Phase transition for the threshold contact process, an approximation of heterogeneous random Boolean networks

• 成果类型: Article
• 署名作者: Chatterjee, Shirshendu
• 署名单位: City University of New York (CUNY) System; City College of New York (CUNY)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-015-0656-2
• 发表日期: 2016
• 页码: 985-1023
• 关键词: topology; graphs; STABILITY; models

摘要

We consider a model for heterogeneous gene regulatory networks that is a generalization of the model proposed by Chatterjee and Durrett (Random Struct Algorithms 39(2):228-246, 2011) as an annealed approximation of Kauffmann's (J Theor Biol 22(3):437-467, 1969) random Boolean networks. In this model, genes are represented by the vertices of a random directed graph on n vertices having specified in-degree distribution (resp. joint distribution of in-degree and out-degree), and the expression bias (the expected fraction of 1's in the Boolean functions) p is same for all vertices. Following Chatterjee and Durrett (Random Struct Algorithms 39(2):228-246, 2011) and a standard practice in the physics literature, we use a discrete-time threshold contact process with parameter (in which vertices are either occupied or vacant, a vertex with at least one (resp. no) occupied input at time t will be occupied at time with probability q (resp. 0)) on to approximate the dynamics of the Boolean network. We find a parameter r, which has an explicit expression in terms of certain moments of (resp. ), such that, with probability tending to 1 as n goes to infinity, if , then starting from all occupied sites the threshold contact process on maintains a positive (quasi-stationary) density (resp. ) of occupied sites till time which is exponential in n, whereas if , then the persistence time of the threshold contact process on is at most logarithmic in n. These two phases correspond to the chaotic and ordered behavior of the gene networks respectively.