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The intrinsic dynamical complexity of classically chaotic systems enforces a universal description of the transport properties of their wave-mechanical analogues. These universal rules have been established within the framework of linear wave transport, where nonlinear interactions are omitted, and are described using Random Matrix Theory (RMT). Here, using a nonlinear complex network of coaxial cables (graphs), we exploit both experimentally and theoretically the interplay of nonlinear interactions and wave chaos. We develop general theories that describe our asymmetric transport (AT) measurements, its universal bound, and its statistical description via RMT. These are controlled by the structural asymmetry factor (SAF) characterizing the structure of the graph. The SAF dictates the asymmetric intensity range (AIR) where AT is strongly present. Contrary to the conventional wisdom that expects losses to deteriorate the transmittance, we identify (necessary) conditions for which the AIR (AT) increases without deteriorating the AT (AIR). Our research initiates the quest for universalities in wave transport of nonlinear chaotic systems and has potential applications for the design of magnetic-free isolators.