title: Islands of stability in motif distributions of random networks.
abstract: We consider random nondirected networks subject to dynamics conserving vertex degrees and study, analytically and numerically, equilibrium three-vertex motif distributions in the presence of an external field h coupled to one of the motifs. For small h, the numerics is well described by the "chemical kinetics" for the concentrations of motifs based on the law of mass action. For larger h, a transition into some trapped motif state occurs in Erdős-Rényi networks. We explain the existence of the transition by employing the notion of the entropy of the motif distribution and describe it in terms of a phenomenological Landau-type theory with a nonzero cubic term. A localization transition should always occur if the entropy function is nonconvex. We conjecture that this phenomenon is the origin of the motifs' pattern formation in real evolutionary networks.
authors.affiliation: Physics Department, Moscow State University, 119992 Moscow, Russia and Department of Applied Mathematics, National Research University Higher School of Economics, 101000 Moscow, Russia.
authors.forename: M V
authors.initials: MV
authors.lastname: Tamm
authors.affiliation: Department of Physics, Yale University, 217 Prospect Street, New Haven, Connecticut 06511, USA.
authors.forename: A B
authors.initials: AB
authors.lastname: Shkarin
authors.affiliation: N.N. Semenov Institute of Chemical Physics of the Russian Academy of Sciences, 119991 Moscow, Russia and Department of Applied Mathematics, National Research University Higher School of Economics, 101000 Moscow, Russia.
authors.forename: V A
authors.initials: VA
authors.lastname: Avetisov
authors.affiliation: Department of Applied Mathematics, National Research University Higher School of Economics, 101000 Moscow, Russia and Université Paris-Sud/CNRS, LPTMS, UMR8626, Bâtiment 100, 91405 Orsay, France and Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Russia.
authors.forename: O V
authors.initials: OV
authors.lastname: Valba
authors.affiliation: Department of Applied Mathematics, National Research University Higher School of Economics, 101000 Moscow, Russia and Université Paris-Sud/CNRS, LPTMS, UMR8626, Bâtiment 100, 91405 Orsay, France and P.N. Lebedev Physical Institute of the Russian Academy of Sciences, 119991 Moscow, Russia.
authors.forename: S K
authors.initials: SK
authors.lastname: Nechaev
mesh_terms: D019277:Entropy; D008962:Models, Theoretical