ARM4SNS:ReputationFunctions: Difference between revisions

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

<math>b=\frac{r}{r+s+2}</math>,<br>
<math>d=\frac{s}{r+s+2}</math>,<br>
<math>b=\frac{2}{r+s+2}</math>,<br>
By using this we obtain the following definition of the discounting operator for reputation functions.<br><br>


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

'''Forgetting'''<br>
'''Forgetting'''<br>

Revision as of 08:01, 28 February 2006

PageRank

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P} : set of hyperlinked webpages
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u,v} : webpages in P
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle N^{-}(u)} : set of webpages pointing to u
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle N^{+}(v)} : set of webpages that v points to
  • the PageRank is: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R(u) = cE(u) + c \sum_{v\in N^{-}(u)} {R(v)\over{|N^{+}(v)|}}} (1.)
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c} is chosen such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sum_{u \in P} R(u) = 1}
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E} is a vector over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P} corresponding to a source of rank and is chosen such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sum_{u \in P} E(u) = 0.15}
  • first term of function (1.) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle cE(u) } gives rank value based on initial rank
  • second term of (1.) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c\sum_{v\in N^{-}(u)} {R(v)\over{|N^{+}(v)|}}} gives rank value as a function of hyperlinks pointing at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u }

Beta

Reputation Function
Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r^{X}_{T}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \varphi(p|r^{X}_{T},s^{X}_{T})} defined by
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (r^{X}_{T},s^{X}_{T})} will be called T's reputation parameters by X.
For simplicity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \varphi^{X}_{T} = \varphi(p|r^{X}_{T},s^{X}_{T})} .

The probability expectation value of the reputation function can be expressed as:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(\varphi(p|r^{X}_{T},s^{X}_{T}))=\frac{r^{X}_{T}+1}{r^{X}_{T}+s^{X}_{T}+2} }

Reputation Rating
Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r^{X}_{T}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Rep(r^{X}_{T},s^{X}_{T})} defined by
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Rep^{X}_{T}=Rep(r^{X}_{T},s^{X}_{T})}

Combining Feedback
Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \varphi(p|r^{X}_{T},s^{X}_{T})} an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \varphi(p|r^{X,Y}_{T},s^{X,Y}_{T})} defined by:
1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r^{X,Y}_{T}=r^{X}_{T}+r^{Y}_{T}}
2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s^{X,Y}_{T}=s^{X}_{T}+s^{Y}_{T}}
is then called T's combined reputation function by X and Y. By using 'Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \otimes} ' to designate this operator, we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega^{A}_{x} = (b,d,u)} , where b, d and u represent belief, disbelief and uncertainty. These parameters satisfy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle b+d+u=1} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle b,d,u \in [0,1]} .
Let X and Y be two agents where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega^{X:Y}_{T}=(b^{X:Y}_{T},d^{X:Y}_{T},u^{X:Y}_{T})} be the opinion such that:
1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle b^{X:Y}_{T}=b^{X}_{Y}b^{Y}_{T}} ,
2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d^{X:Y}_{T}=b^{X}_{Y}d^{Y}_{T}} ,
3. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u^{X:Y}_{T}=d^{X}_{Y}+u^{X}_{Y}+b^{X}_{Y}u^{Y}_{T}} ,
then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega^{X:Y}_{T}} is called the discounting of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega^{Y}_{T}} by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega^{X}_{Y}} expressing X's opinion about T as a result of Y's advice to X. By using 'Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \otimes} ' to designate this operator, we can write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle b=\frac{r}{r+s+2}} ,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=\frac{s}{r+s+2}} ,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \varphi(p|r^{X}_{Y},s^{X}_{Y})} is Y's reputation function by X, and is T's reputation function by Y. Let be the reputation function such that:
1. ,
2. ,
then it is called T's discounted reputation function by X through Y. By using the symbol '' to designate this operator, we can write . In the short notation this can be written as: .

Forgetting

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