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: $R(u)=cE(u)+c\sum _{v\in N^{-}(u)}\frac{R(v)}{|N^{+}(v)|}$ (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.) $c\sum _{v\in N^{-}(u)}\frac{R(v)}{|N^{+}(v)|}$ 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 $\phi (p|r_T^X,s_T^X)$ defined by
$\phi (p|r_T^X,s_T^X)=\frac{\mathrm{\Gamma }(r_T^X+s_T^X+2)}{\mathrm{\Gamma }(r_T^X+1)\mathrm{\Gamma }(s_T^X+1)}{p}^{r_T^X}(1-p{)}^{s_T^X},where0\le p\le 1,0\le r_T^X,0\le s_T^X$
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 ${\phi }_T^X=\phi (p|r_T^X,s_T^X)$.

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

Reputation Rating
For human users is a more simple representation than the reputation function needed.
Let $r_T^X$ 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
$Rep(r_T^X,s_T^X)=(E(\phi (p|r_T^X,s_T^X))-0.5)\cdot 2=\frac{r_T^X-s_T^X}{r_T^X+s_T^X+2}$
is called T's reputation rating by X. For simplicity $Re{p}_T^X=Rep(r_T^X,s_T^X)$

Combining Feedback
Let $\phi (p|r_T^X,s_T^X)$ an $\phi (p|r_T^Y,s_T^Y)$ be two different reputation functions on T resulting from X and Y's feedback respectively. The reputation function $\phi (p|r_T^{X,Y},s_T^{X,Y})$ 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 $\phi (p|r_T^{X,Y},s_T^{X,Y})=\phi (p|r_T^X,s_T^X)\otimes \phi (p|r_T^Y,s_T^Y)$.

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:
$b=\frac{r}{r+s+2}$,
$d=\frac{s}{r+s+2}$,
$u=\frac{2}{r+s+2}$,
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 $\phi (p|r_Y^X,s_Y^X)$ is Y's reputation function by X, and $\phi (p|r_T^Y,s_T^Y)$ is T's reputation function by Y. Let $\phi (p|r_T^{X:Y},s_T^{X:Y})$ 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 $\phi (p|r_T^{X:Y},s_T^{X:Y})=\phi (p|r_Y^X,s_Y^X)\otimes \phi (p|r_T^Y,s_T^Y)$. In the short notation this can be written as: ${\phi }_T^{X:Y}={\phi }_Y^X\otimes {\phi }_T^Y$.

Forgetting