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Prevalence of deficiency-zero reaction networks in an Erdös–Rényi framework

Part of: Limit theorems Graph theory

Published online by Cambridge University Press:  28 January 2022

David F. Anderson Open the ORCID record for David F. Anderson [Opens in a new window]  and
Tung D. Nguyen
David F. Anderson*
Affiliation:
University of Wisconsin-Madison
Tung D. Nguyen*
Affiliation:
University of Wisconsin-Madison, Texas A&M University
*
*Postal address: 617 Van Vleck Hall, 480 Lincoln Drive, Madison, WI 53706, USA. Email address: [email protected]
**Postal address: 601D Blocker Building, 155 Ireland St, College Station, TX 77840, USA. Email address: [email protected]
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Abstract

Reaction networks are commonly used within the mathematical biology and mathematical chemistry communities to model the dynamics of interacting species. These models differ from the typical graphs found in random graph theory since their vertices are constructed from elementary building blocks, i.e. the species. We consider these networks in an Erdös–Rényi framework and, under suitable assumptions, derive a threshold function for the network to have a deficiency of zero, which is a property of great interest in the reaction network community. Specifically, if the number of species is denoted by n and the edge probability by $p_n$ , then we prove that the probability of a random binary network being deficiency zero converges to 1 if $p_n\ll r(n)$ as $n \to \infty$ , and converges to 0 if $p_n \gg r(n)$ as $n \to \infty$ , where $r(n)=\frac{1}{n^3}$ .

Keywords

DeficiencyErdös–Rényireaction networkrandom graphthreshold function

MSC classification

Primary: 60F99: None of the above, but in this section
Secondary: 05C80: Random graphs
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 2 , June 2022 , pp. 384 - 398
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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