Physics – Condensed Matter – Disordered Systems and Neural Networks
Scientific paper
2011-10-19
Nature Communications 3, 787 (2012)
Physics
Condensed Matter
Disordered Systems and Neural Networks
RevTex, 4 pages, 6 eps-figs included. For related work, see http://www.physics.emory.edu/faculty/boettcher/
Scientific paper
10.1038/ncomms1774
Percolation on finitely ramified lattices, those where the loss of a finite number of bonds can forestall end-to-end transport when the number of nodes $N$ increases, is generally considered to be trivial. They percolate only at full bond density, $p=1$. Examples are the one-dimensional lattice and fractals such as the Sierpinski gasket. We show here that by dressing up such lattices with small-world bonds, in a way that the resulting network becomes infinitely ramified, an entirely novel percolation transition at a critical point $p_{c}<1$ emerges. Approaching this transition from below, many clusters of already diverging, yet, sub-extensive, sizes combine explosively at $p_{c}$ such that the resulting cluster becomes extensive. That is, the usual order parameter $P_{\infty}$, describing the probability of any node to be part of the largest cluster, jumps instantly to a finite value at $p_{c}$. Simple examples of this transition are provided by small-world networks consisting of a one-dimensional lattice combined with various hierarchies of long-range bonds that reveal many features of the transition explicitly.
Boettcher Stefan
Singh Vibhor
Ziff Robert M.
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