We show that F-actins form three-dimensional giant network under uni-directional diffusion of polycations, at a dilute actin concentration (0.01 mg/mL) that only bundles are formed by homogeneous mixing with polycations. The mesh size of the actin network depends on polycation concentration and ionic strength, while bundle thickness of network depends only on ionic strength, which indicates that actin network is formed through nucleation-growth mechanism. The mesh size and the bundle thickness are determined by nucleus concentration and nucleus size, respectively. The atomic force microscopy measurement correlates the elasticity of the actin network, E, with the mesh size, xi, as E similar to xi(-1), while the bundle thickness, D dependence of E cannot be described by a simple scaling relation. E similar to D-6.5 when D is small and E similar to D-0.1 when D is large. Our study on the self-assembly of actin network under asymmetric polycation condition would provide the crucial insight into the organization of biopolymers in polarized condition of cell.