Phase transitions in a network with range dependent connection probability

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We consider a one-dimensional network in which the nodes at Euclidean distance l can have long range connections with a probabilty P(l) l⁻δ in addition to nearest neighbour connections. This system has been shown to exhibit small world behaviour for δ< 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 δ= 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.