Fractal Dimension On Networks
Many real networks have two fundamental properties, scale-free property and small-world property. If the degree distribution of the network follows a power-law, the network is scale-free; if any two arbitrary nodes in a network can be connected in a very small number of steps, the network is said to be small-world.
The small-world properties can be mathematically expressed by the slow increase of the average diameter of the network, with the total number of nodes ,
where is the shortest distance between two nodes.
Equivalently, we obtain:
where is a characteristic length.
For a self-similar structure, a power-law relation is expected rather than the exponential relation above. From this fact, it would seem that the small-world networks are not self-similar under a length-scale transformation.
However, analysis of a variety of real complex networks shows they are self-similar on all length scales, a conclusion derived from measuring a power-law relation between the number of boxes needed to cover the network and the size of the box, so called fractal scaling.
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