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The universal fractality of river networks is very well known, however understanding of their underlying mechanisms is still lacking from a stochastic point of view. In this study, we have described the fractal natures of river networks by introducing a stochastic model where the direction of river flow at a site is determined by the dynamical replication probability which depends on the drainage area at the site rather than at random. We found that the probability induces dynamical persistency in river flows resulting in the self-affine properties shown in real river basins.