  The Eden Model in $\mathbb{R}^n$ constructs a blob as follows: initially a
single unit hypercube is infected, and each second a hypercube adjacent to the
infected ones is selected randomly and infected. Manin, Rold\'{a}n, and
Schweinhart investigated the topology of the Eden model in $\mathbb{R}^{n}$ by
considering the possible shapes which can appear on the boundary. In
particular, they give probabilistic lower bounds on the Betti numbers of the
Eden model. In this paper, we prove analogous results for the Eden model on any
infinite, vertex-transitive, locally finite graph: with high probability as
time goes to infinity, every "possible" subgraph (with mild conditions on what
"possible" means) occurs on the boundary of the Eden model at least a number of
times proportional to an isoperimetric profile of the graph. Using this, we can
extend the results about the topology of the Eden model to non-Euclidean
spaces, such as hyperbolic $n$-space and universal covers of certain Riemannian
manifolds.
