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Nonlinear diffusion through large complex networks containing regular subgraphs. (English) Zbl 1147.82343

Summary: Transport through generalized trees is considered. Trees contain the simple nodes and supernodes, either well-structured regular subgraphs or those with many triangles. We observe a superdiffusion for the highly connected nodes while it is Brownian for the rest of the nodes. Transport within a supernode is affected by the finite size effects vanishing as \(N \rightarrow \infty\). For the even dimensions of space, \(d = 2, 4, 6,\ldots\), the finite size effects break down the perturbation theory at small scales and can be regularized by using the heat-kernel expansion.

MSC:

82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
60J65 Brownian motion
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
82C70 Transport processes in time-dependent statistical mechanics
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