Graph colouring is the system of assigning a colour to each vertex of a graph It is done in such a way that adjacent vertices do not have equal colour It is fundamental in graph theory It is often used to solve real-world problems like traffic light signalling, map colouring, scheduling, etc Nowadays, social networks are prevalent systems in our life Here, the users are considered as vertices, and their connections/interactions are taken as edges Some users follow other popular users’ profiles in these networks, and some don’t, but those non-followers are connected directly to the popular profiles That means, along with traditional relationship (information flowing), there is another relation among them It depends on the domination of the relationship between the nodes This type of situation can be modelled as a directed fuzzy graph In the colouring of fuzzy graph theory, edge membership plays a vital role Edge membership is a representation of flowing information between end nodes of the edge Apart from the communication relationship, there may be some other factors like domination in relation This influence of power is captured here In this article, the colouring of directed fuzzy graphs is defined based on the influence of relationship Along with this, the chromatic number and strong chromatic number are provided, and related properties are investigated An application regarding COVID-19 infection is presented using the colouring of directed fuzzy graphs