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In this work, we consider a multi-hop cognitive radio network with multiple flows. The challenges induced by the random behaviors of the primary users are investigated in a stochastic network utility maximization framework. To fully utilize the scarce network resource, we propose an optimal traffic splitting scheme for each source node to explore multiple paths effectively. In addition, the algorithm is fully distributed which provably converges to the global optimum solution with probability one. The analytical results are validated via simulations.