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

References in corpus (7)

• Emergence of scaling in random networks
• Statistical mechanics of complex networks
• Scaling and percolation in the small-world network model
• Epidemics and percolation in small-world networks
• A Measure of data-collapse for scaling
• Small-World Networks: Links with long-tailed distributions
• Small-world phenomena and the statistics of linear polymer networks

Cited by in corpus (24)

• Spatial Networks
• Small-world properties of the Indian Railway network
• Crossover from Scale-Free to Spatial Networks
• Modulated Scale-free Network in the Euclidean Space
• Geographical threshold graphs with small-world and scale-free properties
• An Analysis of the Gel Point of Polymer Model Networks by Computer Simulations
• Two-dimensional SIR epidemics with long range infection
• Geographical networks evolving with optimal policy
• Navigation in a small world with local information
• Phase transitions in Ising model on a Euclidean network
• Promote cooperation by localised small-world communication
• Zero temperature dynamics of Ising model on a densely connected small world network
• Effect of the nature of randomness on quenching dynamics of Ising model on complex networks
• Two-dimensional small-world networks: navigation with local information
• Transitions in spatial networks
• Complex systems: features, similarity and connectivity
• Dynamics on Spatial Networks and the Effect of Distance Coarse Graining
• Phase transition of a one-dimensional Ising model with distance-dependent connections
• Susceptible-Infected-Recovered model on Euclidean network
• Diffusion Processes on Small-World Networks with Distance-Dependent Random-Links
• Complex Networks: effect of subtle changes in nature of randomness
• Degree heterogeneity in spatial networks with total cost constraint
• Self-organized network design by link survivals and shortcuts
• Phase transitions in Ising model induced by weight redistribution on weighted regular networks