7 | $34.795$ | $-37.814$ | 7 | $43.897$ | $-46.904$
8 | $29.940$ | $-31.740$ | 8 | $37.580$ | $-40.672$
9 | $57.664$ | $-62.551$ | 9 | $36.172$ | $-37.726$
|  |  | 10 | $30.597$ | $-32.125$

### Text C. Derivation of the variance of the total intralayer weight for a
community in a multilayer correlation matrix

To derive Eq. (11) in section 2.4, we start with

$\displaystyle\text{Var}\left[\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in S\end{subarray}}^{i-1}C_{ij\alpha}^{\text{con}}\right]$
$\displaystyle=$
$\displaystyle\text{E}\left[\left(\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in
S\end{subarray}}^{i-1}C_{ij\alpha}^{\text{con}}\right)^{2}\right]-\left(\text{E}\left[\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in
S\end{subarray}}^{i-1}C_{ij\alpha}^{\text{con}}\right]\right)^{2}$
$\displaystyle{}=$
$\displaystyle\text{E}\left[\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in
S\end{subarray}}^{i-1}\sum_{\beta=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}k=1\\\
(k,\beta)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\ (r,\beta)\in
S\end{subarray}}^{k-1}C_{ij\alpha}^{\text{con}}C_{kr\beta}^{\text{con}}\right]-$
$\displaystyle\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in
S\end{subarray}}^{i-1}\sum_{\beta=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}k=1\\\
(k,\beta)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\ (r,\beta)\in
S\end{subarray}}^{k-1}C_{ij\alpha}C_{kr\beta}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in
S\end{subarray}}^{i-1}\sum_{\beta=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}k=1\\\
(k,\beta)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\ (r,\beta)\in
S\end{subarray}}^{k-1}\left(\text{E}[C_{ij\alpha}^{\text{con}}C_{kr\beta}^{\text{con}}]-C_{ij\alpha}C_{kr\beta}\right)$
$\displaystyle{}=$
$\displaystyle\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in
S\end{subarray}}^{i-1}\sum_{\beta=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}k=1\\\
(k,\beta)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\ (r,\beta)\in
S\end{subarray}}^{k-1}\left\\{\text{E}\left[\left(\frac{1}{L}\sum_{l=1}^{L}x_{il\alpha}x_{jl\alpha}\right)\left(\frac{1}{L}\sum_{l=1}^{L}x_{kl\beta}x_{rl\beta}\right)\right]-C_{ij\alpha}C_{kr\beta}\right\\}$
$\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in
S\end{subarray}}^{i-1}\sum_{\beta=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}k=1\\\
(k,\beta)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\ (r,\beta)\in
S\end{subarray}}^{k-1}\left(\frac{1}{L^{2}}\text{E}\left[\sum_{l=1}^{L}\sum_{l^{\prime}=1}^{L}x_{il\alpha}x_{jl\alpha}x_{kl^{\prime}\beta}x_{rl^{\prime}\beta}\right]-C_{ij\alpha}C_{kr\beta}\right),$
(S23)

where $L$ is the number of samples we draw from the $N$-variate multivariate
normal distribution. Now, using the fact that different samples are
independent, we obtain

