then take the product of $A_3$, weighted by $\alpha_3$, with the resulting graph. The final result is $$\label{eq:explicit_kron}
  \alpha_3A_3\otimes D_2\otimes D_1 + \alpha_2{\mathbbm{1}}_3\otimes A_2\otimes D_1 + \alpha_1{\mathbbm{1}}_3\otimes{\mathbbm{1}}_2\otimes A_1.$$ In fact, a $k$-fold product, when evaluated this way, matches the right hand side of \[eq:hierarchysum\]. Therefore, the $k$-level weighted hierarchy can also be written unambiguously as $$\label{eq:hpcomposition}
A^{{\mathbin{\Pi_{\vec{\alpha}}}}k} = A_k \mathbin{\Pi_{\alpha_{k}}} A_{k-1} \mathbin{\Pi_{\alpha_{k-1}}} \cdots \mathbin{\Pi_{\alpha_{2}}} (\alpha_1 A_1).$$ Henceforth, the weight $\alpha_1$, which scales the lowest-level adjacency matrix $A_1$, will be dropped due to our normalization choice of $\alpha_1 = 1$.

An important class of hierarchy graphs is one where the level weights follow a geometric progression of weights, i.e., $\alpha_i = \alpha^{i-1}$. We will denote such hierarchies by $G^{{\mathbin{\Pi_\alpha}}k}$, where the scalar subscript $\alpha$ will be understood to mean the mutual weighting between successive hierarchies. For $\alpha > 1$, this leads to a “fat tree” structure, while for $\alpha< 1$, we instead get a “skinny tree” for which the edge weights decrease between consecutive levels from the leaves to the root. These constructions are illustrated in Fig. \[fig:trees\], and mentioned because fat trees are known to be a commonly used architecture in classical networks [@Leiserson1985].

![An illustration of the use of 