# Scale-free network

Free scales or scale-invariant networks or networks are complex networks , the number of links per node to a power law distributed are. Power laws are scale invariant with respect to stretching or compressing the scale of the variable.

Random vs. scale-free network

The proportion of nodes with degrees follows a power law ${\ displaystyle P (k)}$ ${\ displaystyle k}$

${\ displaystyle P (k) \ propto k ^ {- \ gamma}}$,

where is a unitless positive number. ${\ displaystyle \ gamma}$

Rescaling with any factor leads to a proportional power law ${\ displaystyle k \ rightarrow ak}$${\ displaystyle a}$

${\ displaystyle P (ak) \ propto a ^ {- \ gamma} k ^ {- \ gamma} \ propto k ^ {- \ gamma}}$.

## General

Scale-free networks are examined in the theory of complex networks and are considered to be relatively fail-safe . The robustness of such networks, however, only exists in the event of random node failures. With a strategic approach to switching off individual nodes, namely those with a high degree of linking, a scale-free network can quickly break down into small individual networks.

Animation: The growth
stages according to the scale-free Barabási-Albert model

Examples of scale-free and partially scale-free networks are:

• Network of cooperation between actors in films ( ), see also Bacon number${\ displaystyle \ gamma = 3}$
• Power grid - e.g. B. Western USA ( )${\ displaystyle \ gamma = 4}$
• The citation graph (graph of citations ) of scientific articles (k is the number of citations received, )${\ displaystyle \ gamma = 3}$
• Linking graph of the German language Wikipedia

Many small world networks are also scale-free or vice versa, whereby it should be noted that normal random graphs are not scale-free ( Erdős-Rényi - in contrast to Barabási-Albert networks).

Albert-László Barabási and Réka Albert proposed a much-noticed model for generating scale-free networks (cf. Barabási-Albert model ). It starts with a small number of nodes and adds another node at each step. The new node is connected to existing nodes, the probability of the connection being proportional to the number of edges that a node already has. This principle is also known as preferential attachment . It can be shown that this model tends towards the value 3 . ${\ displaystyle m_ {0}}$${\ displaystyle m}$${\ displaystyle \ gamma}$

## Generalizations

Many network probabilities, e.g. B. financial distributions, consist of non - Gaussian distributions with scale-free tail areas (so-called "fat tails"), which quantify the increased risk of extreme profits or losses. With Gaussian distributions, with which the usual standard examples for random processes are formulated, these extreme risk areas are omitted.