$\displaystyle\frac{1}{L^{2}}\text{E}\left[\sum_{l=1}^{L}\sum_{l^{\prime}=1}^{L}x_{il\alpha}x_{jl\alpha}x_{kl^{\prime}\beta}x_{rl^{\prime}\beta}\right]-C_{ij\alpha}C_{kr\beta}$
$\displaystyle=$
$\displaystyle\frac{1}{L^{2}}\sum_{l=1}^{L}\sum_{\begin{subarray}{c}l^{\prime}=1\\\
l^{\prime}\neq
l\end{subarray}}^{L}\text{E}[x_{il\alpha}x_{jl\alpha}x_{kl^{\prime}\beta}x_{rl^{\prime}\beta}]+\frac{1}{L^{2}}\sum_{l=1}^{L}\text{E}[x_{il\alpha}x_{jl\alpha}x_{kl\beta}x_{rl\beta}]-C_{ij\alpha}C_{kr\beta}$
$\displaystyle=$
$\displaystyle\frac{1}{L^{2}}\sum_{l=1}^{L}\text{E}[x_{il\alpha}x_{jl\alpha}]\cdot\sum_{\begin{subarray}{c}l^{\prime}=1\\\
l^{\prime}\neq
l\end{subarray}}^{L}\text{E}[x_{kl^{\prime}\beta}x_{rl^{\prime}\beta}]+\frac{1}{L^{2}}\sum_{l=1}^{L}\text{E}[x_{il\alpha}x_{jl\alpha}x_{kl\beta}x_{rl\beta}]-C_{ij\alpha}C_{kr\beta}$
$\displaystyle=$ $\displaystyle\frac{1}{L^{2}}\cdot
LC_{ij\alpha}\cdot(L-1)C_{kr\beta}+\frac{1}{L^{2}}\sum_{l=1}^{L}\text{E}[x_{il\alpha}x_{jl\alpha}x_{kl\beta}x_{rl\beta}]-C_{ij\alpha}C_{kr\beta}$
$\displaystyle=$
$\displaystyle\frac{(L-1)}{L}C_{ij\alpha}C_{kr\beta}+\frac{1}{L^{2}}\sum_{l=1}^{L}\text{E}[x_{il\alpha}x_{jl\alpha}x_{kl\beta}x_{rl\beta}]-C_{ij\alpha}C_{kr\beta}$
$\displaystyle=$
$\displaystyle-\frac{1}{L}C_{ij\alpha}C_{kr\beta}+\frac{1}{L^{2}}\sum_{l=1}^{L}\text{E}[x_{il\alpha}x_{jl\alpha}x_{kl\beta}x_{rl\beta}].$
(S24)

By substituting Eq. (Text C. Derivation of the variance of the total
intralayer weight for a community in a multilayer correlation matrix) into Eq.
(Text C. Derivation of the variance of the total intralayer weight for a
community in a multilayer correlation matrix), using the fact that the
covariance matrices $C_{\alpha}$ and $C_{\beta}$ are independent when
$\beta\neq\alpha$, and using Isserlis’ Theorem [12], we obtain

