  Networks with long-range connections obeying a distance-dependent power law
of sufficiently small exponent display superdiffusion, L\'evy flights and
robustness properties very different from the scale-free networks. It has been
proposed that these networks, found both in society and biology, be classified
as a new structure, the fractional networks. Particular important examples are
the social networks and the modular hierarchical brain networks where both
short- and long-range connections are present. The anomalous superdiffusive and
the mixed diffusion behavior of these networks is studied here as well as its
relation to the nature and density of the long-range connections.
