  We study the quantum statistical electronic properties of random networks
which inherently lack a fixed spatial dimension. We use tools like the density
of states (DOS) and the inverse participation ratio(IPR) to uncover various
phenomena, such as unconventional properties of the energy spectrum and
persistent localized states(PLS) at various energies, corresponding to quantum
phases with with zero-dimensional(0D) and one-dimensional(1D) order. For small
ratio of edges over vertices in the network $RT$ we find properties resembling
graphene/honeycomb lattices, like a similar DOS containing a linear dispersion
relation at the band center at energy E=0. In addition we find PLS at various
energies including E=-1,0,1 and others, for example related to the golden
ratio. At E=0 the PLS lie at disconnected vertices, due to partial bipartite
symmetries of the random networks (0D order). At E=-1,1 the PLS lie mostly at
pairs of vertices(bonds), while the rest of the PLS at other energies, like the
ones related to the golden ratio, lie at lines of vertices of fixed length(1D
order), at the spatial boundary of the network, resembling the edge states in
confined graphene systems with zig-zag edges. As the ratio $RT$ is increased
the DOS of the network approaches the Wigner semi-circle, corresponding to
random symmetric matrices(Hamiltonians) and the PLS are reduced and gradually
disappear as the connectivity in the network increases. Finally we calculate
the spatial dimension $D$ of the network and its fluctuations, obtaining both
integer and non-integer values and examine its relation to the electronic
properties derived. Our results imply that universal physics can manifest in
physical systems irrespectively of their spatial dimension. Relations to
emergent spacetime in quantum and emergent gravity approaches are also
discussed.
