Characterizing the intrinsic correlations of scale-free networks

Very often, when studying topological or dynamical properties of random scale-free networks, it is tacitly assumed that degree-degree correlations are not present. However, simple constraints, such as the absence of multiple edges and self-loops, can give rise to intrinsic correlations in these structures. In the same way that Fermionic correlations in thermodynamic systems are relevant only in the limit of low temperature, the intrinsic correlations in scale-free networks are relevant only when the extreme values for the degrees grow faster than the square-root of the network size. In this situation, these correlations can significantly affect the dependence of the average degree of the nearest neighbors of a given vertice on this vertices’s degree. Here, we introduce an analytical approach that is capable to predict the functional form of this property. Moreover, our results indicate that random scale-free networks models are not self-averaging, that is, the second moment of their degree distribution may vary orders of magnitude among different realizations. Finally, we argue that the intrinsic correlations investigated here may have profound impact on the critical properties of random scale-free networks.

💡 Research Summary

The paper investigates a subtle but important source of degree‑degree correlations that arise intrinsically in random scale‑free networks when two natural constraints are imposed: the prohibition of self‑loops and multiple edges between the same pair of vertices. While many studies of scale‑free graphs assume that the degree of a vertex is uncorrelated with the degrees of its neighbors, the authors show that the exclusion principle inherent in these constraints generates “intrinsic correlations” that become relevant whenever the maximum degree K_max grows faster than the square‑root of the network size N (i.e., K_max ∝ N^θ with θ > ½).

The authors use the configuration model to generate networks with a prescribed power‑law degree distribution P(k) ∝ k^−γ. Degrees are assigned according to a continuous formula involving a minimum and maximum cutoff (K_min, K_max) and then rounded to integers. Edges are added by repeatedly selecting two vertices with probability proportional to their remaining stubs, but any attempt that would create a self‑loop or a duplicate edge is rejected. For highly connected vertices this leads to a situation where the naïve expected number of edges between i and j, n_ij = k_i k_j/(N⟨k⟩), would exceed one, which is impossible under the constraints. Consequently, many edge‑placement attempts are frustrated, and the actual network exhibits a non‑trivial correlation between the degrees of adjacent vertices.

To describe this analytically, the authors borrow the formalism used for fermionic systems. They rewrite the expected number of edges as

 n_ij = g(k_i) g(k_j) /