Correlations in random Apollonian network

Abstract

In this paper, by both simulations and theoretical predictions we study two and three node (or degree) correlations in random Apollonian network (RAN), which have small-world and scale-free topologies. Using the rate equation approach under the assumption of continuous degree, we first give the analytical solution for two node correlations, expressed by average nearest-neighbor degree (ANND). Then, we revisit the degree distribution of RAN using rate equation method and get the exact connection distribution, based on which we derive a more accurate result for mean clustering coefficient as an average quantity of three degree correlations than the one previously reported. Analytical results reveal that ANND has no correlations with respect to degree, while clustering coefficient is dependent on degree, showing a power-law behavior as $C(k)\sim k^{-1}$. The obtained expressions are successfully contrasted with extensive numerical simulations.

11 pages, 3 figures. Accepted for publication in Physica A

References in corpus (33)

• Emergence of scaling in random networks
• Statistical mechanics of complex networks
• Assortative mixing in networks
• Mixing patterns in networks
• Specificity and stability in topology of protein networks
• Hierarchical Organization in Complex Networks
• Dynamical and correlation properties of the Internet
• Connectivity of Growing Random Networks
• Large-scale topological and dynamical properties of Internet
• Epidemic spreading in correlated complex networks
• Pseudofractal Scale-free Web
• Generic scale of the "scale-free" growing networks
• Maximal planar networks with large clustering coefficient and power-law degree distribution
• Resilience to damage of graphs with degree correlations
• The network topology of a potential energy landscape: A static scale-free network
• Inverted Berezinskii-Kosterlitz-Thouless Singularity and High-Temperature Algebraic Order in an Ising Model on a Scale-Free Hierarchical-Lattice Small-World Network
• Self-similar disk packings as model spatial scale-free networks
• Structural transitions in scale-free networks
• A deterministic small-world network created by edge iterations
• Rate equation approach for correlations in growing network models
• Recursive graphs with small-world scale-free properties
• Coherence in scale-free networks of chaotic maps
• Characterizing the network topology of the energy landscapes of atomic clusters
• High dimensional random Apollonian networks
• High Dimensional Apollonian Networks
• Disease Spreading in Structured Scale-Free Networks
• Magnetic models on Apollonian networks
• Evolving Apollonian Networks with Small-world Scale-free topologies
• A general geometric growth model for pseudofractal scale-free web
• Distance-d covering problems in scale-free networks with degree correlations
• Comment on "Breakdown of the Internet under Intentional Attack"
• Evolving small-world networks with geographical attachment preference
• Walks on Apollonian networks

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