*does not* satisfy many properties which are commonly assumed for operations on matrices. In particular,

1.  Bilinearity: ${\left(A_1+A_2\right)}{\mathbin{\Pi}}B = A_1\otimes D_B + A_2\otimes D_B + {\mathbbm{1}}_{(A_1+A_2)}\otimes B \neq A_1{\mathbin{\Pi}}B + A_2{\mathbin{\Pi}}B$. Similarly, $A{\mathbin{\Pi}}{\left(B_1+B_2\right)}\neq A{\mathbin{\Pi}}B_1 + A{\mathbin{\Pi}}B_2$.

2.  Scalar multiplication: For any scalar $\alpha$, ${\left(\alpha A\right)}{\mathbin{\Pi}}B = \alpha A\otimes D_B + {\mathbbm{1}}_A\otimes B \neq \alpha{\left(A{\mathbin{\Pi}}B\right)} \neq A{\mathbin{\Pi}}{\left(\alpha B\right)}$. Note however that scalar multiplication is distributive in the following way: $\alpha{\left(A{\mathbin{\Pi}}B\right)} = {\left(\alpha A\right)}{\mathbin{\Pi}}{\left(\alpha B\right)}$.

Hierarchical graphs are also instances of hyperbolic graphs. The Gromov-hyperbolicity [@Gromov2007], which measures curvature and is small for a graph with large negative curvature, is only a constant for hierarchical graphs. Since the hyperbolicity in general is at most half the graph diameter, whereas in this case it is independent of the diameter, it is termed *constantly hyperbolic* in the parlance of Ref. [@Chen2013]. Hyperbolic graphs are seen in several real-world complex networks [@Lohsoonthorn2003; @Krioukov2010], most notably the internet [@Montgolfier2011; @Boguna2010]. Hyperbolic lattices have also been realized recently in superconducting circuits [@Kollar2018].

Finally, hierarchies have low tree-, clique- and rank-widths, which are each measures of the decomposibility of a graph [@OUM200579]. These structural properties imply efficient algorithms for optimization problems expressible in monadic second-order (MSO) logic – 