We characterize the rank of edge connection matrices of partition functions of real vertex models, as the dimension of the homogeneous components of the algebra of @math -invariant tensors. Here @math is the sub- group of the real orthogonal group that stabilizes the vertex model. This answers a question of Bal 'azs Szegedy from 2007.
It is shown that a graph parameter can be realized as the number of homomorphisms into a fixed (weighted) graph if and only if it satisfies two linear algebraic conditions: reflection positivity and exponential rank-connectivity. In terms of statistical physics, this can be viewed as a characterization of partition functions of vertex models.