kinds of weights to its edges, and each type of weight might scale differently. For now, we define a generalization of the hierarchical product which will allow us to construct hierarchies that incorporate different weights at different levels of the hierarchy.

\[def:hp\] Given graphs $G$ and $H$, and $\alpha\in\mathbb{R_+}$, the *$\alpha$-weighted hierarchical product* $P = G{\mathbin{\Pi_\alpha}}H$ is a graph on vertices $V_P = V_G\times V_H$ and edges $E_P \subseteq V_P\times V_P$ specified by the adjacency matrix $$A_P = \alpha A_G \otimes D_H + {\mathbbm{1}}_G \otimes A_H,$$ or, equivalently, by the algebraic Laplacian $$L_P = \alpha L_G \otimes D_H + {\mathbbm{1}}_G \otimes L_H .$$ We will often use the shorthand $A_P = A_G{\mathbin{\Pi_\alpha}}A_H$, and $L_P = L_G{\mathbin{\Pi_\alpha}}L_H$.

As before, we may construct a $k$-level, *weighted* hierarchy out of $k$ base graphs $G_1, \ldots, G_k$, and $k$ weights $\alpha_i,\ldots, \alpha_k\equiv \vec{\alpha}$, so that the edges of the $i$-th level graph $G_i$ are weighted by the $i$-th component of $\vec{\alpha}$, $\alpha_i$. The adjacency matrix of such a hierarchy may be written as $$\label{eq:hierarchysum}
A^{{\mathbin{\Pi_{\vec{\alpha}}}}k} := {\sum\limits_{i=1}^{k}\alpha_i {\mathbbm{1}}_{[i+1 {\mathrel{{.}\,{.}}\nobreak}k] }\otimes A_i \otimes D_{[1 {\mathrel{{.}\,{.}}\nobreak}i-1]}},$$ where the subscripts $[a{\mathrel{{.}\,{.}}\nobreak}b]$ on ${\mathbbm{1}}$ and $D$ are shorthand for the Kronecker product of matrices over all descending indices 