Calculations

1. Cell Creation Time

Given the hierarchical nature of the DAG, the creation of a cell, symbolized as $CC$, is contingent upon multiple factors, such as computational speed, data packet size, and I/O operations.

$C(t)=α×P(d)+β×I/O(r)$

Where:

• $α$ and $β$ are proportional constants.

• $P(d)$ represents computational time for processing data of size $d$.

• $I/O(r)$ denotes the I/O operations for data rate $r$.

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2. Equation for Reference Cell Initiation

When data exceeds a certain threshold in a base cell, reference cells are utilized.

$F(D) = \lceil \frac{\omega}{f-\theta} \rceil$

Where:

• $f$ is data size within a cell.

• $θ$ is the reference threshold.

• $ω$ is the maximum reference size.

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3. Dynamic Parameter Updates

Given the dynamic nature, parameters adjust based on efficiency factor $e$:

$α=⌈3e⌉$

$β=⌈5e⌉$

$k=⌈5e⌉$

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4. Adjustment of Efficiency Factor

Incorporating higher-level dynamics, $e$ undergoes adjustments:

$e_{\text{new}} = e \times \left(1 + \gamma \times \frac{\delta f}{f}\right)$

Where:

• $γ$ is a modulation factor.

• $δf$ is the change in the frequency of cell creation.

Following feedback, this is further modulated:

$e=(1−smoothingFactor)×enew+smoothingFactor×min⁡(upperThreshold,enew)$

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5. Time to Consensus

$Tconsensus=Tquery+Tmajority_check+Tfeedback$

Where:

$Tquery=n×avg_delay$

And $n$ is the number of nodes.

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6. Data Retrieval

Considering hierarchical referencing:

$Tretrieval=Tbase_cell+n×Treference_cell$

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7. Throughput

Throughput, $τ$, is the rate of successful message delivery:

$\tau = \frac{n \times f}{T_{\text{consensus}} + T_{\text{retrieval}}}$

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8. Latency

Latency, $L$, can be defined as the time differential from data initiation to consensus:

$L=Tconsensus+Tquery$

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9. Decentralization

A measure, $D$, could quantify decentralization using node distribution:

$D = 1 - \sum_{i=1}^{n} (p_i - \frac{1}{n})^2$

Where $pi$ is the proportion of the $ith$ node's contribution.

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10. Costs

Costs, $Ctotal$, can encompass computational, storage, and network expenses:

$Ctotal=Ccomp+Cstore+Cnet$

Where each component cost can be derived from its respective resources and efficiency parameters.

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11. Liveness ($L$):

Liveness in a decentralized system can be quantified by the percentage of nodes that are active or responsive. If $n$ is the total number of nodes and $a$ is the number of active nodes, then:

$$L = \frac{a}{n}$$

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12. Transaction Fee ($Ftx$):

Transaction fees can be associated with the computational cost ($CC$) and the data size of the transaction ($dtx$). Thus, the formula for the transaction fee can be given by: $F_{\text{tx}}(C, d_{\text{tx}}) = \beta \times C + \gamma \times d_{\text{tx}}$

Where $β$ and $γ$ are constants representing the cost per unit of computation and cost per unit data of the transaction respectively.

12.1. Ensuring Transaction Fee is Below 0.1€:

To ensure that $Ftx$ is naturally below 0.1€, the constants $β$ and $γ$ should be chosen such that even for the maximum values of $C$ and $dtx$, the fee remains below this threshold: $\beta \times C_{\text{max}} + \gamma \times d_{\text{tx, max}} < 0.1$