its binary counterpart ($k_i^{nn}$), ${s}^{nn}_i$ reveals the ‘intensity’ of connectiviy of the neighbours of a node, now taking edge weights into account.

Finally, the Weighted Clustering Coefficient [@Fagiolo2007pre] can be defined as $${c}_i^w(W)=\frac{\sum_{j\neq i}\sum_{k\ne i,j}({w}_{ij}{w}_{jk}{w}_{ki})^{1/3}}{\sum_{j\neq i}\sum_{k\ne i,j}w^0_{ij}w^0_{ki}}
\label{WCC}$$ and measures the propensity of node $i$ to be involved in triangular relations, taking into account the weights of such relations.

In order to compute the expected value of the above properties, it is necessary to compute the expected product of (powers of) distinct matrix entries. The independence of pairs of nodes in the ECM [@mastrandrea2013enhanced] ensures that $$\left\langle \sum_{i\ne j\ne k,\dots}w_{ij}^{\alpha}\cdot w_{jk}^{\beta}\cdot\:\dots\right\rangle=\sum_{i\ne j\ne k,\dots}\langle w_{ij}^{\alpha}\rangle \cdot\langle w_{jk}^{\beta}\rangle\cdot\:\langle\dots\rangle.
%\label{eq:product}
\nonumber$$ Each individual term in the product is given by $$\langle w_{ij}^{\gamma}\rangle=\sum_{w=0}^{+\infty}\! w^{\gamma}q_{ij}(w|\vec{x}^*,\vec{y}^*)=\frac{x^*_{i}x^*_{j}(1-y^*_{i}y^*_{j})\mbox{Li}_{-\gamma}(y^*_{i}y^*_{j})}{1-y^*_{i}y^*_{j}+x^*_{i}x^*_{j}y^*_{i}y^*_{j}}
%\label{eq:li}
\nonumber$$ where $\mbox{Li}_{n}(z)\equiv\sum_{l=1}^{+\infty}z^l/l^n$ is the $n$th polylogarithm of $z$ [@mastrandrea2013enhanced]. The simplest cases $\gamma =1$ and $\gamma=0$ yield the expected weight $$\langle w_{ij} \rangle\equiv \frac{x^*_i x^*_j y^*_i y^*_j }{(1-y^*_i y^*_j)(1-y^*_i y^*_j+x^*_i x^*_jy^*_iy^*_j)}$$ and the connection probability $$p_{ij}\equiv \langle w^0_{ij}\rangle=\frac{x^*_i x^*_j y^*_i y^*_j }{1-y^*_i y^*_j+x^*_i x^*_jy^*_iy^*_j}
\label{pij}$$ respectively [@mastrandrea2013enhanced].

As a result, the expected values of the purely topological (weight-independent) properties can be obtained by simply replacing $w^0_{ij}=a_{ij}$ with $p_{ij}$ [@mastrandrea2013enhanced]: $$\label{ANNDe}
\langle k^{nn}_i(W) \rangle \equiv\frac{\sum_{j\ne i}p_{ij} \langle k_j \rangle}{\langle 