the number of communities (size of $P$) is usually significantly smaller than the size of the entire network. Figure \[fig:threshold-estimation\] presents results of a numerical experiment that compared the estimated largest eigenvalues obtained using this method with the actual ones, showing excellent matching between the two.

![Comparison of estimated and actual largest eigenvalues of adjacency matrices of random networks generated by a stochastic block model. Results were collected from five hundred independent calculations. In each case, the number of communities ($r$) and the size of each community ($m$) were randomly chosen between 3 and 7 and between 40 and 60, respectively. These parameter values were chosen arbitrarily just for illustrative purposes, while the overall trends of results were robust to parameter variations. $P$ and an actual binary adjacency matrix were randomly generated to be symmetric in each case. The estimated values were calculated using Eq. (\[eq:threshold-estimation\]), while the actual ones were explicitly computed from the binary adjacency matrices.[]{data-label="fig:threshold-estimation"}](threshold-estimation.pdf){width=".55\columnwidth"}

We note that other recent studies also discuss spectral properties of stochastic block models, mainly from the viewpoint of community detectability 