  We write the Euler characteristic X(G) of a four dimensional finite simple
geometric graph G=(V,E) in terms of the Euler characteristic X(G(w)) of
two-dimensional geometric subgraphs G(w). The Euler curvature K(x) of a four
dimensional graph satisfying the Gauss-Bonnet relation sum_x K(x) = X(G) can so
be rewritten as an average 1-E[K(x,f)]/2 over a collection two dimensional
"sectional graph curvatures" K(x,f) through x. Since scalar curvature, the
average of all these two dimensional curvatures through a point, is the
integrand of the Hilbert action, the integer 2-2 X(G) becomes an
integral-geometrically defined Hilbert action functional. The result has an
interpretation in the continuum for compact 4-manifolds M: the Euler curvature
K(x), the integrand in the classical Gauss-Bonnet-Chern theorem, can be seen as
an average over a probability space W of 1-K(x,w)/2 with curvatures K(x,w) of
compact 2-manifolds M(w). Also here, the Euler characteristic has an
interpretation of an exotic Hilbert action, in which sectional curvatures are
replaced by surface curvatures of integral geometrically defined random
two-dimensional sub-manifolds M(w) of M.
