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