  The facilitation of Rydberg excitations in a gas of atoms provides an ideal
model system to study epidemic evolution on (dynamic) networks and self
organization of complex systems to the critical point of a non-equilibrium
phase transition. Using Monte-Carlo simulations and a machine learning
algorithm we show that the universality class of this phase transition can be
tuned. The classes include directed percolation (DP), the most common class in
short-range spreading models, and mean-field (MF) behavior, but also different
types of anomalous directed percolation (ADP), characterized by rare long-range
excitation processes. In a frozen gas, ground state atoms that can facilitate
each other form a static network, for which we predict DP universality. Atomic
motion then turns the network into a dynamic one with long-range (Levy-flight
type) excitations. This leads to continuously varying critical exponents
corresponding to the ADP universality class, eventually reaching MF behavior.
These findings also explain the recently observed critical exponent of Rydberg
facilitation in an ultra-cold gas experiment [Helmrich et al., Nature 577, 481
(2020)], which was in between DP and MF values.
