Networks are important representations in computer science to communicate structural aspects of a given system of interacting components. The evolution of a network has several topological properties that can provide us information on the network itself. In this paper, we present a methodology to compare the the topological characteristics of the evolution of a network, encoded into a (persistence) diagram that tracks the lifetimes of those features. This will enable us to classify the evolution of networks based on the distance between the diagrams that represent such network evolution. In that, we also consider complex vectors that bring a complementary perspective to the distance-based classification that is closer to the computational methods, aims to enhance the computational efficiency of those comparisons, and that is by itself a source of open research questions.