# Formulas for Prestige

### General Formula for Prestige

From these assumptions, we derived a generalized formula for the Prestige of a decentralized community:

Where:

$\mathfrak {P}_{\tiny n}$ is the Prestige of a community in the given period.

$\mathfrak {P}_{\tiny (n-1)}$ is the Prestige of a community in the previous period.

$\Delta K$ is the rate of change of each individual parameter, between the current and the previous intervals.

### Calculating K & **ΔK**

**ΔK**

ΔK is the rate of change of individual parameters - to calculate the total change across all parameters, we can simply use the sum of all $\Delta K$, calculated as $\displaystyle \sum_1^i \Delta K_{i}$.

Where the individual parametric change is calculated as:

$\displaystyle \Delta K_{i} = \frac {K_{i_{( n)}}} {K_{i_{( n-1)}}}$

This change rate is determined by the difference in Key Performance Indicators (the *parameters*) between periods, each one individually normalized and weighted according to their importance in each Hub using Archetypes.

$K_{i}$ is the product of each individual parameter multiplied by its custom weight:

$K_{i} = w \times p'$

where:

*w*: the weight assigned to each individual parameter (*p*)*p'*: normalized-p, calculated as $p' = \frac {p} {p_{(max)}}$*p*: the "parameter" of reference.

**Deep-dive**: *Current Parameters*

The system currently uses 5 parameters to calculate a Hub's Prestige and their ability to deliver on operational and business drivers.

Current parameters are:

$\overlinesegment S$ (Size) -> $\displaystyle \frac {TCM_{\tt h\cdot}}{max(TCM)}$ A relative value that compares the

*TCM*(Total Community Members) of a Hub against the TCM of the biggest Hub in the ecosystem.$\overline {PS}$ (Average Member Participation), calculated as: $\displaystyle \sum_{j = 1}^n PS_{j} \times \frac {1}{TCM}$

$\overline {iCL}$ (Average Commitment Level in a Hub), calculated as: $\displaystyle \sum_{j =1}^n iCL_{j} \times \frac {1} {TCM}$

$\overrightarrow P$ (Community Performance), calculated as: $\displaystyle \sum_{j = 1}^n GCp_{j} \times \frac {1} {TCp}$ Given Contributions (from Hub's Participants) over Total Contribution Points.

$\check G$ (Growth of Community), calculated as: $\displaystyle \frac {TCM_{T_{n}} - TCM_{T_{n-1}}}{TCM_{T_{n-1}}}$

### Normalization of *p*

*p*

Normalized-p is calculated through:

It uses the highest value of a specific parameter across all Hubs measured during the given period. The weights for the different Parameters (below) can be set by the individual Hubs - and are set by our Hub for the edge limits.

In total, the sum of the weights must always be equal to 100 - in alignment with the models used for Discrete Commitment and Reputation staking in the Āutonomy Matrix.

The Prestige of a Hub, at the current period ( $\mathfrak P_{\tiny n}$), is updated by multiplying the change rate (*ΔK*) to the previous period's Prestige ($\mathfrak{P}_{\tiny (n-1)}$).

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