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Chemically inspired Erdős–Rényi hypergraphs

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Abstract

High-order structures have been recognised as suitable models for systems going beyond the binary relationships for which graph models are appropriate. Despite their importance and surge in research on these structures, their random cases have been only recently become subjects of interest. One of these high-order structures is the chemical hypergraph, which relates couples of subsets (hypervertices) of an arbitrary number of vertices. Here we develop the Erdős–Rényi model for chemical hypergraphs, which corresponds to the random realisation of edges of the complete chemical hypergraph. A particular feature of random chemical hypergraphs is that the ratio between their expected number of edges and their expected degree or size is 3/2 for large number of vertices. We highlight the suitability of chemical hypergraphs for modelling large collections of chemical reactions and the importance of random chemical hypergraphs to analyse the unfolding of chemistry.

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Notes

  1. Models of the chemical space strongly emphasise the role of reactions as the “gluing” aspect relating substances, which, in turn, endows the set of substances with a structure. This is what actually turn the set of substances into a space [41]. In such a setting, however, non-connected substances, often arising from chemical extractions, play an important role in the space, as they represent new non-synthetic chemicals, which may, or not, remain disconnected in the space, or which require a certain amount of time to be connected to the network. This was the case of the noble gases, for instance, which, for many years, remained as isolated substances of the chemical space. Moreover, determining the average time required to connect a substance to the network of the chemical space is of central importance for studies on the evolution of chemical knowledge, as well for the chemical industry [27].

  2. In formal terms, it can be considered as a mapping to the category of directed graphs.

  3. Formally, and in particular following the notation described in Definition 1, {A, B}-{C, D} can be written down as {{A, B},{C, D}}.

  4. A fairly concise mathematical description on the topic is found in [38]. It may also happen that the edge {A, B}-{C, D}, beyond encoding A + B \(\rightarrow \) C + D and/or C + D \(\rightarrow \) A + B as two reactions occurring at particular conditions, also encodes the same transformation but occurring under different conditions and even through different reaction mechanisms.

  5. Note that stoichiometric coefficients are disregarded in this notation.

  6. Moreover, in this setting we are disregarding the particular reaction conditions at which reactions are carried out. Nonetheless, they can be incorporated as edge labels.

  7. This upper bound holds significance in, for instance, research on the origin of life. A mathematical setting for such studies is provided by Dittrich’s chemical organisation theory [15], where finding sequences of reactions involving a given subset of substances of the chemical space is an important aspect of the approach.

  8. The number of black 0-entries amounts to the unrealised chemical reactions, which together with the 1-entries correspond to the potential chemical space, as called by some philosophers of chemistry [43].

  9. The size of a reaction corresponds to the molecularity of the reaction [2] if the stoichiometric coefficients of the reaction are regarded. As this is not, in general, the case in studies on the chemical space [33, 41], the size of a reaction may be regarded as a proto-molecularity of the reaction. It only accounts for the number of different chemicals reported in the reaction, but not for their actual figures. Often, chemists omit writing, for instance, water or carbon dioxide, as those substances can be inferred from the context of the reaction or because of the tradition to emphasis the target product of a reaction, namely of a chemical synthesis [27].

  10. Bounds for size and degree of chemical hypergraphs are provided in Lemma 7.

  11. See Lemma 8.

  12. A further question is how many of the possible reactions are actually realised by chemists in the chemical space. This is a subject we address in a forthcoming paper.

  13. This indicates that, in the case of a phase transition for this model, the probability P(s) cannot be altered by the criticality.

  14. It is known that for the actual chemical space \(n \sim 10^6\) [33].

  15. Which occurs for low values of n in Fig. 4.

  16. It is known that \(s \ll n\) for actual chemical spaces [33].

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Acknowledgements

The authors thank the feedback from Guido Montufar, Humberto Laguna and Duc H. Luu upon early results of this project.

Funding

MBM thanks the support of the Alexander von Humboldt Foundation.

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Authors and Affiliations

  1. Tecnológico de Monterrey, Escuela de Ingeniería y Ciencias, Ave. Lago de Guadalupe, 52926, Estado de Mexico, Mexico

    Angel Garcia-Chung & Marisol Bermúdez-Montaña

  2. Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

    Angel Garcia-Chung, Marisol Bermúdez-Montaña, Peter F. Stadler, Jürgen Jost & Guillermo Restrepo

  3. Interdisciplinary Center for Bioinformatics, Universität Leipzig, Leipzig, Germany

    Peter F. Stadler & Guillermo Restrepo

  4. Bioinformatics Group, Department of Computer Science, Universität Leipzig, Leipzig, Germany

    Peter F. Stadler

  5. Institute for Theoretical Chemistry, University of Vienna, Vienna, Austria

    Peter F. Stadler

  6. The Santa Fe Institute, Santa Fe, NM, USA

    Peter F. Stadler & Jürgen Jost

  7. Facultad de Ciencias, Universidad Nacional de Colombia, Bogotá, Colombia

    Peter F. Stadler

  8. Center for Scalable Data Analytics and Artificial Intelligence, Universität Leipzig, Leipzig, Germany

    Jürgen Jost

  9. School of Applied Sciences and Engineering, EAFIT University, Medellín, Colombia

    Guillermo Restrepo

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  1. Angel Garcia-Chung
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Contributions

AG-C conducted the research, AG-C and GR conceptualised the project, GR wrote the paper and AG-C, MBM, PFS, JJ and GR discussed and reviewed the edited document.

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Correspondence to Angel Garcia-Chung.

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Garcia-Chung, A., Bermúdez-Montaña, M., Stadler, P.F. et al. Chemically inspired Erdős–Rényi hypergraphs. J Math Chem (2024). https://doi.org/10.1007/s10910-024-01595-8

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