2020/10/14 by David F. Anderson, Anderson, David F., Tung D. Nguyen +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Chemistry · #Chemical Reaction Mechanisms #FOS: Biological sciences #FOS: Mathematics #Gene Regulatory Network Analysis #Molecular Networks (q-bio.MN) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2010.07201
openalex publication_date 2020/10/14 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
Deficiency zero is an important network structure and has been the focus of\nmany celebrated results within reaction network theory. In our previous paper\n\Prevalence of deficiency zero reaction networks in an Erd H os-R 'enyi\nframework, we provided a framework to quantify the prevalence of deficiency\nzero among randomly generated reaction networks. Specifically, given a randomly\ngenerated binary reaction network with n species, with an edge between two\narbitrary vertices occurring independently with probability pn, we\nestablished the threshold function r(n)=\(1)/(n3) such that the\nprobability of the random network being deficiency zero converges to 1 if\n\(pn)/(r(n))\→ 0 and converges to 0 if \(pn)/(r(n))\→\∞, as\nn \→ \∞.\n With the base Erd H os-R 'enyi framework as a starting point, the current\npaper provides a significantly more flexible framework by weighting the edge\nprobabilities via control parameters \αi,j, with i,j\∈ 0,1,2 \nenumerating the types of possible vertices (zeroth, first, or second order).\nThe control parameters can be chosen to generate random reaction networks with\na specific underlying structure, such as "closed" networks with very few inflow\nand outflow reactions, or "open" networks with abundant inflow and outflow.\nUnder this new framework, for each choice of control parameters\n \αi,j , we establish a threshold function r(n, \αi,j )\nsuch that the probability of the random network being deficiency zero converges\nto 1 if fracpnr(n, \αi,j )\→ 0 and converges to 0 if\n fracpnr(n, \αi,j )\→ \∞.\n