Complex network zeta function: Difference between revisions

Revision as of 07:18, 28 February 2008

Revision as of 01:27, 14 March 2008

Line 4:

==Definition==

One usually thinks of dimension for a set which is dense, like the points on a line, for example. Dimension makes sense in a discrete setting, like for graphs, only in the large system limit, as the size tends to infinity. For example, in Statistical Mechanics, one considers discrete points which are located on regular lattices of different dimensions. Such studies have been extended to arbitrary networks, and it is interesting to consider how the definition of dimension can be extended to cover these cases. A very simple and obvious way to extend the definitondefinition of dimension to arbitrary large networks is to consider how the volume (number of nodes within a given distance from a specified node) scales as the distance (shortest path connecting two nodes in the graph) is increased. For many systems arising in physics, this is indeed a useful approach. This definition of dimension could be put on a strong mathematical foundation, similar to the definition of Hausdorff dimension for continuous systems. The mathematically robust definition uses the concept of a zeta function for a graph. The complex network zeta function and the graph surface function were introduced to characterize large graphs. They have also been applied to study patterns in Language Analysis. In this section we will briefly review the definition of the functions and discuss further some of their properties which follow from the definition.

We denote by <math> \textstyle r_{ij} </math> the distance from node <math>\textstyle i</math> to node <math>\textstyle j</math>, i.e., the length of the shortest path connecting the first node to the second node. <math>\textstyle r_{ij}</math> is <math>\textstyle \infty</math> if there is no path from node <math>\textstyle i</math> to node <math>\textstyle j</math>. With this definition, the nodes of the complex network become points in a [[metric space]]<ref name="Shankerb"/>. Simple generalisations of this definition can be studied, e.g., we could consider weighted edges. The graph surface function, <math>\textstyle S(r )</math>, is defined as the number of nodes which are exactly at a distance <math> \textstyle r </math> from a given node, averaged over all nodes of the network. The complex network zeta function <math>\textstyle \zeta_G ( \alpha )</math> is defined as