The dynamical modeling and simulation analysis of the recommendation on the user–movie network

• The dynamical models are established to describe the evolution of a user–movie network with time.
• The personal search and different recommendation methods are studied and compared.
• The search time is considered in our work for the first time.
• The effects of the different recommendation methods depend on the initial degree of the movie.

At present, most research about the recommender system is based on graph theory and algebraic methods, but these methods cannot predict the evolution of the system with time under the recommendation method, and cannot dynamically analyze the long-term utility of the recommendation method. However, these two aspects can be studied by the dynamical method, which essentially investigates the intrinsic evolution mechanism of things, and is widely used to study a variety of actual problems. So, in this paper, network dynamics is used to study the recommendation on the user–movie network, which consists of users and movies, and the movies are watched either by the personal search or through the recommendation. Firstly, dynamical models are established to characterize the personal search and the system recommendation mechanism: the personal search model, the random recommendation model, the preference recommendation model, the degree recommendation model and the hybrid recommendation model. The rationality of the models established is verified by comparing the stochastic simulation with the numerical simulation. Moreover, the validity of the recommendation methods is evaluated by studying the movie degree, which is defined as the number of the movie that has been watched. Finally, we combine the personal search and the recommendation to establish a more general model. The change of the average degree of all the movies is given with the strength of the recommendation. Results show that for each recommendation method, the change of the movie degree is different, and is related to the initial degree of movies, the adjacency matrix AA representing the relation between users and movies, the time tt. Additionally, we find that in a long time, the degree recommendation is not as good as that in a short time, which fully demonstrates the advantage of the dynamical method. For the whole user–movie system, the preference recommendation is the best.