Diffusion on the peer-to-peer network

In a peer-to-peer complex environment, information is permanently diffused. Such an environment can be modeled as a graph, where there are flows of information. The interest of such modeling is that (1) one can describe the exchanges through time from an initial state of the network, (2) the description can be used through the fit of a real-world case and to perform further forecasts, and (3) it can be used to trace information through time. In this paper, we review the methodology for describing diffusion processes on a network in the context of exchange of information in a crypto (Bitcoin) peer-to-peer network. Necessary definitions are posed, and the diffusion equation is derived by considering two different types of Laplacian operators. Equilibrium conditions are discussed, and analytical solutions are derived, particularly in the context of a directed graph, which constitutes the main innovation of this paper. Further innovations follow as the inclusion of boundary conditions, as well as the implementation of delay in the diffusion equation, followed by a discussion when doing approximations useful for the implementation. Numerous numerical simulations additionally illustrate the theory developed all along the paper. Specifically, we validated, through simple examples, the derived analytic solutions, and implemented them in more sophisticated graphs, e.g., the ring graph, particularly important in crypto peer-to-peer networks. As a conclusion for this article, we further developed a theory useful for fitting purposes in order to gain more information on its diffusivity, and through a modeling which the scientific community is aware of.