Small-world network

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== Small-world network ==

A small-world network is a type of mathematical graph in which most nodes are not neighbors of one another, but the neighbors of any given node are likely to be neighbors of each other, and most nodes can be reached from every other by a small number of steps. This concept is a significant part of the study of network theory and has applications in various fields such as sociology, computer science, and biology.

Characteristics

Small-world networks are characterized by two main properties:

• **High clustering coefficient**: This means that nodes tend to create tightly knit groups characterized by a relatively high density of ties.
• **Short average path length**: This indicates that the average number of steps along the shortest paths for all possible pairs of network nodes is small.

These properties are often found in real-world networks, such as social networks, neural networks, and the World Wide Web.

History

The concept of small-world networks was first introduced by Duncan J. Watts and Steven Strogatz in their 1998 paper "Collective dynamics of 'small-world' networks". They demonstrated that many real-world networks exhibit the small-world property.

Examples

• **Social Networks**: In social networks, individuals are connected by a small number of acquaintances, often referred to as the six degrees of separation.
• **Biological Networks**: In biological systems, such as the neural network of the Caenorhabditis elegans, neurons are connected in a small-world topology.
• **Technological Networks**: The structure of the Internet and the World Wide Web also exhibit small-world properties.

Mathematical Models

Several models have been proposed to generate small-world networks:

• **Watts-Strogatz model**: This model starts with a regular lattice and randomly rewires edges to introduce randomness, creating a network with high clustering and short path lengths.
• **Barabási-Albert model**: Although primarily known for generating scale-free networks, this model can also produce networks with small-world properties under certain conditions.

Applications

Small-world networks have numerous applications:

• **Epidemiology**: Understanding the spread of diseases through populations.
• **Sociology**: Analyzing social structures and the spread of information.
• **Neuroscience**: Studying the brain's connectivity and functionality.
• **Engineering**: Designing efficient communication and transportation networks.

Related Concepts

• Scale-free network
• Random graph
• Network theory
• Graph theory
• Clustering coefficient
• Path length

See Also

• Six degrees of separation
• Watts-Strogatz model
• Barabási-Albert model
• Network theory
• Graph theory
• Clustering coefficient
• Path length