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The propagation of electromagnetic wave has been investigated experimentally in 2-D left-handed (LH) transmission line networks that are composed of circuit elements in disordered alignment. The backward wave and the growing evanescent field are observed in the slab-like so-called superlens configuration of the disordered transmission line network. It is found that even though the disorder level of the circuit elements is increased, only a slight fluctuation of the field is observed but no apparent distortion or broadening of the image has been observed. The results imply the potential tolerance of the LH media against disordered periodical structures, which may lead to the possible realization of quasi-random or quasi-periodic LH media.