perform a multiplex analysis on the disaggregated, commodity-specific layers of the WTW.

Analysis of the aggregated network {#wtwagg}
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For simplicity of the notation, in this subsection we indicate with $A$ the adjacency matrix and with $W$ the weighted matrix representing the aggregate network in year 2002, i.e. $w_{ij}\equiv w_{ij}^{AGG}(2002)$. In general, we can relate the entries of the matrices $A$ and $W$ through the zeroth power using the notation $w^0_{ij}=a_{ij}$, where we conventionally define $0^0\equiv0$. This notation will be useful later to calculate the expected values of various structural properties. It also allows us to express the properties of purely topological properties, which in principle depend on the adjacency matrix $A$, as functions of the matrix $W$.

We are interested in assessing to what extent the ECM is able to replicate the higher-order properties characterizing the WTW over time. Therefore we first apply the ECM to the data, thus finding the vectors $\vec{x}^*$ and $\vec{y}^*$, and then we use these vectors to calculate the expected values of the chosen higher-order properties. For consistency with refs. [@Squartini_etal_2011a_pre; @Squartini_etal_2011b_pre], we focus on the Average Nearest Neighbor Degree, the Average Nearest Neighbor Strength (denoted by $k^{nn}_i$ and $s^{nn}_i$ respectively), the Binary Clustering 