  Projected Entangled Pair States (PEPS) are a class of quantum many-body
states that generalize Matrix Product States for one-dimensional systems to
higher dimensions. In recent years, PEPS have advanced understanding of
strongly correlated systems, especially in two dimensions, e.g., quantum spin
liquids. Typically described by tensor networks on regular lattices (e.g.,
square, cubic), PEPS have also been adapted for irregular graphs, however, the
computational cost becomes prohibitive for dense graphs with large vertex
degrees. In this paper, we present a PEPS algorithm to simulate low-energy
states and dynamics defined on arbitrary, fluctuating, and densely connected
graphs. We introduce a cut-off, $\kappa \in \mathbb{N}$, to constrain the
vertex degree of the PEPS to a set but tunable value, which is enforced in the
optimization by applying a simple edge-deletion rule, allowing the geometry of
the PEPS to change and adapt dynamically to the system's correlation structure.
We benchmark our flexible PEPS algorithm with simulations of classical spin
glasses and quantum annealing on densely connected graphs with hundreds of
spins, and also study the impact of tuning $\kappa$ when simulating a uniform
quantum spin model on a regular (square) lattice. Our work opens the way to
apply tensor network algorithms to arbitrary, even fluctuating, background
geometries.
