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4.4 Formalizing the Model
We want a mathematical interpretation of the trust model as defined by the trust evolution function and the trust dynamics.
In this trust model there are two parameters that can be adjusted in order to calculate a trust value. The first is the X value, which is based on the trust dynamics, and adjustments are based on how the interactions with the trustees are interpreted. A positive interaction is defined by the trustee and the trustor give an article a similar recommendation. The N value is based on the experience with the provided recommendation. If the trustor is content with the provided recommendation, those trustees that contributed significantly have their N value adjusted towards a more positive curve.
4.4.1 Trust Evolution Function
As described in section 4.3.3 the initial trust function is determined be the mathematical function f(x) = x. This is assumed in order to have a neutral basis upon initialization of a user in the RoR.
Using the linear formula as suggested does not bring all the possibilities as wanted.
When the user’s profile is changed in to a more optimistic or cautious curve, then this is not possible to make this representation with a linear curve. Another approach would be to represent the functions as a polynomial function with different degrees power. The disadvantage of using a polynomial function is that it is not mirrored inf(x) =−x+ 1 andf(x) =−x−1, as shown in figure 4.7. By having a function expression that is not weighted equally the steps closest to 0 on the X axis is much less significant to that the steps closest to 1 and -1, which would not be fair as step towards trust should only depend on the curves parameters and not the functional expression of the curve.
A third approach (and the approach chosen) is to represent the trust function as
4.4 Formalizing the Model 53
Figure 4.7: Trust Evolution Function represented with a polynomial expression
a superellipse (or Lam´e curve). The superellipse is represented by the formula:
x a
n
+ y b
n
= 1
The superellipse can be adjusted by tweaking the parameters a,b and n. The parameter a and b represent the radius of of the superellipse, and if they are set to 1 then the radius is 1, which fits the function with in the interval defined in section 4.3. In this case it is only needed to operate in the interval -1 to 1, and therefore is is not needed to adjust the aand b parameter. Hence only the parameter n is needed to adjust the n-value in order to update the trust evolution function.
However the superellipse needs some adjustments in order to fit the desired curves as shown on figure4.4. In the original form the super ellipse forn= 4 is plotted on figure4.8 and defined by the expression:
|x|4+|y|4= 1
In order to fit the different part of the superellipse to the trust function, as it is shown on figure 4.4, it needs some adjustments. Four sets of functions are
54 Trust Model
Figure 4.8: The superellipse plotted wherea= 1, b= 1 andn= 4
defined in order to fit the different curves needed:
• Optimistic curve in trust. See Equation4.1.
• Cautious curve in trust. See Equation4.2.
• Optimistic curve in distrust. See Equation 4.3.
• Cautious curve in distrust. See Equation 4.4.
The four different curves are defined like this, where the parameteraandbare set to 1
|x−1|n+|y|n= 1 (4.1)
|x|n+|y−1|n= 1 (4.2)
|x|n+|y+ 1|n= 1 (4.3)
4.4 Formalizing the Model 55
|x+ 1|n+|y|n= 1 (4.4)
Combining equation 4.4,4.2,4.3and 4.1into a piecewise function that is only are defined in 1st and 3rd quadrant, gives the trust function:
T rust(x) =
|x−1|n+|y|n= 1,|x|n+|y−1|n = 1 forx >0 andy >0
|x+ 1|n+|y|n= 1,|x|n+|y+ 1|n = 1 forx <0 andy <0 (4.5) This function plotted looks like figure4.9, whereaandb are 1 andnis 2:
Figure 4.9: The trust function plotted wherea= 1,b= 1 andn= 2 This implementation has the advantage that the neutral trust function, that has been assumed to be linear, can be represented for n=1.
4.4.2 Trust dynamics
As described in section4.3.2trust dynamics are based on forgetability over time.
In our approach to trust dynamics we have made some assumptions based on
56 Trust Model
the experimental research by Jonker, Treur, Theeuwes and Schalken [18]. From the research we see that after 5 successive positive interactions and no previous interactions, the trust value is almost at a maximum. Therefore we make the definition that after 10 successive positive interactions the trust value should be at a maximum.
Based on the coordinate system where the X value goes from -1 to 1, we define a positive experience to an increase to be 101 of the possible interval. As the experiment is based on no previous actions this interval would be from 0 to 1, hence a positive experience should increase the x value with 0.1.
The same experimental research shows that negative interactions reach almost maximum distrust on 5 recommendations as well. Based on this we define a negative interaction should decrease the X value of 0.1.
We define that after 3 months an interaction is only worth 50% of its’ original value and after 9 months it is only 25% worth and after a year it is not worth anything any more. For example, if a person has 2 positive experiences within the last week, a negative experience that is 4 months old, a negative experience that is 10 months old and a positive experience that is 2 years old the X value will be calculated as following
XV alue= 0.1 + 0.1−0.1·50%−0.1·25% + 0.1·0% = 0.125
There is no scientific approach of these time limits and the decrease factors.
Consequently, this area needs more research. Please see section8.1.