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
Famous quotes containing the words dimension of, dimension, complex and/or network:
“Authority is the spiritual dimension of power because it depends upon faith in a system of meaning that decrees the necessity of the hierarchical order and so provides for the unity of imperative control.”
—Shoshana Zuboff (b. 1951)
“God cannot be seen: he is too bright for sight; nor grasped: he is too pure for touch; nor measured: for he is beyond all sense, infinite, measureless, his dimension known to himself alone.”
—Marcus Minucius Felix (2nd or 3rd cen. A.D.)
“All propaganda or popularization involves a putting of the complex into the simple, but such a move is instantly deconstructive. For if the complex can be put into the simple, then it cannot be as complex as it seemed in the first place; and if the simple can be an adequate medium of such complexity, then it cannot after all be as simple as all that.”
—Terry Eagleton (b. 1943)
“Of what use, however, is a general certainty that an insect will not walk with his head hindmost, when what you need to know is the play of inward stimulus that sends him hither and thither in a network of possible paths?”
—George Eliot [Mary Ann (or Marian)