# Modeling Bitcoin Growth with Network Theory

Bitcoin is often viewed as a purely digital asset, primarily understood through its price movements and market dynamics. However, when examined through the lens of network theory, Bitcoin reveals itself to be much more than just a digital currency. Network theory, which studies how nodes (individuals or entities) and links (connections or interactions) interact within a network, provides powerful tools for understanding complex systems. By analyzing Bitcoin’s on-chain data, particularly the number of addresses with non-zero balances and the price, we can model its growth and behavior using principles from network theory. This approach not only enhances our understanding of Bitcoin’s underlying structure but also offers predictive insights into its future development.

Bitcoin operates on a decentralized network, where each address represents a node, and transactions between addresses represent links. The growth of this network, in terms of both the number of active addresses and the transaction volume, can be modeled using differential equations that describe how networks expand and evolve over time. These models capture the essence of Bitcoin’s network dynamics, highlighting how the interplay between nodes and links drives the system’s growth. By applying these models to Bitcoin, we can derive significant insights into its valuation and growth trajectory, demonstrating the applicability of network theory to digital currencies.

Network theory has been successfully applied to various natural and social systems, such as the spread of diseases, the structure of the internet, and social networks like Facebook and Twitter. Applying these concepts to Bitcoin allows us to understand its growth in a structured and mathematical way, moving beyond speculative market analysis to a more scientific approach. This not only solidifies Bitcoin’s standing as a technological innovation but also underscores the robustness of its network structure, which is crucial for its long-term sustainability and value appreciation.

The foundation of our approach is based on the principles outlined by Albert-László Barabási (Barabasi et al, 2002) a pioneer in network science, who has extensively studied the growth and structure of complex networks. His work, particularly in scale-free networks, provides a theoretical backbone for understanding how Bitcoin’s network might evolve. Furthermore, our methodology is inspired by the recent paper “Power Law Growth in Social, Information, and Technological Networks” (Zang et al., 2015), which offers a comprehensive framework for modeling network growth using power laws. This paper serves as a crucial source for our analysis, guiding our application of network theory to Bitcoin.

In this article, we will delve into the specifics of how network theory can be applied to Bitcoin. We will explore the mathematical models that describe the growth of nodes (addresses) and links (transactions), and how these models align with real-world data. By doing so, we aim to provide a comprehensive understanding of Bitcoin’s growth dynamics, supported by empirical evidence and theoretical foundations. The application of Barabási’s network growth principles, combined with the analytical framework from the “Power Law Growth in Social, Information, and Technological Networks” paper, allows us to present a robust model of Bitcoin’s network-driven value proposition.

## The Power-Law Growth Model

In network theory, a power law describes how a relative change in one quantity results in a proportional relative change in another raised to some power. This relationship is found in many natural and social systems, from city populations to internet connectivity. When applied to Bitcoin, the power law can describe how the network of Bitcoin addresses and the price evolve over time.

While we previously demonstrated that the empirical data regarding addresses vs. time, price vs. addresses, hash rate, and price all exhibit power-law relationships, our current work takes a more generalized approach. Rather than assuming the data followed a power-law growth, we used the NETTIDE model from the paper “On Power Law Growth of Social Networks” to find parameters that closely reproduced the empirical data.

The NETTIDE model is versatile and can describe almost all known growth patterns of networks, including sigmoid curves of adoption, exponential growth, logistic curves, and power-law growth, with the appropriate range of parameters. When applied to our data, the model revealed that the best fits were power laws for both the growth of addresses and the growth of links, which in turn accurately reproduced the price evolution over the long term. This application of the NETTIDE model underscores its robustness and ability to capture the dynamics of Bitcoin’s network, illustrating the fundamental nature of power-law growth in both the network’s structure and the market value of Bitcoin.

## Nodes and Links: The Building Blocks of Bitcoin’s Network

In our model, we use the number of Bitcoin addresses with non-zero balances as a proxy for nodes in the network. The price of Bitcoin is then assumed to be proportional to the number of links, which is a concept similar to Metcalfe’s Law. Metcalfe’s Law suggests that the value of a network is proportional to the square of the number of its nodes, but we focus on the simpler assumption that the price scales with the number of connections or interactions in the network.

