ARM4SNS:ReputationFunctions

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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^{X}_{T} and s^{X}_{T} 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^{X}_{T},s^{X}_{T}) defined by

\varphi(p|r^{X}_{T},s^{X}_{T})=\frac{\Gamma(r^{X}_{T}+s^{X}_{T}+2)}{\Gamma(r^{X}_{T}+1)\Gamma(s^{X}_{T}+1)}p^{r^{X}_{T}}(1-p)^{s^{X}_{T}},\qquad where\ 0 \leq p \leq 1,\ 0 \leq r^{X}_{T},\ 0 \leq s^{X}_{T}
is called T's reputation function by X. The tuple (r^{X}_{T},s^{X}_{T}) will be called T's reputation parameters by X.
For simplicity \varphi^{X}_{T} = \varphi(p|r^{X}_{T},s^{X}_{T}).

The probability expectation value of the reputation function can be expressed as:

E(\varphi(p|r^{X}_{T},s^{X}_{T}))=\frac{r^{X}_{T}+1}{r^{X}_{T}+s^{X}_{T}+2}

Reputation Rating
For human users is a more simple representation than the reputation function needed.
Let r^{X}_{T} and s^{X}_{T} 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^{X}_{T},s^{X}_{T}) defined by

Rep(r^{X}_{T},s^{X}_{T})= (E(\varphi(p|r^{X}_{T},s^{X}_{T}))-0.5)\cdot 2 = \frac{r^{X}_{T}-s^{X}_{T}}{r^{X}_{T}+s^{X}_{T}+2}
is called T's reputation rating by X. For simplicity Rep^{X}_{T}=Rep(r^{X}_{T},s^{X}_{T})

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

Belief Discounting
This model uses a metric called opinion to describe beliefs about the truth of statements. An opinion is a tuple \omega^{A}_{x} = (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^{X}_{Y}=(b^{X}_{Y},d^{X}_{Y},u^{X}_{Y}) is X's opinion about Y's advice, and let T be the Target agent where \omega^{Y}_{T}=(b^{Y}_{T},d^{Y}_{T},u^{Y}_{T}) is Y's opinion about T expressed in an advice to X. Let \omega^{X:Y}_{T}=(b^{X:Y}_{T},d^{X:Y}_{T},u^{X:Y}_{T}) be the opinion such that:
1. b^{X:Y}_{T}=b^{X}_{Y}b^{Y}_{T},
2. d^{X:Y}_{T}=b^{X}_{Y}d^{Y}_{T},
3. u^{X:Y}_{T}=d^{X}_{Y}+u^{X}_{Y}+b^{X}_{Y}u^{Y}_{T},
then \omega^{X:Y}_{T} is called the discounting of \omega^{Y}_{T} by \omega^{X}_{Y} 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^{X:Y}_{T}=\omega^{X}_{Y}\otimes\omega^{Y}_{T} .
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 \varphi(p|r^{X}_{Y},s^{X}_{Y}) is Y's reputation function by X, and \varphi(p|r^{Y}_{T},s^{Y}_{T}) is T's reputation function by Y. Let \varphi(p|r^{X:Y}_{T},s^{X:Y}_{T}) be the reputation function such that:
1.  r^{X:Y}_{T}=\frac{2r^{X}_{Y}r^{Y}_{T}}{(s^{X}_{Y}+2)(r^{Y}_{T}+s^{Y}_{T}+2)+2r^{x}_{T}},
2.  s^{X:Y}_{T}=\frac{2r^{X}_{Y}s^{Y}_{T}}{(s^{X}_{Y}+2)(r^{Y}_{T}+s^{Y}_{T}+2)+2r^{x}_{T}},
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^{X:Y}_{T},s^{X:Y}_{T})=\varphi(p|r^{X}_{Y},s^{X}_{Y})\otimes \varphi(p|r^{Y}_{T},s^{Y}_{T})
. In the short notation this can be written as: \varphi^{X:Y}_{T}= \varphi^{X}_{Y} \otimes \varphi^{Y}_{T}.

Forgetting