Coefficient and the Weighted Clustering Coefficient (denoted by $c_i$ and $c^w_i$ respectively). In what follows, we first recall the analytical expressions of these higher-order quantities. Then, for the sake of clarity we write down the explicit expressions for the expected value of the same quantities under the ECM, i.e. the particular form taken by eq. for each property under study.

The Average Nearest Neighbor Degree is defined as $$\label{ANND}
k^{nn}_i(W)\equiv\frac{\sum_{j\ne i}w^0_{ij}k_j}{k_i}=\frac{\sum_{j\ne i}\sum_{k\ne j}w^0_{ij}w^0_{jk}}{\sum_{j\ne i} w^0_{ij}},$$ where $k_i=\sum_{j\ne i} w^0_{ij}$ stands for the $i$-th node degree, and represents the average of the degrees of the partners of node $i$, i.e. it provides a measure of the connectivity of the neighbours of that node.

The Binary Clustering Coefficient has the expression $$\label{BCC}
c_i(W) \equiv\frac{\sum_{j\neq i}\sum_{k\ne i,j}w^0_{ij}w^0_{jk}w^0_{ki}}{\sum_{j\neq i}\sum_{k\ne i,j}w^0_{ij}w^0_{ki}}$$ and measures the tendency of node $i$ to form triangles, i.e. it counts how many closed triangles are attached to node $i$, divided by the maximum number of triangles achievable by a node with degree $k_i$.

The Average Nearest Neighbor Strength is defined as $${s}^{nn}_i(W)\equiv\frac{\sum_{j\ne i}w^0_{ij}{s}_j}{k_i}=\frac{\sum_{j\ne i}\sum_{k\ne j}w^0_{ij}w_{jk}}{\sum_{j\ne i} w^0_{ij}},
\label{ANNS}$$ where $s_i=\sum_{j\ne i} w_{ij}$ stands for the $i$-th node strength, and measures the average strength of the neighbors of vertex $i$. Similarly to 