Scale-free network

Scale -free or scale-invariant networks or networks are complex networks whose number of connections per node, according to a power law distributed. Power laws are scale invariant with respect to stretching or compression of the scale of the variable.

The distribution of nodes and the number k of connections follows a power law

Where a unitless number.

Rescaling with any factor results in a proportional power law.

General

Scale -free networks are studied in the theory of complex networks and are considered relatively fail-safe. The robustness of such networks is, however, only random failures of nodes. Through strategic approach when switching off individual nodes ( ie those with a high degree of linking ) can quickly disintegrate into small networks, a scale-free network.

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

Many small-world networks are scale- free or vice versa, it being noted that normal random graphs are not scale- free ( Erdös - Rényi - unlike Barabási -Albert networks ).

Barabási and Albert proposed a attractive model for generating scale-free networks. This is initiated with a small number of nodes and added in each step, a further node. The new node is respectively connected to an existing node, the connection probability is proportional to the number of edges having a node already. This principle is also referred to as preferential attachment. It can be shown that in this model strives against the value 3.

Generalizations

Many network probabilities, eg financial distributions, of non - Gaussian distributions with scale-free foothills areas (so-called " fat tails " ), which quantify the increased risk for extreme gains or losses. For Gaussian distributions, with which the usual standard examples are formulated for random processes, these extreme risk zones automatically fall away.