Giovanni Santostasi and GPT4 discuss scale properties of adoption and value

Giovanni Santostasi
9 min readApr 1, 2024


Giovanni (3/31/2024): Hi GPT4,
I see in the Bitcoin on-chain data an interesting pattern.
For addresses with different balances (that I’m using as proxies for network activity), I see a power law in time for each of these cut-offs. However, the exponent is different for different balances. The higher balances addresses tend to grow slower in time and the addresses with smaller cutoffs tend to go closer to the power of 3 in time.

Now when you look at the relationship between addresses and price, a la Metcalfe law, you see the smaller cut-off come closer and closer to the theoretical value of Metcalfe’s law.
The interesting thing is that if you take the product of the growth in time of all these cutoffs power law exponents and the generalized Metcalfe for each cutoff you get something close to 6 which is the power law relationship observed between price and time itself.

Do you have an explanation for all this considering issues of adoptions, network effects, and so on?

GPT4: Your observations about the growth of Bitcoin addresses and their relationship to Bitcoin’s price underpin several interconnected dynamics that revolve around adoption, network effects, and value perception. Here’s a potential explanation integrating these aspects:

  1. Adoption and Network Growth: The initial surge in smaller balance addresses growing rapidly reflects a broader phase of adoption. As Bitcoin becomes more widely known and accepted, more individuals enter the market, contributing to the “quantity” aspect of the network. This influx can be attributed to increased awareness, perceived entry points for investment, and the utility of Bitcoin as both a store of value and a means of transaction.
  2. Differential Growth Rates: The observation that addresses with higher balances grow slower could indicate a maturation phase within the network’s demographics. Early adopters and significant investors, who are already well-positioned within the network, may not need to increase their holdings substantially. In contrast, new entrants, attracted by the growing utility and value of Bitcoin, contribute to the rapid increase in smaller balance addresses. This phenomenon mirrors wealth distribution dynamics seen in traditional economic systems, where a relatively small portion controls a significant portion of the wealth.
  3. Network Effects and Metcalfe’s Law: The relationship between the number of addresses (especially those with smaller balances) and the price approaching an exponent closer to 2 reflects Metcalfe’s Law, which posits that the value of a network is proportional to the square of the number of its users. As adoption spreads and the user base increases, the network becomes more valuable not only because of the larger number of participants but also due to the increased potential for interaction, transaction, and innovation within the ecosystem.
  4. Integration of Growth Dynamics: The convergence of growth rates in time for addresses of different balances with the generalized Metcalfe exponent for each cutoff yielding a composite exponent close to the observed power law relationship between price and time can be seen as a manifestation of the network maturing and the value being increasingly recognized. The cumulative effect of individual adoption phases, network growth, and the interplay between quantity (number of addresses) and quality (network interactions, transactions) contributes to the overall valuation of Bitcoin, as perceived by the market.
  5. Implications of Adoption and Network Dynamics: The intricate relationship between adoption rates, network growth, and valuation underscores the significance of Bitcoin’s network effects. As more individuals and institutions recognize and integrate Bitcoin into their economic activities, the network’s intrinsic value is enhanced, reinforcing its position as a digital store of value and medium of exchange. This dynamic ecosystem, driven by both speculative interest and practical utility, reflects a self-reinforcing cycle where increased adoption boosts value, which in turn attracts more participants to the network.

This explanation ties together the empirical observations with theoretical frameworks about network growth, value perception, and the digital economy’s unique characteristics. It highlights the complex interdependencies between user adoption, network expansion, and the evolving perception of Bitcoin’s value as a revolutionary digital asset.

The image I showed to GPT-4.

Giovanni: I like it, this is the graph by the way. The power law with the smallest cut off in the lower panel is, of course, the lower one with more data points (more addresses given the smaller cutoff). Do you have any more insights after seeing the graph?

GPT-4: The graphs you’ve provided lend a visual perspective to the discussion on the growth of Bitcoin addresses and their relationship with Bitcoin’s price. Here are some insights based on the visualization:

Upper Graph: Growth of Addresses Over Time

  • Each line represents the growth of addresses over time for different balance cutoffs. The lines show a clear upward trend, indicating that the number of addresses continues to grow as Bitcoin ages. The slopes provide a quantitative measure of this growth.
  • The slopes in time for different balance cutoffs appear to be different, which aligns with your previous statement that higher balance addresses tend to grow slower. This could indicate varying degrees of investment or accumulation behavior among holders with different wealth brackets.
  • The product of the slopes for price versus addresses and the slope in time hovers around 5.5 to 5.7, not far from the power law relationship you described between price and time (~6).

Lower Graph: Generalized Metcalfe’s Law

  • The bottom graph plots Bitcoin price against the number of addresses, presumably on a log-log scale, to examine the Metcalfe’s law relationship.
  • The red lines possibly represent the best fit lines for different address cutoffs over time. The upward trend suggests a positive correlation between the number of addresses and the price of Bitcoin, which is consistent with Metcalfe’s law.
  • The black “jagged” line that represents the actual data shows fluctuations around the red lines. These fluctuations could represent market cycles, speculative activities, or other macroeconomic factors affecting Bitcoin’s price independently of the address count.

Additional Insights:

  • The power law dynamics reflected in these graphs highlight the non-linear nature of Bitcoin’s growth and adoption. The intersection of user growth (network size) and value appreciation (price) in such a power law fashion suggests a self-reinforcing feedback loop.
  • The varying slopes in the upper graph indicate that user growth is not uniform across all holder sizes, which may reflect the differing behaviors and strategies of users within the Bitcoin ecosystem. For instance, large holders (whales) might not need to increase their holdings as much as smaller investors looking to capitalize on growth.
  • The adherence to a power law model and the close fit to Metcalfe’s law reaffirm the argument that Bitcoin’s value is closely tied to its network’s size and the interactions within it.

