Аннотация
We explore the concepts of self-similarity, dimensionality,
and (multi)scaling in a new family of recursive scale-free nets that
yield themselves to exact analysis through renormalization techniques.
All nets in this family are self-similar and some are fractals
possessing a finite fractal dimension while others are small-world
(their diameter grows logarithmically with their size) and are
infinite-dimensional. We show how a useful measure of transfinite
dimension may be defined and applied to the small-world nets.
Concerning multiscaling, we show how first-passage time for diffusion
and resistance between hubs (the most connected nodes) scale
differently than for other nodes. Despite the different scalings, the
Einstein relation between diffusion and conductivity holds separately
for hubs and nodes. The transfinite exponents of small-world nets obey
Einstein relations analogous to those in fractal nets.
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