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 coupled to one of the motifs. For small , the numerics is well described by the “chemical kinetics” for the concentrations of motifs based on the law of mass action. For larger , 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.
- Received 5 July 2013
DOI:https://doi.org/10.1103/PhysRevLett.113.095701
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