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When network products and services become more valuable as their userbase grows (network effects), this tendency can become a major determinant of how they compete with each other in the market and how the market is structured. Network effects are traditionally linked to high market concentration, early-mover advantages, and entry barriers, and in the market they have also been used as a valuation tool. The recent resurgence of Bitcoin has been partly attributed to network effects, too. We study the existence of network effects in six cryptocurrencies from their inception to obtain a high-level overview of the application of network effects in the cryptocurrency market. We show that, contrary to the usual implications of network effects, they do not serve to concentrate the cryptocurrency market, nor do they accord any one cryptocurrency a definitive competitive advantage, nor are they consistent enough to be reliable valuation tools. Therefore, while network effects do occur in cryptocurrency networks, they are not (yet) a defining feature of the cryptocurrency market as a whole.