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Component systems: do null models explain everything?

Andrea Mazzolini, Mattia Corigliano, Rossana Droghetti, Matteo Osella, Marco Cosentino-Lagomarsino

TL;DR

It is argued that the generality and simplicity of those laws are often due to basic combinatorial or sampling constraints, raising the question of whether such patterns are actually revealing system-specific mechanisms and how to move beyond them.

Abstract

Component systems - ensembles of realizations built from a shared repertoire of modular parts - are ubiquitous in biological, ecological, technological, and socio-cultural domains. From genomes to texts, cities, and software, these systems exhibit statistical regularities that often meet the "bona fide" requirements of laws in the physical sciences. Here, we argue that the generality and simplicity of those laws are often due to basic combinatorial or sampling constraints, raising the question of whether such patterns are actually revealing system-specific mechanisms and how we might move beyond them. To this end, we first present a unifying mathematical framework, which allows us to compare modular systems in different fields and highlights the common "null" trends as well as the system-specific uniqueness, which, arguably, are signatures of the underlying generative dynamics. Next, we can exploit the framework with statistical mechanics and modern machine-learning tools for a twofold objective. (i) Explaining why the general regularities emerge, highlighting the constraints between them and the general principles at their origins, and (ii) "subtracting" them from data, which will isolate the informative features for inferring hidden system-specific generative processes, mechanistic and causal aspects.

Component systems: do null models explain everything?

TL;DR

It is argued that the generality and simplicity of those laws are often due to basic combinatorial or sampling constraints, raising the question of whether such patterns are actually revealing system-specific mechanisms and how to move beyond them.

Abstract

Component systems - ensembles of realizations built from a shared repertoire of modular parts - are ubiquitous in biological, ecological, technological, and socio-cultural domains. From genomes to texts, cities, and software, these systems exhibit statistical regularities that often meet the "bona fide" requirements of laws in the physical sciences. Here, we argue that the generality and simplicity of those laws are often due to basic combinatorial or sampling constraints, raising the question of whether such patterns are actually revealing system-specific mechanisms and how we might move beyond them. To this end, we first present a unifying mathematical framework, which allows us to compare modular systems in different fields and highlights the common "null" trends as well as the system-specific uniqueness, which, arguably, are signatures of the underlying generative dynamics. Next, we can exploit the framework with statistical mechanics and modern machine-learning tools for a twofold objective. (i) Explaining why the general regularities emerge, highlighting the constraints between them and the general principles at their origins, and (ii) "subtracting" them from data, which will isolate the informative features for inferring hidden system-specific generative processes, mechanistic and causal aspects.
Paper Structure (11 sections, 2 figures, 1 table)

This paper contains 11 sections, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The component system and its emerging laws
  3. Key regularities emerge from sampling constraints
  4. Subtracting regularities unveils system-specific mechanistic and causal processes
  5. Conclusions and perspectives
  6. Acknowledgments

Figures (2)

  • Figure 1: Component systems as a unifying representation of modular artifacts. (a) A wide class of systems---including genomes, texts, and LEGO constructions---can be described as component systems, in which each realization is assembled from a shared vocabulary of elementary components that may be reused within and across realizations. (b) This structure is encoded by the component matrix $n_{ij}$, whose entries count how many times component $i$ appears in realization $j$. (c) Simple sums and binarized sums of the matrix elements define a set of fundamental observables that characterize the system at both the component and realization level, including component abundance and occurrence, as well as realization size and vocabulary.
  • Figure 2: Basic statistical laws and their connection due to sampling constraints. (a) Zipf's law for component frequency is reported for the LEGO dataset (https://rebrickable.com/). (b) By extracting components with their empirical probabilities, a random sampling model can generate an artificial ensemble of realizations that can be compared to the empirical ones. (c) The statistics of shared components and the Heaps' law can be often explained by this sampling procedure, although quantitative deviations can reveal system-specific mechanisms.