We revisit the competitive diffusion game on undirected connected graphs and study the complexity of the existence of pure-strategy Nash equilibrium for such games. We first characterize the utility of each player based on the location of its initial seed placements. Using this characterization, we show that the utility of each player is a sub-modular function of its initial seed set. Following this, a simple greedy algorithm provides an initial seed placement within a constant factor of the optimal solution. We show NP-completeness of the decision on the existence of pure-strategy Nash equilibrium for general networks. Finally we provide some necessary conditions for a given profile to be a Nash equilibrium and obtain a lower bound for the maximum social welfare of the game with two players.