Degree Correlation in Scale-Free Graphs

Abstract

We obtain closed form expressions for the expected conditional degree distribution and the joint degree distribution of the linear preferential attachment model for network growth in the steady state. We consider the multiple-destination preferential attachment growth model, where incoming nodes at each timestep attach to $\beta$ existing nodes, selected by degree-proportional probabilities. By the conditional degree distribution $p(\ell |k)$, we mean the degree distribution of nodes that are connected to a node of degree $k$. By the joint degree distribution $p(k,\ell )$, we mean the proportion of links that connect nodes of degrees $k$ and $\ell$. In addition to this growth model, we consider the shifted-linear preferential growth model and solve for the same quantities, as well as a closed form expression for its steady-state degree distribution.

References in corpus (7)

• Langevin approach for the dynamics of the contact process on annealed scale-free networks
• Scaling of degree correlations and the influence on diffusion in scale-free networks
• Random walks on complex trees
• Routes to thermodynamic limit on scale-free networks
• Bosonic reaction-diffusion processes on scale-free networks
• Quasi-stationary analysis of the contact process on annealed scale-free networks
• Effect of node deleting on network structure

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