ARM4SNS:ReputationFunctions
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
For human users is a more simple representation than the reputation function needed.
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 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 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