# Mean-field solution of the small-world network model.

@article{Newman2000MeanfieldSO, title={Mean-field solution of the small-world network model.}, author={Mark E. J. Newman and Cristopher Moore and Duncan J. Watts}, journal={Physical review letters}, year={2000}, volume={84 14}, pages={ 3201-4 } }

The small-world network model is a simple model of the structure of social networks, which possesses characteristics of both regular lattices and random graphs. The model consists of a one-dimensional lattice with a low density of shortcuts added between randomly selected pairs of points. These shortcuts greatly reduce the typical path length between any two points on the lattice. We present a mean-field solution for the average path length and for the distribution of path lengths in the model…

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## References

SHOWING 1-10 OF 16 REFERENCES

Renormalization Group Analysis of the Small-World Network Model

- Physics, Mathematics
- 1999

We study the small-world network model, which mimics the transition between regular-lattice and random-lattice behavior in social networks of increasing size. We contend that the model displays a…

Scaling and percolation in the small-world network model.

- Mathematics, PhysicsPhysical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
- 1999

There is one nontrivial length-scale in the small-world network model of Watts and Strogatz, analogous to the correlation length in other systems, which is well-defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero, giving a normal critical point in this limit.

On the properties of small-world network models

- Physics
- 1999

Abstract:We study the small-world networks recently introduced by Watts and Strogatz [Nature 393, 440 (1998)], using analytical as well as numerical tools. We characterize the geometrical properties…

Collective dynamics of ‘small-world’ networks

- Computer Science, MedicineNature
- 1998

Simple models of networks that can be tuned through this middle ground: regular networks ‘rewired’ to introduce increasing amounts of disorder are explored, finding that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs.

First-order transition in small-world networks

- Physics
- 2000

The small-world transition is a first-order transition at zero density p of shortcuts, whereby the normalised shortest-path distance = /L undergoes a discontinuity in the thermodynamic limit. On…

Spreading and shortest paths in systems with sparse long-range connections.

- Mathematics, MedicinePhysical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
- 1999

Spreading according to simple rules (e.g., of fire or diseases) and shortest-path distances are studied on d-dimensional systems with a small density p per site of long-range connections…

Small-World Networks: Evidence for a Crossover Picture

- Physics
- 1999

Watts and Strogatz [Nature (London) 393, 440 (1998)] have recently introduced a model for disordered networks and reported that, even for very small values of the disorder $p$ in the links, the…

Nonlinear Dynamics and Chaos

- Physics
- 2001

Nonlinear dynamics deals with more-or-less regular fluctuations in system variables caused by feedback intrinsic to the system (as opposed to external forces). Chaos is the most exotic form of…

Phys

- Rev. Lett. 82, 3180
- 1999