$\displaystyle\text{Var}\left[\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in S\end{subarray}}^{i-1}C_{ij\alpha}^{\text{con}}\right]=$
$\displaystyle\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in
S\end{subarray}}^{i-1}\sum_{\beta=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}k=1\\\
(k,\beta)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\ (r,\beta)\in
S\end{subarray}}^{k-1}\left(-\frac{1}{L}C_{ij\alpha}C_{kr\beta}+\frac{1}{L^{2}}\sum_{l=1}^{L}\text{E}[x_{il\alpha}x_{jl\alpha}x_{kl\beta}x_{rl\beta}]\right)$
$\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in S\end{subarray}}^{i-1}\sum_{\begin{subarray}{c}\beta=1\\\
\beta\neq\alpha\end{subarray}}^{\mathcal{L}}\sum_{\begin{subarray}{c}k=1\\\
(k,\beta)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\ (r,\beta)\in
S\end{subarray}}^{k-1}\left(-\frac{1}{L}C_{ij\alpha}C_{kr\beta}+\frac{1}{L^{2}}\sum_{l=1}^{L}\text{E}[x_{il\alpha}x_{jl\alpha}x_{kl\beta}x_{rl\beta}]\right)$
$\displaystyle+\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in S\end{subarray}}^{i-1}\sum_{\begin{subarray}{c}k=1\\\
(k,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\
(r,\alpha)\in
S\end{subarray}}^{k-1}\left(-\frac{1}{L}C_{ij\alpha}C_{kr\alpha}+\frac{1}{L^{2}}\sum_{l=1}^{L}\text{E}[x_{il\alpha}x_{jl\alpha}x_{kl\alpha}x_{rl\alpha}]\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{L}\left(\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in
S\end{subarray}}^{i-1}C_{ij\alpha}\right)\left(\sum_{\begin{subarray}{c}\beta=1\\\
\beta\neq\alpha\end{subarray}}^{\mathcal{L}}\sum_{\begin{subarray}{c}k=1\\\
(k,\beta)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\ (r,\beta)\in
S\end{subarray}}^{k-1}C_{kr\beta}\right)$
$\displaystyle+\frac{1}{L^{2}}\sum_{l=1}^{L}\left(\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in
S\end{subarray}}^{i-1}\text{E}[x_{il\alpha}x_{jl\alpha}]\right)\left(\sum_{\begin{subarray}{c}\beta=1\\\
\beta\neq\alpha\end{subarray}}^{\mathcal{L}}\sum_{\begin{subarray}{c}k=1\\\
(k,\beta)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\ (r,\beta)\in
S\end{subarray}}^{k-1}\text{E}[x_{kl\beta}x_{rl\beta}]\right)$
$\displaystyle-\frac{1}{L}\sum_{\alpha=1}^{\mathcal{L}}\left(\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in
S\end{subarray}}^{i-1}C_{ij\alpha}\right)\left(\sum_{\begin{subarray}{c}k=1\\\
(k,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\
(r,\alpha)\in S\end{subarray}}^{k-1}C_{kr\alpha}\right)$
$\displaystyle+\frac{1}{L^{2}}\sum_{l=1}^{L}\left\\{\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in S\end{subarray}}^{i-1}\sum_{\begin{subarray}{c}k=1\\\
(k,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\
(r,\alpha)\in
S\end{subarray}}^{k-1}(\text{E}[x_{il\alpha}x_{jl\alpha}]\cdot\text{E}[x_{kl\alpha}x_{rl\alpha}]\right.$
$\displaystyle\left.+\text{E}[x_{il\alpha}x_{kl\alpha}]\cdot\text{E}[x_{jl\alpha}x_{rl\alpha}]+\text{E}[x_{il\alpha}x_{rl\alpha}]\cdot\text{E}[x_{jl\alpha}x_{kl\alpha}])\right\\}$
$\displaystyle=$
$\displaystyle-\frac{1}{L}\left(\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in
S\end{subarray}}^{i-1}C_{ij\alpha}\right)\left(\sum_{\begin{subarray}{c}\beta=1\\\
\beta\neq\alpha\end{subarray}}^{\mathcal{L}}\sum_{\begin{subarray}{c}k=1\\\
(k,\beta)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\ (r,\beta)\in
S\end{subarray}}^{k-1}C_{kr\beta}\right)$ $\displaystyle+\frac{1}{L^{2}}\cdot
L\left(\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in
S\end{subarray}}^{i-1}C_{ij\alpha}\right)\left(\sum_{\begin{subarray}{c}\beta=1\\\
\beta\neq\alpha\end{subarray}}^{\mathcal{L}}\sum_{\begin{subarray}{c}k=1\\\
(k,\beta)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\ (r,\beta)\in
S\end{subarray}}^{k-1}C_{kr\beta}\right)$
$\displaystyle-\frac{1}{L}\sum_{\alpha=1}^{\mathcal{L}}\left(\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in
S\end{subarray}}^{i-1}C_{ij\alpha}\right)\left(\sum_{\begin{subarray}{c}k=1\\\
(k,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\
(r,\alpha)\in S\end{subarray}}^{k-1}C_{kr\alpha}\right)$
$\displaystyle+\frac{1}{L^{2}}\cdot
L\left[\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in S\end{subarray}}^{i-1}\sum_{\begin{subarray}{c}k=1\\\
(k,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\
(r,\alpha)\in
S\end{subarray}}^{k-1}(C_{ij\alpha}C_{kr\alpha}+C_{ik\alpha}C_{jr\alpha}+C_{ir\alpha}C_{jk\alpha})\right].$
(S25)

