PageRank

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

Beta

Reputation Function
Let ${\displaystyle r_{T}^{X}}$ and ${\displaystyle s_{T}^{X}}$ 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 ${\displaystyle \varphi (p|r_{T}^{X},s_{T}^{X})}$ defined by
${\displaystyle \varphi (p|r_{T}^{X},s_{T}^{X})={\frac {\Gamma (r_{T}^{X}+s_{T}^{X}+2)}{\Gamma (r_{T}^{X}+1)\Gamma (s_{T}^{X}+1)}}p^{r_{T}^{X}}(1-p)^{s_{T}^{X}},\qquad where\ 0\leq p\leq 1,\ 0\leq r_{T}^{X},\ 0\leq s_{T}^{X}}$
is called T's reputation function by X. The tuple ${\displaystyle (r_{T}^{X},s_{T}^{X})}$ will be called T's reputation parameters by X.
For simplicity ${\displaystyle \varphi _{T}^{X}=\varphi (p|r_{T}^{X},s_{T}^{X})}$.

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

Reputation Rating
For human users is a more simple representation than the reputation function needed.
Let ${\displaystyle r_{T}^{X}}$ and ${\displaystyle s_{T}^{X}}$ 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 ${\displaystyle Rep(r_{T}^{X},s_{T}^{X})}$ defined by
${\displaystyle Rep(r_{T}^{X},s_{T}^{X})=(E(\varphi (p|r_{T}^{X},s_{T}^{X}))-0.5)\cdot 2={\frac {r_{T}^{X}-s_{T}^{X}}{r_{T}^{X}+s_{T}^{X}+2}}}$
is called T's reputation rating by X. For simplicity ${\displaystyle Rep_{T}^{X}=Rep(r_{T}^{X},s_{T}^{X})}$

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

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

Forgetting