  A graph profile records all possible densities of a fixed finite set of
graphs. Profiles can be extremely complicated; for instance the full profile of
any triple of connected graphs is not known, and little is known about
hypergraph profiles. We introduce the tropicalization of graph and hypergraph
profiles. Tropicalization is a well-studied operation in algebraic geometry,
which replaces a variety (the set of real or complex solutions to a finite set
of algebraic equations) with its "combinatorial shadow". We prove that the
tropicalization of a graph profile is a closed convex cone, which still
captures interesting combinatorial information. We explicitly compute these
tropicalizations for arbitrary sets of complete and star hypergraphs. We show
they are rational polyhedral cones even though the corresponding profiles are
not even known to be semialgebraic in some of these cases. We then use
tropicalization to prove strong restrictions on the power of the sums of
squares method, equivalently Cauchy-Schwarz calculus, to test (which is weaker
than certification) the validity of graph density inequalities. In particular,
we show that sums of squares cannot test simple binomial graph density
inequalities, or even their approximations. Small concrete examples of such
inequalities are presented, and include the famous Blakley-Roy inequalities for
paths of odd length. As a consequence, these simple inequalities cannot be
written as a rational sum of squares of graph densities.
