Phase transitions in a network with range dependent connection probability

Details Phase transitions in a network with range dependent connection probability Phase transitions in a network with range dependent connection probability

2002-06-28

arxiv.org/abs/cond-mat/0206570v1

Physics

Condensed Matter

Statistical Mechanics

4 pages, revtex4, submitted to Physical Review E

Scientific paper

10.1103/PhysRevE.66.037102

We consider a one-dimensional network in which the nodes at Euclidean distance $l$ can have long range connections with a probabilty $P(l) \sim l^{-\delta}$ in addition to nearest neighbour connections. This system has been shown to exhibit small world behaviour for $\delta < 2$ above which its behaviour is like a regular lattice. From the study of the clustering coefficients, we show that there is a transition to a random network at $\delta = 1$. The finite size scaling analysis of the clustering coefficients obtained from numerical simulations indicate that a continuous phase transition occurs at this point. Using these results, we find that the two transitions occurring in this network can be detected in any dimension by the behaviour of a single quantity, the average bond length. The phase transitions in all dimensions are non-trivial in nature.

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