Statistical ensemble of scale-free random graphs

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  • 9 April 2001
Abstract
The statistical ensemble of random graphs is defined using methods borrowed from field theory. A computer algorithm to generate these random graphs is proposed. It is shown how to insure that they are scale-free, for trees and for degenerate graphs, respectively. The analytic discussion focuses on scale-free tree graphs. The behavior of the ensemble of trees is characterized by two global parameters, the fractal and the spectral dimensions, which are explicitly calculated. It is discussed in detail how the geometry of the trees varies when the weights of the nodes are modified. The stability of the scale-free regime is also considered: when it breaks down, either a scale is spontaneously generated or else, a "singular" node appears and the graphs become crumpled. Possible generalizations are discussed.

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