# Fractal Dimension On Networks

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.

Read more about Fractal Dimension On NetworksThe Methods For Calculation of The Dimension, Real-world Fractal Networks, Other Definitions For Network Dimensions

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Fractal Dimension On Networks - Other Definitions For Network Dimensions
... The best definition of dimension for a complex network or graph depends on the application ... For example, metric dimension is defined in terms of the resolving set for a graph ... scaling property of the "mass" as defined above with distance, or based on the complex network zeta function have also been studied ...

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