On the other hand, graph theory has shown to have applicability in representing and modeling various real-world systems as random graphs, as many of their crucial features are challenging to capture using deterministic models. In particular, a model of random graphs with a fixed number of vertices n is a stochastic mechanism for determining which of the n(n -1)/2 potential edges actually arise. In 1959, the Erdős-Rényi (ER) random graph was defined. In this graph, every pair of vertices is independently linked with probability p. When p remains constant, it is termed as a dense ER graph. Conversely, if p = O(1/n), it is referred to as a sparse ER graph. The properties of ER graphs have been thoroughly examined, including their largest independent sets, chromatic number, Hamiltonian paths, connected components, automorphisms, among others . Additionally, in terms of applications, ER is essentially akin to an elementary model of an epidemic known as the Reed-Frost model, the findings regarding ER are beneficial for understanding evolutionarily stable strategies and, recently, some slight modifications of the original model serve as a significant foundation to model interactions within communities . In simpler terms, communities represent tightly-knit groups of nodes where interactions are frequent among members but less so with nodes outside the community. Hence, community analysis can reveal important patterns, decomposing large collections of interactions into more meaningful components. In general, although ER graphs are commonly recognized as suboptimal models for interaction networks, their simplicity, mathematical manageability, and relations with alternative models possessing real-world properties render them a valuable asset in network research and analysis.