Formulæ for γ & deep-dives

3. Variety (γ\gamma)

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

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

Deep-dive: calculating rγr\gamma

As a reminder, rγr\gamma is calculated by:

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

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

fCT=uCTj,hmax(uCTk,h)\displaystyle fCT = \frac {uCT_{j, h}} {max(uCT_{k, h})}

where:

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

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

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

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

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

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

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

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

Once we have rγj,h  hin Hjr\gamma_{j, h} \text { } \forall \text { } \tt h \cdot in \text { } \tt H \cdot_{j}, then we can calculate γ\gamma as:

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

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