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arXiv · 0802.1055

Unusual percolation in simple small-world networks

Reuven Cohen·Daryush Jonathan Dawid·Mehran Kardar·Yaneer Bar-Yam
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Abstract

We present an exact solution of percolation in a generalized class of Watts-Strogatz graphs defined on a 1-dimensional underlying lattice. We find a non-classical critical point in the limit of the number of long-range bonds in the system going to zero, with a discontinuity in the percolation probability and a divergence in the mean finite-cluster size. We show that the critical behavior falls into one of three regimes depending on the proportion of occupied long-range to unoccupied nearest-neighbor bonds, with each regime being characterized by different critical exponents. The three regimes can be united by a single scaling function around the critical point. These results can be used to identify the number of long-range links necessary to secure connectivity in a communication or transportation chain. As an example, we can resolve the communication problem in a game of "telephone".

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Reuven Cohen, Daryush Jonathan Dawid, Mehran Kardar, Yaneer Bar-Yam. 2008-02-07. Unusual percolation in simple small-world networks. https://doi.org/10.1103/physreve.79.066112

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