Abstract
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.
Highlights
This document aims at describing the diffusion of information in the peer-to-peer (P2P)network related to a crypto, for instance Bitcoin, through a fundamental diffusion approach.The information could be anything, but the object of communication between agents, and, to add context, it could be the hash calculation
Miners have the purpose of hash transactions within a list of transactions—the memory pool—which appear in the block in the blockchain Lipton and Treccani (2021)
Each transaction circulates from node to node in the memory pool used for remembering all new transactions, and miners receiving new transaction lists, supposing no double spending, start to generate hashes to create the new block
Summary
This document aims at describing the diffusion of information in the peer-to-peer (P2P). Considering that a directed graph is essential if we need to describe the interactions within a P2P network, this is what we do in this paper, and this constitutes the main innovation of the present study Another hypothesis is made in this modeling: when a vertex agent receives a piece of information, he/she immediately treats it and diffuses it to its neighbors. Introducing Heaviside functions in the Laplacian operator is a much easier task in the context of graph modeling, for the reason that there are no discontinuity issues for the variable x, usually continuous, and here being discrete (and representing the node index) We engage this point of view at the end of the study.
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