ARM4SNS:ReputationFunctions - Revision history

Dato: /* Beta */

2006-02-28T08:28:25Z

Beta

← Older revision		Revision as of 08:28, 28 February 2006
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	</math><br><br>		</math><br><br>
	'''Reputation Rating'''<br>		'''Reputation Rating'''<br>
			For human users is a more simple representation than the reputation function needed.<br>
	Let <math>r^{X}_{T}</math> and <math>s^{X}_{T}</math> 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 <math>Rep(r^{X}_{T},s^{X}_{T})</math> defined by<br>		Let <math>r^{X}_{T}</math> and <math>s^{X}_{T}</math> 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 <math>Rep(r^{X}_{T},s^{X}_{T})</math> defined by<br>
	<math>		<math>

Dato: /* Beta */

2006-02-28T08:02:53Z

Beta

← Older revision		Revision as of 08:02, 28 February 2006
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	<math>b=\frac{r}{r+s+2}</math>,<br>		<math>b=\frac{r}{r+s+2}</math>,<br>
	<math>d=\frac{s}{r+s+2}</math>,<br>		<math>d=\frac{s}{r+s+2}</math>,<br>
	<math>b=\frac{2}{r+s+2}</math>,<br>		<math>u=\frac{2}{r+s+2}</math>,<br>
	By using this we obtain the following definition of the discounting operator for reputation functions.<br><br>		By using this we obtain the following definition of the discounting operator for reputation functions.<br><br>

Dato: /* Beta */

2006-02-28T08:01:09Z

Beta

← Older revision		Revision as of 08:01, 28 February 2006
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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>

Dato: /* Beta */

2006-02-28T07:27:20Z

Beta

← Older revision		Revision as of 07:27, 28 February 2006
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	1. <math>r^{X,Y}_{T}=r^{X}_{T}+r^{Y}_{T}</math><br>		1. <math>r^{X,Y}_{T}=r^{X}_{T}+r^{Y}_{T}</math><br>
	2. <math>s^{X,Y}_{T}=s^{X}_{T}+s^{Y}_{T}</math><br>		2. <math>s^{X,Y}_{T}=s^{X}_{T}+s^{Y}_{T}</math><br>
	is then called T's combined reputation function by X and Y. By using '<math>\~~oplus~~</math>' to designate this operator, we get		is then called T's combined reputation function by X and Y. By using '<math>\otimes</math>' to designate this operator, we get
	<math>		<math>
	\varphi(p\|r^{X,Y}_{T},s^{X,Y}_{T})=\varphi(p\|r^{X}_{T},s^{X}_{T}) \~~oplus~~ \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>		'''Belief Discounting'''<br><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>
			2. <math>d^{X:Y}_{T}=d^{X}_{Y}d^{Y}_{T}</math>,<br>
			3. <math>u^{X:Y}_{T}=u^{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>.


	'''Reputation Discounting'''<br>		'''Reputation Discounting'''<br>
	'''Forgetting'''<br>		'''Forgetting'''<br>

Dato: /* Beta */

2006-02-27T19:18:56Z

Beta

← Older revision		Revision as of 19:18, 27 February 2006
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	=Beta=		=Beta=
	'''Reputation Function'''<br>		'''Reputation Function'''<br>
	Let <math>r^{X}_{T}</math> and <math>s^{X}_{T}</math> 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 <math>\varphi(p\|r^{X}_{T},s^{X}_{T})</math> defined by~~<br>~~<br>		Let <math>r^{X}_{T}</math> and <math>s^{X}_{T}</math> 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 <math>\varphi(p\|r^{X}_{T},s^{X}_{T})</math> defined by<br>
			<math>
	~~<math>~~\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}~~</math><br><br>~~		\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}
			</math><br>
	is called T's reputation function by X. The tuple <math>(r^{X}_{T},s^{X}_{T})</math> will be called T's reputation parameters by X. <br>For simplicity <math>\varphi^{X}_{T} = \varphi(p\|r^{X}_{T},s^{X}_{T})</math>.<br><br>		is called T's reputation function by X. The tuple <math>(r^{X}_{T},s^{X}_{T})</math> will be called T's reputation parameters by X. <br>For simplicity <math>\varphi^{X}_{T} = \varphi(p\|r^{X}_{T},s^{X}_{T})</math>.<br><br>
	The '''probability expectation value''' of the reputation function can be expressed as:<br>		The '''probability expectation value''' of the reputation function can be expressed as:<br>
			<math>
	~~<math>~~E(\varphi(p\|r^{X}_{T},s^{X}_{T}))=\frac{r^{X}_{T}+1}{r^{X}_{T}+s^{X}_{T}+2}~~</math> <br><br>~~		E(\varphi(p\|r^{X}_{T},s^{X}_{T}))=\frac{r^{X}_{T}+1}{r^{X}_{T}+s^{X}_{T}+2}
			</math><br><br>
	'''Reputation Rating'''<br>		'''Reputation Rating'''<br>
	Let <math>r^{X}_{T}</math> and <math>s^{X}_{T}</math> 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 <math>Rep(r^{X}_{T},s^{X}_{T})</math> defined by~~<br>~~<br>		Let <math>r^{X}_{T}</math> and <math>s^{X}_{T}</math> 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 <math>Rep(r^{X}_{T},s^{X}_{T})</math> defined by<br>
	<math>		<math>
	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}		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}
	</math~~><br~~><br>		</math><br>
	is called T's reputation rating by X. For simplicity <math>Rep^{X}_{T}=Rep(r^{X}_{T},s^{X}_{T})</math><br><br>		is called T's reputation rating by X. For simplicity <math>Rep^{X}_{T}=Rep(r^{X}_{T},s^{X}_{T})</math><br><br>
	'''Combining Feedback'''<br>		'''Combining Feedback'''<br>
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	1. <math>r^{X,Y}_{T}=r^{X}_{T}+r^{Y}_{T}</math><br>		1. <math>r^{X,Y}_{T}=r^{X}_{T}+r^{Y}_{T}</math><br>
	2. <math>s^{X,Y}_{T}=s^{X}_{T}+s^{Y}_{T}</math><br>		2. <math>s^{X,Y}_{T}=s^{X}_{T}+s^{Y}_{T}</math><br>
	is then called T's combined reputation function by X and Y. By using '<math>\oplus</math>' to designate this operator, we get ~~<math>\varphi(p\|r^{X,Y}_{T},s^{X,Y}_{T})=\varphi(p\|r^{X}_{T},s^{X}_{T}) \oplus \varphi(p\|r^{Y}_{T},s^{Y}_{T})</math>.~~		is then called T's combined reputation function by X and Y. By using '<math>\oplus</math>' to designate this operator, we get
			<math>
			\varphi(p\|r^{X,Y}_{T},s^{X,Y}_{T})=\varphi(p\|r^{X}_{T},s^{X}_{T}) \oplus \varphi(p\|r^{Y}_{T},s^{Y}_{T})
			</math>.<br><br>
			'''Belief Discounting'''<br>
			'''Reputation Discounting'''<br>
			'''Forgetting'''<br>

