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 '$\otimes$' to designate this operator, we get $\varphi (p|r_{T}^{X,Y},s_{T}^{X,Y})=\varphi (p|r_{T}^{X},s_{T}^{X})\otimes \varphi (p|r_{T}^{Y},s_{T}^{Y})$.

Belief Discounting
This model uses a metric called opinion to describe beliefs about the truth of statements. An opinion is a tuple $\omega _{x}^{A}=(b,d,u)$, where b, d and u represent belief, disbelief and uncertainty. These parameters satisfy $b+d+u=1$ where $b,d,u\in [0,1]$.
Let X and Y be two agents where $\omega _{Y}^{X}=(b_{Y}^{X},d_{Y}^{X},u_{Y}^{X})$ is X's opinion about Y's advice, and let T be the Target agent where $\omega _{T}^{Y}=(b_{T}^{Y},d_{T}^{Y},u_{T}^{Y})$ is Y's opinion about T expressed in an advice to X. Let $\omega _{T}^{X:Y}=(b_{T}^{X:Y},d_{T}^{X:Y},u_{T}^{X:Y})$ be the opinion such that:
1. $b_{T}^{X:Y}=b_{Y}^{X}b_{T}^{Y}$,
2. $d_{T}^{X:Y}=b_{Y}^{X}d_{T}^{Y}$,
3. $u_{T}^{X:Y}=d_{Y}^{X}+u_{Y}^{X}+b_{Y}^{X}u_{T}^{Y}$,
then $\omega _{T}^{X:Y}$ is called the discounting of $\omega _{T}^{Y}$ by $\omega _{Y}^{X}$ expressing X's opinion about T as a result of Y's advice to X. By using '$\otimes$' to designate this operator, we can write $\omega _{T}^{X:Y}=\omega _{Y}^{X}\otimes \omega _{T}^{Y}$.
The author of BETA provides a mapping between the opinion metric and the beta function defined by:
$b={\frac {r}{r+s+2}}$,
$d={\frac {s}{r+s+2}}$,
$b={\frac {2}{r+s+2}}$,
By using this we obtain the following definition of the discounting operator for reputation functions.

Reputation Discounting
Let X, Y and T be three agents where $\varphi (p|r_{Y}^{X},s_{Y}^{X})$ is Y's reputation function by X, and $\varphi (p|r_{T}^{Y},s_{T}^{Y})$ is T's reputation function by Y. Let $\varphi (p|r_{T}^{X:Y},s_{T}^{X:Y})$ be the reputation function such that:
1. $r_{T}^{X:Y}={\frac {2r_{Y}^{X}r_{T}^{Y}}{(s_{Y}^{X}+2)(r_{T}^{Y}+s_{T}^{Y}+2)+2r_{T}^{x}}}$,
2. $s_{T}^{X:Y}={\frac {2r_{Y}^{X}s_{T}^{Y}}{(s_{Y}^{X}+2)(r_{T}^{Y}+s_{T}^{Y}+2)+2r_{T}^{x}}}$,
then it is called T's discounted reputation function by X through Y. By using the symbol '$\otimes$' to designate this operator, we can write $\varphi (p|r_{T}^{X:Y},s_{T}^{X:Y})=\varphi (p|r_{Y}^{X},s_{Y}^{X})\otimes \varphi (p|r_{T}^{Y},s_{T}^{Y})$. In the short notation this can be written as: $\varphi _{T}^{X:Y}=\varphi _{Y}^{X}\otimes \varphi _{T}^{Y}$.

Forgetting