By combining the first and third terms in Eq. (Text C. Derivation of the
variance of the total intralayer weight for a community in a multilayer
correlation matrix), we obtain

$\displaystyle\text{Var}\left[\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in S\end{subarray}}^{i-1}C_{ij\alpha}^{\text{con}}\right]=$
$\displaystyle-\frac{1}{L}\left(\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in
S\end{subarray}}^{i-1}C_{ij\alpha}\right)\left(\sum_{\beta=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}k=1\\\
(k,\beta)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\ (r,\beta)\in
S\end{subarray}}^{k-1}C_{kr\beta}\right)$
$\displaystyle+\frac{1}{L}\left(\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in
S\end{subarray}}^{i-1}C_{ij\alpha}\right)\left(\sum_{\begin{subarray}{c}\beta=1\\\
\beta\neq\alpha\end{subarray}}^{\mathcal{L}}\sum_{\begin{subarray}{c}k=1\\\
(k,\beta)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\ (r,\beta)\in
S\end{subarray}}^{k-1}C_{kr\beta}\right)$
$\displaystyle+\frac{1}{L}\left[\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in S\end{subarray}}^{i-1}\sum_{\begin{subarray}{c}k=1\\\
(k,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\
(r,\alpha)\in
S\end{subarray}}^{k-1}(C_{ij\alpha}C_{kr\alpha}+C_{ik\alpha}C_{jr\alpha}+C_{ir\alpha}C_{jk\alpha})\right]$
$\displaystyle=$
$\displaystyle-\frac{1}{L}\left(\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in S\end{subarray}}^{i-1}C_{ij\alpha}\right)^{2}$
$\displaystyle+\frac{1}{L}\left(\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in
S\end{subarray}}^{i-1}C_{ij\alpha}\right)\left(\sum_{\begin{subarray}{c}\beta=1\\\
\beta\neq\alpha\end{subarray}}^{\mathcal{L}}\sum_{\begin{subarray}{c}k=1\\\
(k,\beta)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\ (r,\beta)\in
S\end{subarray}}^{k-1}C_{kr\beta}\right)$
$\displaystyle+\frac{1}{L}\left(\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in S\end{subarray}}^{i-1}\sum_{\begin{subarray}{c}k=1\\\
(k,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\
(r,\alpha)\in S\end{subarray}}^{k-1}C_{ij\alpha}C_{kr\alpha}\right)$
$\displaystyle+\frac{1}{L}\left[\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in S\end{subarray}}^{i-1}\sum_{\begin{subarray}{c}k=1\\\
(k,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\
(r,\alpha)\in
S\end{subarray}}^{k-1}(C_{ik\alpha}C_{jr\alpha}+C_{ir\alpha}C_{jk\alpha})\right]$
$\displaystyle=$
$\displaystyle-\frac{1}{L}\left(\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in
S\end{subarray}}^{i-1}C_{ij\alpha}\right)^{2}+\frac{1}{L}\left(\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in S\end{subarray}}^{i-1}C_{ij\alpha}\right)^{2}$
$\displaystyle+\frac{1}{L}\left[\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in S\end{subarray}}^{i-1}\sum_{\begin{subarray}{c}k=1\\\
(k,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\
(r,\alpha)\in
S\end{subarray}}^{k-1}(C_{ik\alpha}C_{jr\alpha}+C_{ir\alpha}C_{jk\alpha})\right]$
$\displaystyle=$
$\displaystyle\frac{1}{L}\left[\sum_{\alpha=1}^{\mathcal{L}}\sum_{\begin{subarray}{c}i=1\\\
(i,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}j=1\\\
(j,\alpha)\in S\end{subarray}}^{i-1}\sum_{\begin{subarray}{c}k=1\\\
(k,\alpha)\in S\end{subarray}}^{N}\sum_{\begin{subarray}{c}r=1\\\
(r,\alpha)\in
S\end{subarray}}^{k-1}(C_{ik\alpha}C_{jr\alpha}+C_{ir\alpha}C_{jk\alpha})\right].$
(S26)