Dato: /* Beta */

2006-02-27T19:10:47Z

Beta

← Older revision		Revision as of 19:10, 27 February 2006
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	=Beta=		=Beta=
			'''Reputation Function'''<br>
			Let <math>r^{X}_{T}</math> and <math>s^{X}_{T}</math> 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 <math>\varphi(p\|r^{X}_{T},s^{X}_{T})</math> defined by<br><br>
			<math>\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}</math><br><br>
			is called T's reputation function by X. The tuple <math>(r^{X}_{T},s^{X}_{T})</math> will be called T's reputation parameters by X. <br>For simplicity <math>\varphi^{X}_{T} = \varphi(p\|r^{X}_{T},s^{X}_{T})</math>.<br><br>
			The '''probability expectation value''' of the reputation function can be expressed as:<br>
			<math>E(\varphi(p\|r^{X}_{T},s^{X}_{T}))=\frac{r^{X}_{T}+1}{r^{X}_{T}+s^{X}_{T}+2}</math> <br><br>
			'''Reputation Rating'''<br>
			Let <math>r^{X}_{T}</math> and <math>s^{X}_{T}</math> 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 <math>Rep(r^{X}_{T},s^{X}_{T})</math> defined by<br><br>
			<math>
			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}
			</math><br><br>
			is called T's reputation rating by X. For simplicity <math>Rep^{X}_{T}=Rep(r^{X}_{T},s^{X}_{T})</math><br><br>
			'''Combining Feedback'''<br>
			Let <math>\varphi(p\|r^{X}_{T},s^{X}_{T})</math> an <math>\varphi(p\|r^{Y}_{T},s^{Y}_{T})</math> be two different reputation functions on T resulting from X and Y's feedback respectively. The reputation function <math>\varphi(p\|r^{X,Y}_{T},s^{X,Y}_{T})</math> defined by:<br>
			1. <math>r^{X,Y}_{T}=r^{X}_{T}+r^{Y}_{T}</math><br>
			2. <math>s^{X,Y}_{T}=s^{X}_{T}+s^{Y}_{T}</math><br>
			is then called T's combined reputation function by X and Y. By using '<math>\oplus</math>' to designate this operator, we get <math>\varphi(p\|r^{X,Y}_{T},s^{X,Y}_{T})=\varphi(p\|r^{X}_{T},s^{X}_{T}) \oplus \varphi(p\|r^{Y}_{T},s^{Y}_{T})</math>.

Dato: /* PageRank */

2006-02-27T18:48:05Z

PageRank

← Older revision		Revision as of 18:48, 27 February 2006
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	*first term of function (1.) <math> cE(u) </math> gives rank value based on initial rank <br>		*first term of function (1.) <math> cE(u) </math> gives rank value based on initial rank <br>
	*second term of (1.) <math>c\sum_{v\in N^{-}(u)} {R(v)\over{\|N^{+}(v)\|}}</math> gives rank value as a function of hyperlinks pointing at <math> u </math><br>		*second term of (1.) <math>c\sum_{v\in N^{-}(u)} {R(v)\over{\|N^{+}(v)\|}}</math> gives rank value as a function of hyperlinks pointing at <math> u </math><br>

			=Beta=

Dato at 18:41, 27 February 2006

2006-02-27T18:41:45Z

New page

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