“Living” polymeric networks made by noncovalent reversible bonds exhibit gel-like elasticity at time scales shorter than the lifetime of the network bridge, but show sol-like fluidity at longer time scales. To explore sol–gel transition of such fluid living networks, we studied rheology of the poly(vinyl alcohol)-borax solutions and diffusion of Brownian particles dispersed in the solutions by using diffusing-wave spectroscopy microrheology over a wide range of frequency (approximately 0.1–105 rad/s). At a certain borax concentration Cb = Cb*, the microrheologically estimated dynamic modulus exhibits a power-law behavior in terms of the frequency ω at ω above a terminal flow regime (>100 rad/s). We developed a theory to describe the linear viscoelasticity of critical physical gels by extending a known theory of chemical gelation. The theoretically derived dynamic modulus agrees well with the experimental results. Also the time-cure-superposition is experimantally satisfied for the mean-square-displacement of the Brownian particles at borax concentrations around Cb*. The shift factors to construct the master curves obey the power-law if plotted against the relative distance from this particular concentration ϵ ≔ (Cb–Cb*)/Cb*. All these facts indicate that a percolated network is formed at the borax concentrations above Cb*. We found an anomalous domain around the thus-estimated gel point in which the viscosity is nearly independent of the extent of cross-linking. We argue that the plateau viscosity is inherent in flowable weak physical gels.