Formulæ for γ & deep-dives

3. Variety ($\gamma$)

The concept of Variety is quite intuitive: it calculates how "diverse" are someone's contributions across all the different Hubs they are part of. This keeps into consideration the Role of the individual in each one of their Hubs, as well as the Market in which each Hub operates, and the different Contributions they commit to complete, and actually submit. In one sentence: someone's Variety score is based on the amount of "unique" contributions (= contributions which do not repeat) they deliver across all their Hubs, divided by the total amount of Contributions submitted (= how many of the Contributions submitted are different from the others).

It's calculated using the formula:

$$\gamma = \displaystyle \sum_{h = 1}^n r\gamma_{\tiny {j, h}} \times \frac {100} {\tt H \cdot}$$

where $r\gamma$ is the "credible relative variety" of j’s contribution at the local Hub’s level.

Deep-dive: calculating $r\gamma$

As a reminder, $r\gamma$ is calculated by:

1. Checking the number of unique contribution types (uCT) completed by j in $[\tt h \cdot]$

2. Calculating the fractional Contribution Types of j in $[\tt h \cdot]$, as:

$\displaystyle fCT = \frac {uCT_{j, h}} {max(uCT_{k, h})}$

where:

• fCT is the fractional value of j’s Contribution Types in the Hub $[\tt h \cdot]$

• uCT is the amount of Unique Contribution Types completed by j in the Hub $[\tt h \cdot]$

• $max(uCT)$ is the highest amount of Unique Contribution Types completed by any Participant in the Hub $[\tt h \cdot]$

• To properly evaluate $r\gamma_{j, h}$we need to consider the credibility of a Hub as well:

$r\gamma{j, h} = fCT \times \mathfrak P'_{(\tt h \cdot)}$

This will ensure that the Hubs considered are credible and genuine communities.

$\mathfrak P'_{(\tt h \cdot)}$ is the normalized value of Prestige for a Hub $h$, calculated as:

$\displaystyle\mathfrak P'{(\tt h \cdot)} = \frac {\mathfrak P{(\tt h \cdot)}} {max(\mathfrak P_{(\tt N \cdot_{j})})}$

Once we have $r\gamma_{j, h} \text { } \forall \text { } \tt h \cdot in \text { } \tt H \cdot_{j}$, then we can calculate $\gamma$ as:

$$\gamma = \displaystyle \sum_{h = 1}^n r\gamma_{\tiny {j, \tt H \cdot}} \times \frac {100} {\tt H \cdot}$$