in the integer interval $[a{\mathrel{{.}\,{.}}\nobreak}b]$. For instance, $D_{[1{\mathrel{{.}\,{.}}\nobreak}i-1]} := D_{G_{i-1}}\otimes D_{G_{i-2}}\otimes \cdots \otimes D_{G_1}$.

Defined as above, a weighted hierarchy $G^{{\mathbin{\Pi_{\vec{\alpha}}}}k}$ is uniquely and efficiently specified by a real vector of weights $\vec{\alpha}\in{\mathbbm{R_+}}^k$ and an ordered tuple of graphs ${\left(G_1,\ldots, G_k\right)}$. It will be the case that our analyses are unaffected by an overall scaling of the weight vector, so that one may identify $\vec{\alpha} \equiv c\vec{\alpha}$ for any real scalar $c$. As convention, we will always normalize by setting $\alpha_1=1$, which corresponds to assigning a unit-weight multiplicative factor to the lowest-level graphs in the hierarchy.

We can construct the adjacency matrix of the graph $G^{{\mathbin{\Pi_{\vec{\alpha}}}}k}$ by repeated application of the two-fold product (Def. \[def:hp\]) in some well-defined way, analogous to \[eq:unwHPcomp\]. However, unlike before, the weighted product is non-associative, so we must first define an order of operations for manifold weighted products. Unless otherwise specified, we will always evaluate a manifold product from *right to left*, which corresponds to building the hierarchies from the bottom up, and is required in order to ensure that this definition matches \[eq:hierarchysum\]. For example, in the 3-fold product $A_3{\mathbin{\Pi}}_{\alpha_3} A_2 {\mathbin{\Pi}}_{\alpha_2} (\alpha_1A_1)$, we will first evaluate the product $A_2 {\mathbin{\Pi}}_{\alpha_2} (\alpha_1 A_1)$, and 