title: The analysis of bi-level evolutionary graphs.
abstract: Evolutionary graphs (EGs), in which evolutionary dynamic is arranged on a graph, were initially proposed by Lieberman et al. [Lieberman, E., Hauert, C., Nowak, M.A., 2005. Evolutionary dynamics on graphs. Nature 433, 312-316] in the biological field and many biological phenomena are successfully explained. EGs on two levels (or bi-level EGs), based on some biological phenomena, are considered in this paper. The bi-level EGs are compared with the one-rooted EGs in two cases. One has the identical numbers of the followers, the other with the same numbers of total individuals. Then, some properties of the bi-level EGs are obtained. It is showed that bi-level EGs are more stable, and the bi-level EGs with just two leaders are the most stable, if they have identical followers respectively. The bi-level EGs theory can successfully explain the phenomena of symbiosis in biology.
authors.affiliation: Department of Mathematics, Jinan University, Guangzhou 510632, PR China.
authors.forename: Pei-ai
authors.initials: PA
authors.lastname: Zhang
authors.forename: Pu-yan
authors.initials: PY
authors.lastname: Nie
authors.forename: Dai-qiang
authors.initials: DQ
authors.lastname: Hu
authors.forename: Fei-yan
authors.initials: FY
authors.lastname: Zou
mesh_terms: D000818:Animals; D005075:Biological Evolution; D005716:Game Theory; D006801:Humans; D008433:Mathematics; D008957:Models, Genetic; D008962:Models, Theoretical; D049490:Systems Biology