To understand Bitcoin’s growth dynamics, we employed the NETTIDE model, solving its differential equations numerically. These equations model the growth of nodes (addresses with non-zero balances) and links (connections or transactions between addresses). The growth of links is constrained by the empirical data of the addresses, ensuring that our model closely follows the observed network behavior.

By varying the parameters in different regimes we can reproduce different network growth patterns.

The value N represents the saturation point, the maximum number of nodes the network can reach. However, our data suggests that we are far from this saturation point, implying that Bitcoin’s network is still in an expansive growth phase. This makes a power-law model with θ=1 the best fit for both the growth of nodes and links.

## Simulation and Empirical Data

The beauty of this approach is that it’s not just a fit but a simulation that uses real data parameters to match empirical data. By solving these differential equations with initial conditions and parameters derived from actual Bitcoin network data, we obtain a theoretical model that aligns closely with the observed growth of Bitcoin addresses and price.

This approach provides several advantages:

**Comprehensive Modeling**: The NETTIDE model encompasses a wide range of growth patterns observed in network dynamics. It can describe almost all known growth patterns of networks, including sigmoid curves of adoption, exponential growth, logistic curves, and power laws, depending on the parameter settings. This flexibility makes it particularly powerful in modeling complex systems like Bitcoin.**Data-Driven**: The parameters β, β′, θ, and N are not arbitrarily chosen but are derived from the actual data. This ensures that the model is firmly grounded in reality, enhancing its predictive accuracy.**Predictive Power**: By accurately modeling the growth of Bitcoin addresses and the number of transactions, the NETTIDE model allows us to predict future trends in Bitcoin’s network development and price evolution. This can be invaluable for investors, developers, and policymakers.**Insightful Analysis**: The model helps in understanding the fundamental drivers of Bitcoin’s growth. By examining how changes in network parameters affect the overall growth, we gain insights into what factors are most influential in Bitcoin’s development.

In summary, the NETTIDE model not only validates the observed power-law growth in Bitcoin’s network but also provides a robust framework for predicting future trends based on empirical data. This aligns well with the principles of network theory and underscores the significance of Bitcoin as a network-driven phenomenon.

## Results

As shown in the graph, the black lines represent the actual data for the number of addresses with non-zero balances and the Bitcoin price over time. The red lines represent the solutions to our differential equations. The close alignment between the empirical data and our model underscores the robustness of this approach.

## Densification of the Bitcoin network

The concept of densification in network theory refers to how the number of connections (links) within a network increases in relation to the number of nodes (addresses). In simpler terms, as a network grows, it can either spread out, adding new nodes with relatively few connections, or it can densify, where existing nodes form more connections. Densification indicates a network that is becoming more interconnected over time.

In the context of Bitcoin, densification can be understood through the relationship between the number of addresses with non-zero balances (nodes) and the price (which we equate to the number of connections or links). The red line in the graph represents the theoretical model derived from the NETTIDE framework, while the black line represents the actual empirical data.

## Introducing Periodicity to Model Bitcoin Price Bubbles

To capture another critical aspect of Bitcoin’s behavior — the periodic bubbles associated with the halving cycles — we introduced periodicity into the model. Specifically, we made the parameter α\alphaα periodic with a period of four years. This modification reflects the cyclical nature of Bitcoin halvings, which occur approximately every four years and significantly impact Bitcoin’s price dynamics.

By incorporating periodicity into the parameter α, the degree of freedom, the modified model now accounts for both the long-term power-law trend and the short-term fluctuations (bubbles) in Bitcoin’s price.

In the context of the NETTIDE (Network TIDE) model from the paper, α represents a parameter that influences the growth rate and the nature of the network’s connections or edges. It plays a critical role in determining how the network evolves over time.

## Real Network Grow Like Power Laws Not S-Curves

The original NETTIDE paper demonstrates the applicability of their model by fitting the growth dynamics of several real-world social networks: WeChat, arXiv, Enron, and Weibo. Each of these networks exhibits distinct characteristics, yet the model successfully captures their growth patterns using a power law relationship.