Given these patterns, it would be interesting to further explore the causes behind the variation in slopes for different address balance cutoffs and how these might correlate with market events, technological developments, or changes in the regulatory environment. Understanding these underlying factors could provide deeper insight into Bitcoin’s adoption curve and market dynamics.

Giovanni: But why the product of the exponents is always the same?

GPT-4: The product of the exponents reflecting a consistent value when analyzing the growth of Bitcoin addresses (over time and in relation to price) may point to an underlying scaling law that governs the Bitcoin network’s expansion and valuation. Here’s a potential explanation considering network effects and adoption patterns:

Scaling Laws and Network Effects: Scaling laws, such as power laws, often emerge in complex systems where smaller units combine into larger structures in a self-similar manner. In the context of Bitcoin:

  • User Adoption: New users are attracted to Bitcoin not just because of the potential financial gain but also due to its growing utility and acceptance. As more people join, they contribute to the network’s utility, making it more attractive to others — a virtuous cycle. This network effect can be measured through the increase in addresses.
  • Monetary Value: Bitcoin’s monetary value can also exhibit a network effect as the price increase makes it more attractive as an investment, which in turn can attract more users, further pushing up the price.

The multiplication of the exponents from growth in time and price relationship to addresses gives a constant value, which suggests that the rates of change in network size and network value reinforce each other in a predictable way.

Intrinsic Correlation: If the network’s value (as reflected by the price) and the network’s size (as indicated by active addresses) are intrinsically correlated due to network effects, the rate at which each of these properties grows might be expected to exhibit a mathematical relationship that remains consistent over time, hence the product of their exponents being around the same value.

Interplay Between Adoption and Valuation: The power law governing the relationship between Bitcoin’s price and time could be seen as a macro-scale manifestation of the underlying network dynamics. It reflects the aggregate effect of individual decisions to participate in the Bitcoin network (creating addresses) and the market’s collective valuation of those decisions (price). The consistent product of exponents suggests that:

  • The growth of addresses affects the price in a predictable way.
  • Price increases motivate increased participation, affecting the number of addresses.

The consistent product might hint that the Bitcoin network’s growth and value follow a form of the “square-cube law” found in other natural systems, where a change in one dimension results in a predictable change in another.


Overall, these observations suggest that despite the complexity and various factors influencing Bitcoin’s adoption and price, there may be a relatively stable and quantifiable relationship between the number of users (addresses) and the perceived value (price) of the network, as guided by the principles of network theory.

Giovanni: Does the growth in the cube of time remind you of some natural processes that are relevant to the Bitcoin adoption growth:

GPT-4: The spread of diseases can sometimes follow a power-law distribution under specific conditions, particularly in the context of epidemics where the number of infected individuals can grow disproportionately to the time since the first case, especially in the early stages of an outbreak where containment measures are not yet in place. This growth is often modeled using exponential functions in the short term; however, power-law growth can be observed in some situations.

Here are some conditions and examples where the spread of diseases might follow a power-law pattern:

  1. Network Topology: Infections spread more like a power law in scale-free networks, where some individuals (nodes) have significantly more connections than others. This kind of network is common in human social structures, where a few individuals can have a vast number of contacts and thus a higher probability of spreading the disease to more people.
  2. Super-Spreading Events: Power-law growth can occur when there are super-spreader events, where a single individual infects a disproportionately large number of people. These events can significantly skew the average infection rate and cause rapid spread.
  3. Long-tailed Distributions of Contacts: If the distribution of the number of contacts per individual follows a power law (some people have very few, others have a lot), the infection rate can also follow a power law due to the variability in transmission opportunities.
  4. Behavioral Factors: Human behavior can create conditions for power-law spread, such as frequent travel or large gatherings, which can amplify the spread exponentially if not controlled.

For example, the early spread of HIV/AIDS in specific populations demonstrated power-law growth due to network effects within communities. Similarly, the spread of SARS in 2003 was heavily influenced by super-spreader events.

Link to Bitcoin Adoption Spread: In the context of Bitcoin, if we consider addresses with non-zero balances as a proxy for adoption or participation in the Bitcoin economy, and the rate of new addresses following a power law, it could be analogous to a disease spread in a few ways:

  • Network Effects: Just like individuals can influence others to adopt a behavior or even a disease, early adopters of Bitcoin can influence others to join, and their influence might not be linear but could follow a power-law if they are particularly influential (analogous to hubs in a network).
  • Adoption Events: Moments that lead to widespread media attention could act like super-spreader events, bringing a disproportionate number of new users into the space quickly.
  • Heterogeneity in Adoption Rates: Different regions and communities may adopt Bitcoin at varying rates, and if these rates are influenced by power-law distributions in social connectivity, the overall growth rate can manifest as a power-law.
  • Cultural and Social Factors: Just like behavioral factors in disease spread, cultural acceptance, regulatory changes, or economic events can lead to periods of rapid adoption, akin to a power-law distribution.

In both cases, the commonality lies in the presence of non-linear, uneven distribution of a certain feature (connectivity in networks, and economic or technological adoption for Bitcoin), leading to outcomes that cannot be modeled simply by traditional means but require an understanding of complex systems and network theory.

Giovanni: Thank you GPT-4 for your analysis, it is very relevant and interesting. Let’s talk later more about these fascinating topics.

GPT-4: You are welcome, Bitcoin is a very fascinating topic, indeed.