## Семинар лаборатории Нелинейных, неравновесных и сложных систем 11.11.2013

## Распределение мотивов (концентраций малых подграфов) в случайных сетях

### М.В. Тамм (физический факультет МГУ)

### Аннотация

We consider random non-directed Erdős–Rényi networks subject to a
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 magnitude of the
external
field the numerics is well described by chemical kinetics equations based
on the law of mass action for the concentrations of motifs. For larger
external fields a transition into a state with some trapped motif
distribution occurs. 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. We
argue that the localization transition should always occur if the entropy
function is non-convex. We conjecture that this phenomenon may be the
reason for motifs' pattern formation in real networks.