  Complex natural and technological systems can be considered, on a
coarse-grained level, as assemblies of elementary components: for example,
genomes as sets of genes, or texts as sets of words. On one hand, the joint
occurrence of components emerges from architectural and specific constraints in
such systems. On the other hand, general regularities may unify different
systems, such as the broadly studied Zipf and Heaps laws, respectively
concerning the distribution of component frequencies and their number as a
function of system size. Dependency structures (i.e., directed networks
encoding the dependency relations between the components in a system) were
proposed recently as a possible organizing principles underlying some of the
regularities observed. However, the consequences of this assumption were
explored only in binary component systems, where solely the presence or absence
of components is considered, and multiple copies of the same component are not
allowed. Here, we consider a simple model that generates, from a given ensemble
of dependency structures, a statistical ensemble of sets of components,
allowing for components to appear with any multiplicity. Our model is a minimal
extension that is memoryless, and therefore accessible to analytical
calculations. A mean-field analytical approach (analogous to the "Zipfian
ensemble" in the linguistics literature) captures the relevant laws describing
the component statistics as we show by comparison with numerical computations.
In particular, we recover a power-law Zipf rank plot, with a set of core
components, and a Heaps law displaying three consecutive regimes (linear,
sub-linear and saturating) that we characterize quantitatively.
