  A majority of studied models for scale-free networks have degree
distributions with exponents greater than $2$. Real networks, however, can
demonstrate essentially more heavy-tailed degree distributions. We explore two
models of scale-free equilibrium networks that have the degree distribution
exponent $\gamma = 1$, $P(q) \sim q^{-\gamma}$. Such degree distributions can
be identified in empirical data only if the mean degree of a network is
sufficiently high. Our models exploit a rewiring mechanism. They are local in
the sense that no knowledge of the network structure, apart from the immediate
neighbourhood of the vertices, is required. These models generate uncorrelated
networks in the infinite size limit, where they are solved explicitly. We
investigate finite size effects by the use of simulations. We find that both
models exhibit disassortative degree-degree correlations for finite network
sizes. In addition, we observe a markedly degree-dependent clustering in the
finite networks. We indicate a real-world network with a similar degree
distribution.
