[hal-01184363] Degree-correlation of Scale-free graphs

Barabási and Albert [1] suggested modeling scale-free networks by the following random graph process: one node is added at a time and is connected to an earlier node chosen with probability proportional to its degree. A recent empirical study of Newman [5] demonstrates existence of degree-correlation between degrees of adjacent nodes in real-world networks. Here we define the \textitdegree correlation―-correlation of the degrees in a pair of adjacent nodes―-for a random graph process. We determine asymptotically the joint probability distribution for node-degrees, d and d', of adjacent nodes for every 0≤d≤ d'≤n^1 / 5, and use this result to show that the model of Barabási and Albert does not generate degree-correlation. Our theorem confirms the result in [KR01], obtained by using the mean-field heuristic approach.

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