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# Random networks with q-exponential degree distribution

Dec 2022

We use the configuration model to generate networks having a degreedistribution that follows a $q$-exponential,$P_q(k)=(2-q)\lambda[1-(1-q)\lambda k]^{1/(q-1)}$, for arbitrary values of theparameters $q$ and $\lambda$. We study the assortativity and the shortest pathof these networks finding that the more the distribution resembles a pure powerlaw, the less well connected are the corresponding nodes. In fact, the averagedegree of a nearest neighbor grows monotonically with $\lambda^{-1}$. Moreover,our results show that $q$-exponential networks are more robust against randomfailures and against malicious attacks than standard scale-free networks.Indeed, the critical fraction of removed nodes grows logarithmically with$\lambda^{-1}$ for malicious attacks. An analysis of the $k_s$-coredecomposition shows that $q$-exponential networks have a highest $k_s$-core,that is bigger and has a larger $k_s$ than pure scale-free networks. Being atthe same time well connected and robust, networks with $q$-exponential degreedistribution exhibit scale-free and small-world properties, making them aparticularly suitable model for application in several systems.

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