Cerutti-AT2013-Graphical Subjective Logic

An Empirical Evaluation ofGeometric Subjective Logic

Operators

Federico Cerutti, Alice Toniolo, Nir Oren, Timothy J. Norman

AT-2013Friday 2nd August, 2013

c© 2013 Federico Cerutti <[email protected]>

Summary

Basic concepts of Subjective Logic and its usage in Trust SystemsProposal of two new operators for Subjective LogicDescription of the designed experimentResults of the empirical evaluation

c© 2013 Federico Cerutti <[email protected]> 2 of 19

Trustworthiness and Reputation

Trust, trustworthiness, and reputation have different meanings indifferent approaches [Castelfranchi and Falcone, 2010]Trustworthiness: property of an agent representing its willingnessto share information in a trustworthy mannerReputation: property of an agent representing the subjective viewof its trustworthiness obtained from an agent with which we candirectly communicateSubjective Logic as the way for expressing both the degree oftrustworthiness and the degree of reputation


Introduction to Subjective Logic

A subjective logic opinion is a triple ωX = 〈bX , dX , uX〉bX : belief that X holds;dX : disbelief that X holds;uX : uncertainty that X holds;bX , dX , uX ∈ [0, 1] and bX + dX + uX = 1.


Opinions and Beta Distribution

Beta(p|α, β) =Γ(α+ β)

Γ(α) Γ(β)pα−1ββ−1

α = r + 1β = s+ 1r: number of observations

for xs: number of observations

for ¬xDerived subjective logicopinion:

b = rr+s+2

d = sr+s+2

u = 2r+s+2


p

Subjective Logic Opinions and Trust Modelling:Bootstrap

Alice has a history of interactions with BobAlice counts the positive (r) and negative (s) interactions withBobAlice derives the trustworthiness degree of Bob as

OBobAlice

b = r

r+s+2

d = sr+s+2

u = 2r+s+2

Similarly with John (OJohnAlice)Similarly the other agents

Bob derives OAliceBob and OBillBob

John derives OAliceJohn and OBillJohn

Bill derives OJohnBill and OBobBill


Alice

Bob

John

BillBob

John

Alice Bill

John

Bob

BillAlice

Subjective Logic Opinions and Trust Modelling:Transitivity

Alice wants to know what is the reputation degree for BillAlice asks Bob and John what is the reputation degree of Bill(their opinion on his trustworthiness)Either Bob or John can:

answer that he does not know Billrespectively return OBillBob or OBillJohn,each of which can beeither the actual opinionthey have on Bill, orsomething different

That’s why Alice shoulddiscount what she receivesgiven the trustworthinessdegree of Bob and John she derived:OBillAlice = (OBobAlice ⊗OBillBob)⊕ (OJohnAlice ⊗OBillJohn).


Alice

Bob

John

Bill

Subjective Logic Opinions and Trust Modelling:Transitivity operators

Definition (Former Def. 5 of [Jøsang et al., 2006])

Let A,B be two agents where A’s opinion about B’s recommendations isexpressed as ωAB = 〈bAB , dAB , uAB , aAB〉 and let x be a proposition where B’sopinion about x (e.g. the degree of trustworthiness of a third agent [ed.]) isrecommended to A with the opinion ωBX = 〈bBx , dBx , uBx , aBx 〉. LetωA:Bx = 〈bA:B

x , dA:Bx , uA:B

x , aA:Bx 〉 be the opinion such that:

bA:Bx = bAB bBxdA:Bx = bAB dBxuA:Bx = dAB + uAB + bAB uBxaA:Bx = aBx

then ωA:Bx is called the uncertainty favouring discounted opinion of A. By

using the symbol ⊗ to designate this operation, we get ωA:Bx = ωAB ⊗ ωBx .


Subjective Logic Opinions and Trust Modelling:Transitivity operatorsDefinition (Former Thm. 1 of [Jøsang et al., 2006])

Let ωAx = 〈bAx , dAx , uAx , aAx 〉 and ωBx = 〈bBx , dBx , uBx , aBx 〉 be trust in x from Aand B respectively. The opinion ωAB

x = 〈bABx , dAB

x , uABx , aAB

x 〉 is thencalled the consensus between ωAx and ωBx , denoting the trust that animaginary agent [A,B] would have in x, as if that agent represented both Aand B. In case of Bayesian (totally certain) opinions, their relative weightcan be defined as γA/B = lim (uBx /u

Ax ).

Case I: uAx + uBx − uAx uBx 6= 0bABx =

bAx uBx +bBx uA

x

uAx +uB

x −uAx uB

x

dABx =

dAx uBx +dBx uA

x

uAx +uB

x −uAx uB

x

uABx =

uAx uB

x

uAx +uB

x −uAx uB

x

aABx =

aAx uBx +aBx uA

x −(aAx +aBx ) uAx uB

x

uAx +uB

x −2 uAx uB

x

Case II: uAx + uBx − uAx uBx = 0bABx =

(γA/B bAx +bBx )

(γA/B+1)

dABx =

(γA/B dAx +dBx )

(γA/B+1)

uABx = 0

aABx =

(γA/B aAx +aBx )

(γA/B+1)

By using the symbol ‘⊕’ to designate this operator, we can writeωABx = ωAx ⊕ ωBx .


