Flexible construction of hierarchical scale-free networks with general exponent

Extensive studies have been done to understand the principles behind architectures of real networks. Recently, evidence for hierarchical organization in many real networks has also been reported. Here, we present a hierarchical model that reproduces the main experimental properties observed in real networks: scale-free of degree distribution P(k) [frequency of the nodes that are connected to k other nodes decays as a power law P(k)∼k^−γ] and power-law scaling of the clustering coefficient C(k)∼k⁻¹. The major points of our model can be summarized as follows. (a) The model generates networks with scale-free distribution for the degree of nodes with general exponent γ>2, and arbitrarily close to any specified value, being able to reproduce most of the observed hierarchical scale-free topologies. In contrast, previous models cannot obtain values of γ>2.58. (b) Our model has structural flexibility because (i) it can incorporate various types of basic building blocks (e.g., triangles, tetrahedrons, and, in general, fully connected clusters of n nodes) and (ii) it allows a large variety of configurations (i.e., the model can use more than n−1 copies of basic blocks of n nodes). The structural features of our proposed model might lead to a better understanding of architectures of biological and nonbiological networks.