### Text D. Graphical lasso

For multivariate Gaussian distributions, a zero in the precision matrix (i.e.,
inverse covariance matrix) is equivalent to conditional independence of two
variables, which one can relate to the absence of an edge in the network.
Therefore, it is natural to use the non-zero entries of the estimated
precision matrix to determine the edges [13]. However, when the number of
variables is larger than the number of samples, the empirical covariance
matrix is not full rank. In this case, the empirical covariance matrix is
singular, meaning that its condition number is infinite, so estimating the
precision matrix becomes difficult [14]. The graphical lasso addresses this
problem by regularizing the maximum likelihood estimator with a lasso penalty
enforcing sparsity [15, 16, 14].

Let $\vec{y}$ be a $p$-variate Gaussian random column vector, with
distribution $\mathcal{N}(\mu,C)$, where $\mu$ is the $p$-dimensional mean
vector and $C$ is the $p\times p$ covariance matrix. Given $n$ independently
drawn samples $\\{\vec{y_{1}},\ldots,\vec{y_{n}}\\}$ of this random vector,
the sample covariance matrix can be written as

$\hat{C}=\frac{1}{n-1}\sum_{k=1}^{n}(\vec{y_{k}}-\hat{\mu})(\vec{y_{k}}-\hat{\mu})^{\top},$
(S27)

where $\hat{\mu}=\frac{1}{n}\sum_{k=1}^{n}\vec{y_{k}}$, and ⊤ represents the
transposition. Let the inverse covariance matrix be denoted as
$C^{-1}=\Theta$. We consider a generalized $\ell_{1}$ regularization given by
$\lambda\sum_{i=1}^{p}\sum_{j=1}^{i-1}|\Theta_{ij}|$, where $\lambda$ is the
penalizing parameter. Then, the problem is to maximize the lasso regularized
log-likelihood to obtain the graphical lasso estimator, i.e.,

$\Theta^{*}=\text{arg}\min_{\Theta\succ
0}\left\\{-\log\det\Theta+\text{tr}(\hat{C}\Theta)+\lambda\sum_{i=1}^{p}\sum_{j=1}^{i-1}|\Theta_{ij}|\right\\},$
(S28)

where $\Theta\succ 0$ signifies that $\Theta$ is a positive definite matrix
[14].

Using the TPM data for each of the four tissues separately, we first
calculated the $203\times 203$ empirical covariance matrix. Then, we applied
the GraphicalLassoCV function from the Python package scikit-learn version
1.0.2 [17, 18] to estimate a precision matrix, which is a sparsified co-
expression network, for each tissue. This graphical lasso algorithm
incorporates a cross-validated choice of the $\ell_{1}$ penalty. To simplify
analysis, we regard the generated networks as unsigned and unweighted network.
The results for the unsigned weighted networks are similar to those for the
unsigned unweighted networks, as we will show.

### Text E. Z scores for the average distance between pairs of genes on each
chromosome separately in each community

We show in Table B the Z scores for the average distance between pairs of
genes on each chromosome and each community with $\gamma=1$. We show the
corresponding results with $\gamma=3$ in Table C. We only calculated the Z
scores for the chromosome-community pairs with at least 3 genes. In these
tables, N/A implies that either there are less than 3 genes, or the standard
deviation of the average distance is equal to 0 because every random selection
of genes is the same gene set. The latter event occurs when all the genes on
that chromosome are associated with the same community.

