Abstract
Many information-technology innovations are driven, in their early stages, by an egalitarian ethos that empowers individuals through dis-intermediation. Bitcoin and peer to peer financial systems were inspired by these egalitarian ambitions. However, in bitcoin we have recently witnessed a strong centralization around a few large mining pools, which puts control of most of the system in the hands of a few. In this chapter we investigate the physical limits of distributed consensus mechanisms over networks, and discuss whether there are scalability and efficiency reasons that incentivize centralization. We compute the time to reach majority consensus in a variety of settings, comparing egalitarian networks with centralized networks, and quantifying the effect of network topology on the propagation of information.
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Notes
- 1.
More in general, it is possible to show that the initial phases of the propagation process are characterized by an exponential behavior for the fraction of informed nodes over time. This increases with a characteristic time \( \tau = 1/\beta \),with \( \beta = \frac{{cp_{0} }}{\langle l\rangle }\frac{{\langle k^{2} \rangle - \langle k\rangle }}{\langle k\rangle } \). Note that, for a scale-free network with a tail exponent \( 2\, < \gamma \, \le \,3 \), if we consider the natural cut-off \( N^{1/(\gamma - 1)} \) for the degree distribution (Sergey 2008), we find that \( \beta \) scales as \( \beta { \sim }\langle k^{2} \rangle /\langle k\rangle { \sim }N^{1/(\gamma - 1)} \) for large values of \( N \). This would alter the dependence of \( T(0.5) \) on \( N \) in Eq. (12), speeding up the propagation.
- 2.
Note here that the analytical equations are not defined for \( n^{*} = 1 \).
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Caccioli, F., Livan, G., Aste, T. (2016). Scalability and Egalitarianism in Peer-to-Peer Networks. In: Tasca, P., Aste, T., Pelizzon, L., Perony, N. (eds) Banking Beyond Banks and Money. New Economic Windows. Springer, Cham. https://doi.org/10.1007/978-3-319-42448-4_11
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