Small worlds. (English) Zbl 0988.60100
Consider a circle \(C\) of circumference 1 and a Poisson process of intensity \(n\) on \(C\). From each of its points a random chord is drawn to a uniformly distributed point. These chords we call “shortcuts” and let them all have “length” 0. A “small worlds” network is created with local links and long range shortcuts. Let \(D\) be the distance between two random points on \(C\) using arcs and shortcuts. One result proved in this nice paper is the convergence of \(nD - (1/4)\log n\) to a certain non-degenerate distribution as \(n\) tends to infinity; the same result holds if the number of chords is \(n\) (non-random). Also higher-dimensional results are derived. Models of this type have recently received considerable attention and have been investigated by physicists in heuristic ways, giving in some cases non-correct results, contrary to the rigorous and interesting methods of the present paper.
Reviewer: Lars Holst (Stockholm)
MSC:
| 60K99 | Special processes |
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