  We formulate a Euclidean theory of edge length dynamics based on a notion of
Ricci curvature on graphs with variable edge lengths. In order to write an
explicit form for the discrete analog of the Einstein-Hilbert action, we
require that the graph should either be a tree or that all its cycles should be
sufficiently long. The infinite regular tree with all edge lengths equal is an
example of a graph with constant negative curvature, providing a connection
with $p$-adic AdS/CFT, where such a tree takes the place of anti-de Sitter
space. We compute simple correlators of the operator holographically dual to
edge length fluctuations. This operator has dimension equal to the dimension of
the boundary, and it has some features in common with the stress tensor.
