Islands of stability in motif distributions of random networks

M. V. Tamm 1, 2 A. B. Shkarin 3 V. A. Avetisov 2, 4 O. V. Valba 2, 5, 6 S. K. Nechaev 2, 6, 7

Physical Review Letters, American Physical Society, 2014, 113, pp.095701

We consider random non-directed 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 Erd\H{o}s-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 non-zero cubic term. A localization transition should always occur if the entropy function is non-convex. We conjecture that this phenomenon is the origin of the motifs' pattern formation in real evolutionary networks.

  • 1. Physics Department
  • 2. Department of Applied Mathematics
  • 3. Department of Physics
  • 4. The Semenov Institute of Chemical Physics
  • 5. MIPT - Moscow Institute of Physics and Technology
  • 6. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
  • 7. P. N. Lebedev Physical Institute