Dimension of Complex Network
Usually, dimension is defined based on the scaling exponent of some property in the appropriate limit. One property one could use is the scaling of volume with distance. For regular lattices the number of nodes within a distance of node scales as .
For systems which arise in physical problems one usually can identify some physical space relations among the vertices. Nodes which are linked directly will have more influence on each other than nodes which are separated by several links. Thus, one could define the distance between nodes and as the length of the shortest path connecting the nodes.
For complex networks one can define the volume as the number of nodes within a distance of node, averaged over, and the dimension may be defined as the exponent which determines the scaling behaviour of the volume with distance. For a vector, where is a positive integer, the Euclidean norm is defined as the Euclidean distance from the origin to, i.e.,
However, the definition which generalises to complex networks is the norm,
The scaling properties hold for both the Euclidean norm and the norm. The scaling relation is
where d is not necessarily an integer for complex networks. is a geometric constant which depends on the complex network. If the scaling relation Eqn. holds, then one can also define the surface area as the number of nodes which are exactly at a distance from a given node, and scales as
A definition based on the complex network zeta function generalises the definition based on the scaling property of the volume with distance and puts it on a mathematically robust footing.
Read more about this topic: Shortcut Model
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