We propose a new model for the analysis of network recurrent event data. Our model can flexibly accommodate homophily and degree heterogeneity, and can provide a natural explanation for node popularity. Theoretically, we show that the proposed method can handle the sparse network to some extent. Our method is based on multivariate counting processes, in which each subject is assigned two individual-specific time-varying degree parameters. It results in a high-dimensional estimation problem. We develop new strategies to establish the consistency and asymptotic normality of the estimators obtained by a kernel smoothing method. Simulation studies are provided to assess the finite performance of the proposed method and a real data analysis illustrates its practical utility.