Graph information ratio

We introduce the notion of information ratio Ir(H/G) between two (simple, undirected) graphs G and H, which characterizes the maximal number of source symbols per channel use that can be reliably sent over a channel with confusion graph H, where reliability is measured w.r.t. a source confusion graph G. Many different results are provided, including in particular lower and upper bounds on Ir(H/G) in terms of various graph properties, inequalities and identities for behavior under strong product and disjoint union, relations to graph cores, and notions of graph criticality. Informally speaking, 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 intuitions are discussed.