Table B: Z scores for the average distance between pairs of genes on each
chromosome and each significant community detected with $\gamma=1$. Comm.
denotes community, and Chr denotes chromosome. $\gamma=1$
---
| Comm.
Chr | 1 | 2 | 3 | 4 | 5
1 | $-2.093$ | $-1.526$ | $-0.122$ | $-1.466$ | $1.272$
2 | N/A | $-0.767$ | $0.061$ | $2.129$ | $-0.725$
3 | N/A | N/A | $0.113$ | $1.101$ | N/A
4 | N/A | N/A | N/A | $-0.110$ | $-2.573$
5 | N/A | N/A | N/A | N/A | N/A
6 | $0.606$ | N/A | N/A | $0.564$ | $-0.065$
7 | N/A | $-1.938$ | $-1.842$ | $0.968$ | $-0.302$
8 | N/A | N/A | N/A | N/A | N/A
9 | N/A | N/A | N/A | $-0.722$ | N/A
10 | N/A | $-2.336$ | $0.413$ | N/A | N/A
11 | N/A | N/A | $-2.156$ | $0.077$ | $0.454$
12 | N/A | N/A | $-1.376$ | $-0.905$ | $-0.260$
13 | N/A | N/A | N/A | N/A | N/A
14 | N/A | N/A | $0.438$ | N/A | $0.180$
15 | N/A | N/A | N/A | N/A | N/A
16 | $-0.091$ | $0.737$ | $0.100$ | N/A | N/A
17 | $1.535$ | N/A | $-2.160$ | $-0.579$ | $-0.787$
18 | N/A | N/A | N/A | N/A | N/A
19 | $0.235$ | N/A | $-0.521$ | $1.592$ | $1.894$
20 | N/A | N/A | N/A | N/A | $0.791$
21 | N/A | N/A | N/A | N/A | N/A
22 | N/A | N/A | N/A | N/A | $-1.780$
X | N/A | N/A | N/A | N/A | N/A
Y | N/A | N/A | N/A | N/A | N/A
M | N/A | N/A | N/A | N/A | $-2.061$
Table C: Z scores for the average distance between pairs of genes on each
chromosome and each significant community detected with $\gamma=3$. Comm.
denotes community, and Chr denotes chromosome. $\gamma=3$
---
| Comm.
Chr | 1 | 2 | 3 | 4 | 5 | 6 | 7
1 | $-2.808$ | $0.116$ | $-0.447$ | N/A | $-0.482$ | N/A | N/A
2 | $-0.671$ | N/A | N/A | N/A | $2.094$ | $-2.220$ | N/A
3 | N/A | N/A | N/A | N/A | N/A | N/A | N/A
4 | N/A | N/A | N/A | N/A | $-1.993$ | N/A | N/A
5 | N/A | N/A | N/A | N/A | N/A | N/A | N/A
6 | N/A | $0.573$ | N/A | N/A | N/A | N/A | N/A
7 | $-1.643$ | N/A | N/A | N/A | $-2.616$ | N/A | N/A
8 | N/A | N/A | N/A | N/A | N/A | N/A | N/A
9 | N/A | N/A | N/A | N/A | N/A | N/A | N/A
10 | N/A | N/A | N/A | N/A | $0.293$ | N/A | N/A
11 | N/A | N/A | N/A | N/A | $-2.902$ | N/A | N/A
12 | N/A | $1.729$ | N/A | N/A | $-0.577$ | N/A | N/A
13 | N/A | N/A | N/A | N/A | N/A | N/A | N/A
14 | N/A | N/A | N/A | N/A | N/A | $-1.756$ | N/A
15 | N/A | N/A | N/A | N/A | N/A | N/A | N/A
16 | $-0.416$ | $0.920$ | N/A | N/A | $1.016$ | N/A | N/A
17 | N/A | $1.639$ | N/A | N/A | $-2.175$ | N/A | N/A
18 | N/A | N/A | N/A | N/A | N/A | N/A | N/A
19 | N/A | $-1.142$ | N/A | N/A | $-0.179$ | N/A | N/A
20 | N/A | N/A | N/A | N/A | N/A | N/A | N/A
21 | N/A | N/A | N/A | N/A | N/A | N/A | N/A
22 | N/A | N/A | N/A | N/A | N/A | N/A | N/A
X | N/A | N/A | N/A | N/A | N/A | N/A | N/A
Y | N/A | N/A | N/A | N/A | N/A | N/A | N/A
M | N/A | N/A | N/A | N/A | N/A | N/A | N/A

