**Fractal Dimension On Networks**

Many real networks have two fundamental properties, scale-free property and small-world property. If the degree distribution of the network follows a power-law, the network is scale-free; if any two arbitrary nodes in a network can be connected in a very small number of steps, the network is said to be small-world.

The small-world properties can be mathematically expressed by the slow increase of the average diameter of the network, with the total number of nodes ,

where is the shortest distance between two nodes.

Equivalently, we obtain:

where is a characteristic length.

For a self-similar structure, a power-law relation is expected rather than the exponential relation above. From this fact, it would seem that the small-world networks are not self-similar under a length-scale transformation.

However, analysis of a variety of real complex networks shows they are self-similar on all length scales, a conclusion derived from measuring a power-law relation between the number of boxes needed to cover the network and the size of the box, so called fractal scaling.

Read more about Fractal Dimension On Networks: The Methods For Calculation of The Dimension, Real-world Fractal Networks, Other Definitions For Network Dimensions

### Other articles related to "fractal dimension on networks, dimension, network":

**Fractal Dimension On Networks**- Other Definitions For Network Dimensions

... The best definition of

**dimension**for a complex

**network**or graph depends on the application ... For example, metric

**dimension**is defined in terms of the resolving set for a graph ... scaling property of the "mass" as defined above with distance, or based on the complex

**network**zeta function have also been studied ...

### Famous quotes containing the words networks and/or dimension:

“To be perfectly, brutally honest, those of us who are still carrying diaper everywhere we go are not at our most scintillating time of life....We need to remember that at one time in our lives, we all had senses of humor and knew things that were going on in the world. And if we just keep our social *networks* open, there will be people ready to listen when we once again have intelligent things to say.”

—Louise Lague (20th century)

“By intervening in the Vietnamese struggle the United States was attempting to fit its global strategies into a world of hillocks and hamlets, to reduce its majestic concerns for the containment of communism and the security of the Free World to a *dimension* where governments rose and fell as a result of arguments between two colonels’ wives.”

—Frances Fitzgerald (b. 1940)