title: Scale-free networks with exponent one.
abstract: A majority of studied models for scale-free networks have degree distributions with exponents greater than two. 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 γ=1, P(q)∼q^{-γ}. 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 neighborhood 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.
authors.affiliation: Departamento de Física da Universidade de Aveiro and I3N, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal.
authors.forename: G
authors.initials: G
authors.lastname: Timár
authors.affiliation: Departamento de Física da Universidade de Aveiro and I3N, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal.|A. F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia.
authors.forename: S N
authors.initials: SN
authors.lastname: Dorogovtsev
authors.affiliation: Departamento de Física da Universidade de Aveiro and I3N, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal.
authors.forename: J F F
authors.initials: JF
authors.lastname: Mendes