Question

To quantify network structure using FRACTAL DIMENSION: please I need suggestions on how best to introduce the topic, description of network structure and fractal dimension application to network.

Answer 1

FRACTAL DIMENSION

Today the method of determining the dimension is being applied to many kinds of observations, especially to time series in physics, engineering, and in fields as far apart as meteorology and neurophysiology. A fractal dimension of a set is a number that tells us how densely the set occupies the metric space in which it lies. It is invariant under various stretching and squeezing of the underlying space.This makes the fractal dimension meaningful as an experimental observable, it possesses a certain robustness and independent of the measurement units.

Complex networks are important since they describe efficiently many social, biological and communication systems .There exist many types of networks and characterizing their topology is very important for a wide range of static and dynamic properties.

Complex networks have been studied extensively in interdisciplinary fields including mathematic, statistical physics, computer science, sociology, economics, biology, etc. Complex networks are ubiquitous in the real world, e.g., there are technological networks such as the power grid, biological networks such as the protein interaction networks,and social networks such as scientific collaboration networks,and human communication networks, to name a few.

Many real complex networks share distinctive characteristic properties that differ in many ways from the random and regular networks. One such property is the “small-world effect”, which means that the average shortest path length between vertices in network is short, usually scaling logarithmically with the size N of network, while maintaining high clustering coefficient. A famous example is the so-called “six degrees of separa- tion” in social 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 .

To investigate self-similarity in networks, we use the box-counting method and renormalization.

Application

Many natural phenomena are better described using a fractional dimension, and fractals are thus used as descriptive models for the growth of plants, particle aggregation, river cartography, realistic images, and similar phenomena. Their fractal dimension characterizes most of these fractal models. In physical systems, the fractal dimension reflects some properties of the system. The physical characteristics of some bodies are related to the fractal dimension of their surfaces.

In analytical chemistry, the fractal dimension is used as a tool to characterize chemical patterns and problems of sample homogeneity. A given fractal dimension makes it possible to simulate a variety of systems: fluid extraction or contaminant mitigation techniques, the hybrid orbital model of proteins, or the growth of conflict rate in aircraft flight schedules.

Fractal dimension provides an objective meaning for comparing fractals and they can be attached to clouds, trees, coastlines, feathers, networks of neurons in the body, dust in the air at an instant in time, the clothes we are wearing, the distribution of frequencies of the light reflected by a flower, the colors emitted by the sun, and the wrinkled surface of the sea during a storm.