To improve the convergence speed of distributed online optimization algorithms, a fast first-order Online Distributed Dual Average optimization (FODD) algorithm was proposed by sequentially adding edges to the underlying network topology. Firstly, aiming at solving the problem of the online distributed optimization to make the selected edge and network model mix quickly by using the method of edge addition, a mathematical model was established and solved by FODD. Secondly, the relationship between network topology designed and the convergence rate of the online distributed dual average algorithm was revealed, which clearly showed that, by improving the algebraic connectivity of the underlying topology network, the Regret bound could also be greatly improved. The Online Distributed Dual Average (ODDA) algorithm was extended from static networks to time-varying networks. Meanwhile, the proposed FODD algorithm was proved to be convergent and the convergence rate was specified. Finally, the results of numerical simulations show that, compared with existing algorithms such as ODDA, the proposed FODD algorithm has better convergence performance.