Heaps' law, statistics of shared components and temporal patterns from a sample-space-reducing process
arXiv:1806.09419 · doi:10.1103/PhysRevE.98.052139
Abstract
Zipf's law is a hallmark of several complex systems with a modular structure, such as books composed by words or genomes composed by genes. In these component systems, Zipf's law describes the empirical power law distribution of component frequencies. Stochastic processes based on a sample-space-reducing (SSR) mechanism, in which the number of accessible states reduces as the system evolves, have been recently proposed as a simple explanation for the ubiquitous emergence of this law. However, many complex component systems are characterized by other statistical patterns beyond Zipf's law, such as a sublinear growth of the component vocabulary with the system size, known as Heap's law, and a specific statistics of shared components. This work shows, with analytical calculations and simulations, that these statistical properties can emerge jointly from a SSR mechanism, thus making it an appropriate parameter-poor representation for component systems. Several alternative (and equally simple) models, for example based on the preferential attachment mechanism, can also reproduce Heaps' and Zipf's laws, suggesting that additional statistical properties should be taken into account to select the most-likely generative process for a specific system. Along this line, we will show that the temporal component distribution predicted by the SSR model is markedly different from the one emerging from the popular rich-gets-richer mechanism. A comparison with empirical data from natural language indicates that the SSR process can be chosen as a better candidate model for text generation based on this statistical property. Finally, a limitation of the SSR model in reproducing the empirical "burstiness" of word appearances in texts will be pointed out, thus indicating a possible direction for extensions of the basic SSR process.
arXiv comment14 pages, 4 figures
References in corpus (22)
- Power laws, Pareto distributions and Zipf's law
- Are biological systems poised at criticality?
- Collaborative Tagging and Semiotic Dynamics
- Beyond word frequency: Bursts, lulls, and scaling in the temporal distributions of words
- Network dynamics of innovation processes
- Are there laws of genome evolution?
- Stochastic model for the vocabulary growth in natural languages
- Zipf's Law Leads to Heaps' Law: Analyzing Their Relation in Finite-Size Systems
- On the origin of long-range correlations in texts
- Zipf's law unzipped
- Complex systems: physics beyond physics
- Understanding scaling through history-dependent processes with collapsing sample space
- Statistical laws in linguistics
- Scaling laws and fluctuations in the statistics of word frequencies
- A Yule-Simon process with memory
- Universal distribution of component frequencies in biological and technological systems
- Log-log Convexity of Type-Token Growth in Zipf's Systems
- Zipf and Heaps laws from dependency structures in component systems
- Statistics of shared components in complex component systems
- Sample space reducing cascading processes produce the full spectrum of scaling exponents
- Size dependent word frequencies and translational invariance of books
- Extreme robustness of scaling in sample space reducing processes explains Zipf's law in diffusion on directed networks