1307.0113 (M. V. Tamm et al.)

Islands of stability in motif distributions of random networks    [PDF]

M. V. Tamm, A. B. Shkarin, V. A. Avetisov, O. V. Valba, S. K. Nechaev

We consider non-directed Erdos-Renyi random networks subject to a dynamics conserving the vertex degrees and study numerically and analytically the equilibrium three-vertex motif distributions in the presence of the external field coupled with one of the motifs. For small values of external fields the numerical data is well described by a chemical kinetics based on the law of mass actions for the concentrations of motifs. For larger external fields a transition into a state with some trapped motif distribution occurs. We present the description of this transition in terms of a phenomenological Landau--type theory with non-zero cubic term, and explain it by employing the notion of the entropy of the motif distribution. We argue that the localization transition should always occur if the entropy function is non-convex, and conjecture that this phenomenon may be the reason for motifs' pattern formation in real networks.

View original: http://arxiv.org/abs/1307.0113