Abstract: In recent years, in a market environment where there is a competitive relationship between updated products and old products, a delayed launch of updated products has become the product operation strategy of many enterprises. Current competitive diffusion studies focus on the problems of diffusion maximization of one product under external competition, but have not considered the problems of diffusion maximization of all products under internal competition (such as competition between old products and updated products). The optimization of launching time and seed selection for updated products under a non-delisting strategy is considered: in a social network G(N，E) with existing old products, products spread their influence in the form of diffusion model P, and the old products will keep on spreading when updated products are launched, and an optimal choosing on the stage t and the p seeds will be found for launching updated products to maximize the sum of the profits from both old and updated products. In this paper, an integer programming model which is based on competition is established, and a multi-stage greedy algorithm is designed to solve large-scale instances. The basic idea of the algorithm is: select new product seeds iteratively in a certain stage of the planning period, and calculate the marginal influence of all other candidate seeds, calculate the total income of all cases, and find out the total profit of all cases. To maximize the total revenue, the seed delivery stage and seeds of the product are renewed. It is tested on scale-free network with n=1000, m=3 and a real social network data set Facebook. Computational experiments show that this algorithm has higher solution efficiency, which is 96% higher than the traditional greedy algorithm; And this algorithm has better solution quality, which is 253% better than random algorithm and 14% better than degree discount algorithm. At the same time, with simulations in a real network, it is found that the larger number of seeds of the updated products, or the larger planning stage, or the higher profit of the updated products tends to a later launching time to make the manufacturer obtain higher profits, and it can provide decision-making for enterprises to carry out network marketing.

Key words: social network analysis; influence maximization; update products; launch timing