# Crypto indexes and benchmarks: impact of an ever-increasing number of cryptocurrencies on Bitcoin dominance

#### Variables and Definitions
- Let $$( B(t) )$$ represent Bitcoin's market capitalization at time $$ t $$.
- Let $$( M(t) )$$ denote the total market capitalization of all cryptocurrencies at time $$t $$.
- Assume $$( n(t) )$$ represents the number of cryptocurrencies at time $$t$$, which approaches infinity as $$t$$ increases.

#### Differential Equations and Asymptotic Behavior
1. **Differential Equation for Bitcoin's Market Cap**:
$$
\frac{dB}{dt} = rB(t) - \delta B(t) \sum_{i=1}^{n(t)} \frac{1}{i^\alpha}
$$
Here, \( r \) is the intrinsic growth rate of Bitcoin and \( \delta \) scales the impact of competition from an increasing number of cryptocurrencies. The summation term models the dilution effect, with \( \alpha > 1 \) adjusting the influence of each additional cryptocurrency.

2. **Total Market Cap Growth**:
$$
\frac{dM}{dt} = aM(t) + \beta \int_{1}^{n(t)} \frac{1}{x^\gamma} \, dx
$$
\( a \) represents the base growth rate of the total market cap. The integral term accounts for the cumulative effect of new cryptocurrencies, with \( \beta \) scaling this contribution and \( \gamma > 1 \) controlling the diminishing influence of each successive cryptocurrency.

3. **Asymptotic Analysis for Bitcoin Dominance**:
$$
D(t) = \frac{B(t)}{M(t)} \approx \lim_{n(t) \to \infty} \frac{rB(t) - \delta B(t) \sum_{i=1}^{n(t)} \frac{1}{i^\alpha}}{aM(t) + \beta \int_{1}^{n(t)} \frac{1}{x^\gamma} \, dx}
$$

### New Index Methodology Development

To develop a new index that better reflects the expanding landscape:
- **Incorporate Total Cryptocurrency Contributions**:
Construct an index where each cryptocurrency's contribution decays according to its market cap ranking, ensuring smaller cryptocurrencies are also represented. Mathematically, this can be formulated as:
$$
I(t) = \sum_{i=1}^{n(t)} \frac{C_i(t)}{i^\theta}
$$
where $$C_i(t)$$ is the market cap of the $$i-th$$ cryptocurrency, and $$theta > 1$$ determines the rate at which the influence of lower-ranked cryptocurrencies decays.

- **Use of Integral and Summation**:
For computational simplicity and to handle the infinite series as \( n(t) \to \infty \), approximations such as integral estimates of summations can be utilized:
$$
\sum_{i=1}^{n(t)} \frac{1}{i^\alpha} \approx \int_{1}^{n(t)} \frac{1}{x^\alpha} \, dx
$$

### Conclusion

This expanded mathematical model captures the complexities of an evolving cryptocurrency market, where Bitcoin's dominance is rigorously analyzed against a backdrop of increasing competition. The new index methodology suggested will provide a more holistic and dynamic understanding of market shifts, thereby fostering a more equitable and insightful benchmarking system. This approach allows for continuous adaptation to the changing landscape of cryptocurrency investments and ensures a robust analytical tool for investors and analysts alike.