### Text F. Results of the gene set enrichment analysis

We show the two most significant GO:BP and HP results from g:Profiler for the
top 50 highly expressed genes out of the $203$ genes in the network in each
tissue in Table D. See [19] for the entire output from g:Profiler, i.e., the
list of all significant GO:BP and HP results, for the top 50 genes in
pancreas; see [20] for the top 50 genes in salivary gland; see [21] for the
top 50 genes in mammary gland; and see [22] for the top 50 genes in skin.

We show the two most significant GO:BP and HP results from g:Profiler for each
community in the partition of the multilayer correlation matrix with
$\gamma=3$ in Table E. See [23] for the entire output from g:Profiler, i.e.,
the list of all significant GO:BP and HP results, for the genes in community
1; see [24] for community 2; see [25] for community 3; see [26] for community
4; see [27] for community 5; see [28] for community 6; and see [29] for
community 7.

We show the two most significant GO:BP and HP results from g:Profiler for the
top 50 highly connected genes (i.e., top 50 hub genes) out of the $203$ genes
in the network in each tissue in Table F. See [30] for the entire output from
g:Profiler, i.e., the list of all significant GO:BP and HP results, for the
top 50 genes in pancreas; see [31] for the top 50 genes in salivary gland; see
[32] for the top 50 genes in mammary gland; and see [33] for the top 50 genes
in skin.

