A small-world graph (or small-world network) is a type of graph in network theory that exhibits two key properties:
- High Clustering – Nodes tend to form tightly knit groups where neighbors of a node are also likely to be connected to each other.
- Short Average Path Length – Despite the high clustering, the average number of steps required to get from one node to another is relatively low.
Key Characteristics:
- High clustering coefficient: This means that if node A is connected to node B and node C, then B and C are also likely to be connected.
- Small average shortest path: Most nodes can be reached from any other in just a few steps, even in large networks.
- Presence of "shortcuts": These are long-range connections that significantly reduce the distance between different parts of the network.
Example:
A classic example is social networks, where friends of your friends are often also your friends, but you can still connect to distant individuals through just a few acquaintances (the famous "six degrees of separation" concept).
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The Watts-Strogatz Model:
Duncan Watts and Steven Strogatz formalized the small-world property in 1998 using a model that interpolates between a regular lattice and a random graph:
- Start with a regular ring lattice where each node is connected to its k nearest neighbors.
- Randomly rewire some edges with a certain probability, introducing shortcuts that reduce the average shortest path length without destroying clustering.
Real-World Examples:
- Social networks (Facebook, LinkedIn, Twitter)
- Biological networks (neural connections in the brain)
- Technological networks (the power grid, the internet)
- Transportation systems (airline routes)
Would you like a visualization or code example to generate a small-world network?