**WeChat**

**Dataset**: WeChat is a widely-used social media platform in China.**Findings**: The model shows that WeChat’s cumulative node and link growth can be accurately predicted, with the densification plot displaying a clear power law relationship.

**arXiv**

**Dataset**: arXiv is a repository for research papers, mainly in the fields of physics, mathematics, and computer science.**Findings**: Similar to WeChat, the model predicts the cumulative growth and rate of new connections. The densification plot again reveals a power law, demonstrating the robustness of the NETTIDE model across different types of networks.

**Enron**

**Dataset**: The Enron email dataset includes email communications among Enron employees, providing insights into corporate communications.**Findings**: The model captures the cumulative growth and rate of email exchanges, even accounting for anomalies like the company’s bankruptcy. The densification plot fits well with a power law.

**Weibo**

**Dataset**: Weibo is another major social media platform in China.**Findings**: The model accurately predicts Weibo’s growth in terms of nodes and links, with the densification plot showing a power law.

**Bitcoin**

**Dataset**: On-chain data.**Findings:**Bitcoin is now part of the set of networks that exhibit power law growth as modeled by the NETTIDE framework. However, Bitcoin stands out in a unique way. Unlike traditional networks where value is often derived from user engagement or content, Bitcoin’s value is intrinsically linked to the growth of its network. The number of addresses with a non-zero balance (nodes) and the transactions between them (links) directly influence its price. This dual nature of Bitcoin — as both a network and an asset — distinguishes it from other networks and financial assets. The power law relationship between Bitcoin’s addresses and price underscores its unique position in the landscape of digital assets and social networks.

## Significance

This model demonstrates that Bitcoin’s growth can be understood and predicted using principles of network theory. The power-law relationship is not just a statistical fit but a reflection of the underlying dynamics of Bitcoin’s network. By modeling Bitcoin in this way, we gain deeper insights into its long-term behavior and value, grounded in the well-established principles of network growth.

**Predictive Power**: The close match between the theoretical model and the empirical data shows that our model can effectively capture the underlying dynamics of Bitcoin’s growth. This implies that the parameters used in the model are robust and accurately reflect the real-world behavior of the Bitcoin network.**Network Robustness**: Densification is a sign of a robust network. In a densifying network, nodes become more interconnected, which can enhance the resilience of the network. For Bitcoin, this means increased security and stability, as more connections can help prevent single points of failure.**Value Correlation**: The proportional relationship between the number of addresses and the price suggests that the value of Bitcoin is closely tied to the network’s structure. As the network grows and becomes more interconnected, the value of Bitcoin increases. This supports the idea that Bitcoin’s price is not just driven by speculative trading but is also fundamentally linked to the network’s growth.**Scalability**: Understanding densification helps in assessing the scalability of the Bitcoin network. If the network continues to densify, it suggests that Bitcoin can handle more transactions and users without degrading performance. This is crucial for its long-term adoption and success.**Economic Implications**: The power-law relationship found in the model indicates a scale-free nature of the Bitcoin network. This means that the network exhibits properties similar to other complex systems, such as the internet or social networks, where a few nodes (addresses) hold a significant number of connections. This has implications for understanding the distribution of wealth and influence within the Bitcoin ecosystem.

## Relevance of Our Result:

Our model shows that the densification of the Bitcoin network follows a power-law distribution, which is a significant finding for several reasons:

**Validation of Network Theory**: The application of network theory to Bitcoin and the successful modeling of its growth dynamics validate the idea that Bitcoin is not a normal asset but follows dynamics more similar to physical systems.**Strategic Insights**: For investors, developers, and policymakers, understanding the power law trend provides strategic insights into the potential future growth of Bitcoin. It can guide decisions on investment, development priorities, and regulatory approaches.**Long-Term Vision**: The power-law fit suggests that Bitcoin’s growth will continue to follow a predictable pattern, barring any major disruptions. This long-term vision can help stakeholders make informed decisions about their involvement in the Bitcoin network.

## Conclusion

Bitcoin’s value and growth are deeply rooted in its network structure. By applying network theory and power-law modeling, we can better understand and predict its behavior. This approach highlights the interconnected nature of Bitcoin and provides a powerful tool for analyzing its long-term potential.

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