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This research aims to describe the transition between different connectivity regimes in small-world networks through exact mathematical formulations.

Exactly solvable small-world network

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This research aims to describe the transition between different connectivity regimes in small-world networks through exact mathematical formulations.
Analyzed a model with sites arranged on a circle, connected based on a probability distribution.
Determined the relationship between the average shortest distances and the connection probability.
Derived the distance distribution and scaling behavior of the network.
As connection probability p varies from 0 to 1, average shortest distance changes from n to 1.
Found that the distribution of shortest distances closely matches numerical results for standard small-world networks.

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Analytical exploration reveals distance distribution and scaling in small-world networks, indicating valuable insights into connectivity dynamics.

Dorogovt︠s︡ev et al. (2000) studied this question.

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Analytical exploration reveals distance distribution and scaling in small-world networks, indicating valuable insights into connectivity dynamics.