Table D: Results of the gene set enrichment analysis for the top 50 highly expressed genes out of the $203$ genes in the network in each tissue. Tissue |  Top significant terms from GO:BP |  $p$ value |  Top significant terms from HP |  $p$ value
---|---|---|---|---
pancreas | oxidative phosphorylation | $6.39\cdot 10^{-15}$ | recurrent pancreatitis | $1.44\cdot 10^{-22}$
aerobic electron transport chain | $4.69\cdot 10^{-14}$ | mitochondrial inheritance | $1.79\cdot 10^{-20}$
salivary gland | oxidative phosphorylation | $7.02\cdot 10^{-16}$ | mitochondrial inheritance | $6.33\cdot 10^{-21}$
aerobic electron transport chain | $7.37\cdot 10^{-15}$ | centrocecal scotoma | $3.23\cdot 10^{-20}$
mammary gland | oxidative phosphorylation | $6.35\cdot 10^{-15}$ | mitochondrial inheritance | $9.67\cdot 10^{-20}$
mitochondrial ATP synthesis coupled electron transport | $4.77\cdot 10^{-14}$ | centrocecal scotoma | $3.63\cdot 10^{-19}$
skin | oxidative phosphorylation | $7.28\cdot 10^{-15}$ | mitochondrial inheritance | $5.06\cdot 10^{-19}$
aerobic electron transport chain | $5.35\cdot 10^{-14}$ | centrocecal scotoma | $1.59\cdot 10^{-18}$
Table E: Results of the gene set enrichment analysis for the communities of the multilayer correlation matrix with $\gamma=3$. Comm. denotes community. Comm. |  Top significant terms from GO:BP |  $p$ value |  Top significant terms from HP |  $p$ value
---|---|---|---|---
1 | oxidative phosphorylation | $5.90\cdot 10^{-11}$ | recurrent pancreatitis | $1.78\cdot 10^{-19}$
aerobic electron transport chain | $1.05\cdot 10^{-10}$ | mitochondrial inheritance | $1.06\cdot 10^{-17}$
2 | keratinocyte differentiation | $8.26\cdot 10^{-25}$ | palmoplantar keratoderma | $3.05\cdot 10^{-7}$
epidermis development | $1.90\cdot 10^{-24}$ | hyperkeratosis | $3.05\cdot 10^{-7}$
3 | retina homeostasis | $2.52\cdot 10^{-10}$ | leber optic atophy | $6.65\cdot 10^{-15}$
oxidative phosphorylation | $2.23\cdot 10^{-9}$ | mitochondrial inheritance | $6.65\cdot 10^{-15}$
4 | skin development | $3.23\cdot 10^{-13}$ | alopecia | $8.30\cdot 10^{-5}$
intermediate filament organization | $2.64\cdot 10^{-12}$ | nail dystrophy | $8.30\cdot 10^{-5}$
5 | keratinization | $1.95\cdot 10^{-19}$ | palmoplantar blistering | $1.71\cdot 10^{-6}$
epidermis development | $1.95\cdot 10^{-19}$ | palmoplantar keratoderma | $2.02\cdot 10^{-5}$
6 | positive regulation of respiratory burst | $5.36\cdot 10^{-8}$ | N/A | N/A
regulation of respiratory burst | $2.60\cdot 10^{-7}$ | N/A | N/A
7 | adaptive thermogenesis | $1.73\cdot 10^{-2}$ | N/A | N/A
fatty acid biosynthesis process | $1.73\cdot 10^{-2}$ | N/A | N/A
Table F: Results of the gene set enrichment analysis for the top 50 highly connected genes out of the $203$ genes in the single-layer network of each tissue. Tissue |  Top significant terms from GO:BP |  $p$ value |  Top significant terms from HP |  $p$ value
---|---|---|---|---
pancreas | cytoplasmic translation | $1.68\cdot 10^{-10}$ | mutism | $1.77\cdot 10^{-3}$
sequestering of metal ion | $2.49\cdot 10^{-8}$ | arterial rupture | $1.93\cdot 10^{-3}$
salivary gland | digestion | $3.05\cdot 10^{-8}$ | pancreatic calcification | $5.28\cdot 10^{-9}$
cytoplasmic translation | $4.52\cdot 10^{-5}$ | pancreatic pseudocyst | $5.66\cdot 10^{-8}$
mammary gland | antibacterial humoral response | $3.23\cdot 10^{-10}$ | nail dystrophy | $9.67\cdot 10^{-5}$
defense response to bacterium | $3.36\cdot 10^{-8}$ | palmoplantar blistering | $4.37\cdot 10^{-4}$
skin | digestion | $1.59\cdot 10^{-8}$ | pancreatic calcification | $4.68\cdot 10^{-6}$
proteolysis | $4.89\cdot 10^{-6}$ | recurrent pancreatitis | $4.82\cdot 10^{-5}$

### Text G. Tissue-specific hub genes versus gene communities

To compare the overlap of the top 50 highly connected genes in each tissue and
the gene communities identified by our algorithm with $\gamma=3$, we calculate
the Jaccard index for each pair of the set of the 50 most connected genes in
one of the four tissues and one of the seven gene communities identified by
our algorithm. The Jaccard index is equal to $1$ if the two sets perfectly
overlap and $0$ if the two sets are disjoint. We show the $4\times 7=28$
Jaccard index values in Fig B. The largest Jaccard index is $0.324$, revealing
the lack of notable similarity between all of the $28$ pairs of gene sets.
Therefore, we conclude that our multilayer community detection method uncovers
sets of genes that are different from top hub genes in each layer.

Figure B: Jaccard index between the set of tissue-specific hub genes and the
set of genes in a community. Each row corresponds to the top 50 hub genes in
each layer (i.e., tissue), where “panc” denotes pancreas, “sal” denotes
salivary gland, “mamm” denotes mammary gland, and “skin” denotes skin (not sun
exposed). Each column corresponds to a community identified with $\gamma=3$.

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