Chemically inspired Erdős-Rényi oriented hypergraphs

TL;DR

This work introduces an Erdős-Rényi-like model for oriented hypergraphs to capture high-order chemical space representations, where hypervertices encode educts and products and hyperedges represent reactions with directionality. By deriving exact counts for the complete oriented hypergraph (e.g., $u_r(n)=\frac12(3^n-2^{n+1}+1)$) and per-vertex participation $u(n)=3^{n-1}-2^{n-1}$, the authors establish a framework linking the random wiring probability $p$ to the expected number of reactions and their sizes. A key result is that, for large $n$, the ratio of expected total reactions to expected per-vertex degree satisfies $\frac{\mathbb{E}[R]}{\mathbb{E}[D]} \to \frac{3}{2}$, providing a robust null model for assessing real chemical spaces; the model also yields p-independent size distributions $P(s)=\frac{u_s}{u_r}$ and various growth regimes $\mathbb{E}[R]\sim p\,u_r$ under parametrizations like $p= n^\alpha/\beta^n$. The framework paves the way for phase-transition analysis in higher-order random structures and offers a quantitative benchmark for comparing actual chemical spaces against random wiring, with potential extensions to directed hypergraphs and stoichiometric constraints.

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 oriented hypergraph, which relates couples of subsets of an arbitrary number of vertices. Here we develop the Erdős-Rényi model for oriented hypergraphs, which corresponds to the random realisation of oriented hyperedges of the complete oriented hypergraph. A particular feature of random oriented hypergraphs is that the ratio between their expected number of oriented hyperedges and their expected degree or size is 3/2 for large number of vertices. We highlight the suitability of oriented hypergraphs for modelling large collections of chemical reactions and the importance of random oriented hypergraphs to analyse the unfolding of chemistry.

Figures (6)

• Figure 1: Chemical reactions as graphs and hypergraphs. All structures, from b to j correspond to (hyper)graph models for the chemical reactions in a, which constitute a chemical space of seven substances and three reactions.
• Figure 2: Toy chemical space constituted by four substances {A,B,C,D} and four reactions $r_i$. On the left, reactions are presented in chemical notation and on the right the chemical space is depicted as an oriented hypergraph.
• Figure 3: Amount of possible and impossible reactions. Visual depiction of adjacency matrices ${\bf M}$ for chemical spaces of a) $n=4$, b) $n=7$ and c) $n=10$ substances (vertices). Possible reactions (black entries) correspond to oriented hyperedges, where the related hypervertices (sets of substances) are disjoint. Impossible reactions (red entries) are the oriented hyperedges relating non-disjoint sets of substances.
• Figure 4: Effects of the probability of triggering chemical reactions upon the expected number of reactions of randomly wired chemical spaces. Probability is expressed as $p=n^\alpha/\beta^n$ and the plots show how the expected number of reactions $\hbox{E}[R]$ varies with the selection of $\alpha$ and $\beta$. In a) $\beta = 3$ and $\alpha$ takes different values, which show the decreasing power law ($\alpha = -1$), the constant ($\alpha=0$), linear ($\alpha=1$) and quadratic ($\alpha=2$) growth of $\hbox{E}[R]$ for large values of the number of substances $n$. In all these chemical spaces, where $\beta=3$, $\alpha \leq (n\ln{3})/(\ln{n})$ to warranty that $0 \leq p \leq 1$. In b) $\beta=1$ and $\alpha \leq 0$ to secure that $0 \leq p \leq 1$. These plots correspond to exponential-like growths of $\hbox{E}[R]$ for large values of $n$, where the slope of the linear fit tend to $\ln 3 \approx 1.099$. Plots in a) and b) were obtained for different values of $n$ in $\hbox{E}[R] = \frac{n^\alpha}{2\beta^n}(3^n-2^{n+1}+1)$.
• Figure 5: Effects of the probability of triggering reactions upon the expected number of reactions of different sizes in a randomly wired chemical space. Probability is given by $p=n^\alpha /\beta^n$. a) Behaviour at $\alpha = -2$ and $\beta = 1$, which corresponds to a chemical space whose number of reactions exponentially expand with the number of substances only in case $s > |\alpha|$. b) Distribution of number of reactions of size $s = 50, 100, 150$ and $200$ for chemical spaces with $\alpha = 2$ and $\beta=3$, corresponding to spaces whose number of reactions grows at a power law of the number of substances. Maximum values are given at $n_{max} = 76, 151, 226$ and $301$ respectively.

Theorems & Definitions (23)

• Definition 1
• Definition 2
• Definition 3
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4