Conventional modelling of networks evolving in time focuses on capturing variations in the network structure. However, the network might be static from the origin or experience only deterministic, regulated changes in its structure, providing either a physical infrastructure or a specified connection arrangement for some other processes. Thus, to detect the change in network use, we need to focus on the processes happening on the network. In this work, we present the concept of monitoring random temporal edge network processes that take place on the edges of a graph with a fixed structure. Our framework is based on the generalized network autoregressive statistical models with time-dependent exogenous variables (GNARX models) and cumulative sum control charts. To demonstrate its effective detection of various types of changes, we conduct a simulation study and monitor cross-border physical electricity flows in Europe.