This chapter analyzes the anonymity of the Bitcoin system, focusing on the topological structure of two networks derived from Bitcoin's public transaction history: the transaction network and the user network. Bitcoin users are identified only by public keys, and the system attempts to prevent de-anonymization by storing user-to-public-key mappings only on the user's node and allowing users to generate multiple public keys. The transaction network represents the flow of Bitcoins between transactions over time, while the user network represents the flow of Bitcoins between users. Both networks have a non-trivial topological structure and provide complementary views of the Bitcoin system, with implications for anonymity. By combining these structures with external information and techniques such as context discovery and flow analysis, the chapter investigates an alleged Bitcoin theft, which had a market value of approximately half a million U.S. dollars at the time. The analysis reveals that the user network has considerable cyclic structure, which, combined with other aspects of the Bitcoin system, poses risks to anonymity. The chapter also discusses the integration of off-network information, such as IP addresses and public keys, to identify users. It highlights the potential for de-anonymization through the use of network visualization and analysis tools, as well as the risks associated with the use of centralized Bitcoin services. The case study illustrates the potential risks to the anonymity of a user who has good reason to remain anonymous. The analysis also shows that the Bitcoin system is vulnerable to de-anonymization through the use of network structures and external information. The chapter concludes that the Bitcoin system's anonymity is limited, and that further research is needed to fully understand the implications of the system's design for user privacy.This chapter analyzes the anonymity of the Bitcoin system, focusing on the topological structure of two networks derived from Bitcoin's public transaction history: the transaction network and the user network. Bitcoin users are identified only by public keys, and the system attempts to prevent de-anonymization by storing user-to-public-key mappings only on the user's node and allowing users to generate multiple public keys. The transaction network represents the flow of Bitcoins between transactions over time, while the user network represents the flow of Bitcoins between users. Both networks have a non-trivial topological structure and provide complementary views of the Bitcoin system, with implications for anonymity. By combining these structures with external information and techniques such as context discovery and flow analysis, the chapter investigates an alleged Bitcoin theft, which had a market value of approximately half a million U.S. dollars at the time. The analysis reveals that the user network has considerable cyclic structure, which, combined with other aspects of the Bitcoin system, poses risks to anonymity. The chapter also discusses the integration of off-network information, such as IP addresses and public keys, to identify users. It highlights the potential for de-anonymization through the use of network visualization and analysis tools, as well as the risks associated with the use of centralized Bitcoin services. The case study illustrates the potential risks to the anonymity of a user who has good reason to remain anonymous. The analysis also shows that the Bitcoin system is vulnerable to de-anonymization through the use of network structures and external information. The chapter concludes that the Bitcoin system's anonymity is limited, and that further research is needed to fully understand the implications of the system's design for user privacy.