ARM4SNS:ReputationFunctions

PageRank

• $P$: set of hyperlinked webpages
  
• $u,v$: webpages in P
  
• $N^{-}(u)$: set of webpages pointing to u
  
• $N^{+}(v)$: set of webpages that v points to
  
• the PageRank is: $R(u)=cE(u)+c\sum _{v\in N^{-}(u)}{R(v) \over {|N^{+}(v)|}}$ (1.)
• $c$ is chosen such that $\sum _{u\in P}R(u)=1$
  
• $E$ is a vector over $P$ corresponding to a source of rank and is chosen such that $\sum _{u\in P}E(u)=0.15$
  
• first term of function (1.) $cE(u)$ gives rank value based on initial rank
  
• second term of (1.) $c\sum _{v\in N^{-}(u)}{R(v) \over {|N^{+}(v)|}}$ gives rank value as a function of hyperlinks pointing at $u$
  

Beta

Reputation Function
Let $r_{T}^{X}$ and $s_{T}^{X}$ represent the collective amount of positive and negative feedback about a Target T provided by an agent (or collection of agents) denoted by X, then the function $\varphi (p|r_{T}^{X},s_{T}^{X})$ defined by

$\varphi (p|r_{T}^{X},s_{T}^{X})={\frac {\Gamma (r_{T}^{X}+s_{T}^{X}+2)}{\Gamma (r_{T}^{X}+1)\Gamma (s_{T}^{X}+1)}}p^{r_{T}^{X}}(1-p)^{s_{T}^{X}},\qquad where\ 0\leq p\leq 1,\ 0\leq r_{T}^{X},\ 0\leq s_{T}^{X}$

is called T's reputation function by X. The tuple $(r_{T}^{X},s_{T}^{X})$ will be called T's reputation parameters by X.
For simplicity $\varphi _{T}^{X}=\varphi (p|r_{T}^{X},s_{T}^{X})$.

The probability expectation value of the reputation function can be expressed as:
$E(\varphi (p|r_{T}^{X},s_{T}^{X}))={\frac {r_{T}^{X}+1}{r_{T}^{X}+s_{T}^{X}+2}}$

Reputation Rating
Let $r_{T}^{X}$ and $s_{T}^{X}$ represent the collective amount of positive and negative feedback about a Target T provided by an agent (or collection of agents) denoted by X, then the function $Rep(r_{T}^{X},s_{T}^{X})$ defined by

$Rep(r_{T}^{X},s_{T}^{X})=(E(\varphi (p|r_{T}^{X},s_{T}^{X}))-0.5)\cdot 2={\frac {r_{T}^{X}-s_{T}^{X}}{r_{T}^{X}+s_{T}^{X}+2}}$

is called T's reputation rating by X. For simplicity $Rep_{T}^{X}=Rep(r_{T}^{X},s_{T}^{X})$

Combining Feedback
Let $\varphi (p|r_{T}^{X},s_{T}^{X})$ an $\varphi (p|r_{T}^{Y},s_{T}^{Y})$ be two different reputation functions on T resulting from X and Y's feedback respectively. The reputation function $\varphi (p|r_{T}^{X,Y},s_{T}^{X,Y})$ defined by:
1. $r_{T}^{X,Y}=r_{T}^{X}+r_{T}^{Y}$
2. $s_{T}^{X,Y}=s_{T}^{X}+s_{T}^{Y}$
is then called T's combined reputation function by X and Y. By using '$\oplus$' to designate this operator, we get $\varphi (p|r_{T}^{X,Y},s_{T}^{X,Y})=\varphi (p|r_{T}^{X},s_{T}^{X})\oplus \varphi (p|r_{T}^{Y},s_{T}^{Y})$.