Phase transitions in a network with range dependent connection probability

Abstract

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.

4 pages, revtex4, submitted to Physical Review E

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