A random network is defined as a random array of waveguide segments with variable lengths. Each waveguide segment provides a continuous one-dimensional propagation channel. Wave scattering occurs only at the nodes of the network. The network model is shown to be equivalent to a zero-energy state of an equivalent tight-binding Hamiltonian. There can be two kinds of randomness in a network. The coordination number at each node and the length of each waveguide segment can be varied independently. A network becomes ordered only when each node has the same coordination number and all the segments are either equal in length or differ by some integral multiple of the wavelength. A uniformly extended state is found for any random network provided all the segments are of some integral multiple of the wavelength, modulo 2\ensuremath{\pi}. Both analytical and numerical approaches are used to investigate the critical localization behavior in the vicinity of either the uniform state or the ordered network. Our results show that the network model belongs to the same universality class as the disordered tight-binding Anderson model, but with special tunable characteristics. It is thus suggested that a random network of optical fibers can be an effective system for the observation of light localization.