Random networks with q-exponential degree distribution

Abstract

We use the configuration model to generate random networks having a degree distribution that follows a q-exponential, P_{q}(k)=(2−q)λ[1−(1−q)λk]^{−1/(q−1)}, for arbitrary values of the parameters q and λ. Typically, for small values of λ, this distribution crosses over from a plateau at small k's to a power-law decay at large values of the node degrees. Furthermore, by sufficiently increasing λ, we can continuously narrow this plateau, getting closer and closer to a pure power-law degree distribution. As a generalization of the pure scale-free networks, therefore, q-exponentials display a rich variety of behavior in terms of their topological and transport properties. This is substantiated here by investigating their average degree, assortativity, small-world behavior, resilience to random and malicious attacks, and k-core decomposition. Our results show that the more the degree distribution resembles a pure power law, the less well connected the networks. As a consequence, their average degree follows 〈k〉∼λ^{−1} for λ<1, and the expected average degree k_{nn} of the nearest neighbors of a given node with degree k generally decreases with λ. Moreover, random q-exponential networks exhibit small-world behavior for any λ, but with an average shortest path that becomes smaller as λ decreases and q increases. Finally, q exponentials become more resilient to random and malicious attacks as their degree distribution systematically deviates from the pure power law. Being at the same time well-connected and robust, networks with q-exponential degree distribution exhibit scale-free and small-world properties, making them a particularly suitable model for applications in several systems.