# Risk Equation Derivation

## Mathematical Derivation of Risk Equations

The mathematical framework for assessing the risk associated with relay families within the Anyone Network is critical for understanding the probability of network compromise due to relay concentration within a single family. This appendix provides a detailed explanation of the equations used to calculate these risks and the simplifications applied for practical implementation.

#### Basic Definitions and Formula

Relay Family ($F$): Represents the number of relays controlled by a single entity or group within the total network.

Total Relays ($R$): The total number of operational relays within the Anyone Network.

Proportion ($s$): Defined as the ratio of the number of relays in a family to the total number of relays, represented by $s = \frac{F}{R}$.

#### Probability Calculation for Multiple Hops within One Family

The primary concern is calculating the probability that two or more network hops occur within the same relay family, which could potentially compromise the networkās integrity. The probability $P(>=2)$ of having at least two hops within the same family is derived from combinatorial probabilities as follows:

$P(>=2) = 1 - \left[\frac{(R - sR)(R - sR - 1)}{(R-1)(R-2)}\right]$

This formula calculates the probability by first determining the likelihood that the first two hops are not within the family and then subtracting this value from 1.

#### Simplification and Approximation

The detailed formula can be cumbersome for computation, especially when needing quick calculations within large networks. Thus, a simplification is applied using approximations based on the assumption that the subtraction of 1 and 2 in the denominator & numerator has a minimal impact when compared to the total number of relays $R$, especially as $R$ becomes large. These approximations lead to a simplified formula:

Assuming: $R - sR - 1 \approx R - sR$ $R - 1 \approx R$ $R - 2 \approx R$$R - 1 \approx R R - 2 \approx R$

The equation simplifies to: $P(>=2) = 1 - \left[\frac{(R - sR)^2}{R^2}\right]$ $= 1 - (1 - s)^2$$= 2s - s^2$ $= s(2 - s)$

This simplified equation, $P(>=2) = s(2 - s)$, offers a clean and efficient way to estimate the risk, especially useful in scenarios where $R$ is large, and quick estimations are needed.

#### Graphical Analysis and Validation

To validate this approximation, we can plot the function $1-(1-s)^2$ against a direct calculation from the more complex combinatorial formula, particularly for different values of $R$. As shown through calculations:

For $s=0.05$ or 5%, the probability $P(>=2)$ is approximately 9.75%, which aligns closely with empirical data as $R$ increases.

Lower values of $s$, such as 0.01, show the approximation becomes more accurate as $R$ increases, reflecting its utility in large-scale networks.

#### Application and Boost Calculation

This mathematical framework supports the calculation of Family Boosts, which are designed to incentivize transparency and declaration of family status by relay operators. Boosts are calculated directly from the value $s$, with different tiers of boosts applied depending on the size of $s$ to mitigate the risks associated:

Boost for $s \leq 0.05$: A linear increase in boosts for smaller $s$ values encourages disclosure without disproportionately rewarding larger families.

Boost for $s > 0.05$: The growth of the boost amount slows down to prevent excessive concentration of benefits.

### Conclusion

The mathematical construction of risk equations provides a robust basis for implementing security measures in the Anyone Network. By utilizing these calculations, the network can effectively manage and mitigate risks associated with relay family concentrations, ensuring a secure and reliable decentralized network environment. This appendix serves as a reference point for understanding the theoretical underpinnings and practical applications of risk assessments within the Anyone Networkās operational framework.

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