Zipf and Heaps laws from dependency structures in component systems

Abstract

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, sublinear, and saturating) that we characterize quantitatively.

Heaps' law, statistics of shared components, and temporal patterns from a sample-space-reducing process

“…Similarly, the theoretical relation that is often used in linguistics to connect Zipf's law and Heaps' law is based on an equivalent random sampling framework [33][34][35][36]. Interestingly, also when these statistical patterns are generated with more complex models explicitly based on networks of component dependencies [23,24], thus with a strong intrinsic correlation structure, they do not significantly deviate from the random sampling prediction [24]. This surprising phenomenology suggests that average statistical laws, such as Zipf's and Heaps' laws, do not contain enough information about the microscopic dynamics to clearly distinguish between alternative generative mechanisms.…”

“…This surprising phenomenology suggests that average statistical laws, such as Zipf's and Heaps' laws, do not contain enough information about the microscopic dynamics to clearly distinguish between alternative generative mechanisms. High-order statistical observables, such as two-point correlations between components [24] or fluctuation scalings [52] could thus be necessary to actually select the more appropriate model for a given empirical component system.…”

“…Without invoking criticality, the random-group-formation model instead tries to explain the widespread emergence of Zipf's law from an entropy maximization principle in the general process of partitioning of elements into categories (or balls into boxes) [22]. Zipf's law can also emerge from more complex models based on the idea that components have specific networks of dependencies and that these relations determine their co-occurence in a realization [23,24].…”

Taxonomic classification method for metagenomics based on core protein families with Core-Kaiju

“…Introduction comparison method and its parameters. For example in k-mer based methods, the 48 choice of k determines sensitivity and precision of the classification, such that sensitivity 49 increases and precision decreases with increasing values for k, and vice versa. As we will 50 show, false positive predictions often need to be corrected heuristically by removing all 51 species/taxa with abundance below a given arbitrary threshold (see Materials and 52 Methods section for an overview on different algorithms of taxonomy classification).…”

“…Our primary goal was to find the so-called 183 core families [47], i.e. the protein domains which are present in the overwhelming 184 majority of the bacterium proteomes but occurring just few times in each of 185 them [40,48]. In order to analyze the occurrences of PFAM in proteomes, we converted 186…”