# Islands of stability in motif distributions of random networks

Source: [Islands of stability in motif distributions of random networks](<https://www.sota2.com/research/paper/68fa30e11e209131ce879515>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-24T18:29:29.035Z

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

Published: 2013-06-29T15:17:01.000Z

## Abstract

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-R\\'enyi 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.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1307.0113>)

[PDF](<https://arxiv.org/pdf/1307.0113.pdf>)