  We introduce the notion of information ratio $\text{Ir}(H/G)$ between two
(simple, undirected) graphs $G$ and $H$, defined as the supremum of ratios
$k/n$ such that there exists a mapping between the strong products $G^k$ to
$H^n$ that preserves non-adjacency. Operationally speaking, the information
ratio is the maximal number of source symbols per channel use that can be
reliably sent over a channel with a confusion graph $H$, where reliability is
measured w.r.t. a source confusion graph $G$. Various results are provided,
including in particular lower and upper bounds on $\text{Ir}(H/G)$ in terms of
different graph properties, inequalities and identities for behavior under
strong product and disjoint union, relations to graph cores, and notions of
graph criticality. Informally speaking, $\text{Ir}(H/G)$ can be interpreted as
a measure of similarity between $G$ and $H$. We make this notion precise by
introducing the concept of information equivalence between graphs, a more
quantitative version of homomorphic equivalence. We then describe a natural
partial ordering over the space of information equivalence classes, and endow
it with a suitable metric structure that is contractive under the strong
product. Various examples and open problems are discussed.