An Alternative Perspective

xO ,dO + uO cos(π3 )

sin(π3 )

yO , uO


A Graphical Discount Operator

Admissible space of opinions


Alice

Bob

John

Bill


Given a reputation opinion. . .


Alice

Bob

John

Bill


. . . we project it into the admissible space of opinions s.t.

τ ∝ α and |# »

OBobAliceOBillBob | ∝ |

# »

BOBillBob |


Alice

Bob

John

Bill


Finally the discounted opinion is the sum of the two vectors


Alice

Bob

John

Bill

A Graphical Fusion Operator of DiscountedOpinions

Requirements:

the fusion of an opinion Wi = Ti Ci must be balanced usingKi = f(Ti) for some function f(·): e.g. if Alice trusts Bob morethan John, then it seems reasonable that the received opinionsare evaluated differently;if ∀i, j Ki = Kj , then the graphical fusion operator onW1,W2, . . . ,Wn, F(W1,W2, . . . ,Wn) is the centroid of thepolygon determined by n opinions;if ∃i Ki = 0, thenF(W1, . . . ,Wn) = F(W1, . . . ,Wi−1,Wi+1, . . . ,Wn).


Alice

Bob

John

Bill

A Graphical Fusion Operator of DiscountedOpinions

Formally:

bF(W1,...,Wn) =1∑n

i=1Ki

(n∑i=1

Ki bWi

)

dF(W1,...,Wn) =1∑n

i=1Ki

(n∑i=1

Ki dWi

)

uF(W1,...,Wn) =1∑n

i=1Ki

(n∑i=1

Ki uWi

)We considered only the case Ki = bTi +

uTi2 .


Alice

Bob

John

Bill

Empirical Evaluation: Trust System Construction

Set of 50 agents A = ag1, . . . , ag50Each agent agx is characterised by the probability of respondingtruthfully to another agent’s query, namely P Tagx ∈ [0 . . . 1](randomly selected)Each agent knows Ω = >For each agent agx, we determine if it can communicate withagy 6= agx according to PL: if agy is connected to agx, then wesay that agy is a connection of agx (agy ∈ Nagx).


Empirical Evaluation: Bootstrapping

Following [Ismail and Jøsang, 2002] a Beta distribution is usedfor analysing repetitive experiments and deriving a subjectivelogic opinionEach agent agx asks each agent agy ∈ Nagx about Ω #B timesGiven the number of exchanges when agent agy answeredtruthfully (#>) and when it lied (#⊥)

Oagyagx = 〈 #>

#B + 2,

#⊥#B + 2

,2

#B + 2〉

Opinion derived by the (omniscient) experimenter

OagyExp = 〈P Tagy , (1− P

Tagy), 0〉


Empirical Evaluation: Network Exploration

An “explorer” agS ∈ A is randomly selectedagS asks each agent in its connections about the trustworthinessit has of its connections (e.g. Alice asks both Bob and John(separately) about which agents they know and what is theiropinion of them)For each agent agz they claim to know (e.g. Bill), agS computesOagzagS |J = (O

agy1agS ⊗O

agzagy1

)⊕ . . .⊕ (OagynagS ⊗O

agzagyn ) (Jøsang

operators), and OagzagS |G = F((Oagy1agS O

agzagy1

), . . . , (OagynagS O

agzagyn )

(the graphical operators)Now agz is known and can be directly questioned about otheragents it might nowThe explorations ends when no new agents are discovered


Empirical Evaluation: Results

For each run, for each value of PL, 250 explorations over 10different networks were carried out (overall 12500 explorationsacross 500 different networks)For small values of PL, the graphical operators return opinionscloser to the “ideal” one than Jøsang’s by a factor 2 (on theaverage dJ

dG' 2), or, in other terms, approx. 50% closer;

The greater the PL, the more similar the performance of the twosets of operators, the smaller the standard deviation on theresults obtained by the experiments;The overall average on the logarithmic scale is 1.56 (36% on thelinear scale).


Empirical Evaluation: Results

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 5 10 15 20 25 30

Aver

age

of

the

scal

arlo

gar

itm

icco

mp

aris

on

val

ue

PL

Average Logaritmic Comparison with respect to the Probability of Initial Connection PL

Run01Run02Run03Run04

Run05Run06Run07Run08

Run09Run10


Conclusions

Empirical investigation on discount and fusion operators forSubjective LogicProposal of two new operators for Subjective Logic based ongraphical propertiesEmpirical comparison of the new proposal w.r.t. Jøsangoperators, with improving of the results of 36% (on the average)What’s next:

Study of random graph generation (e.g. scale-free networks. . . )Experimental evaluation in presence of more uncertainty (loweringthe number of interactions in the bootstrap phase)Reaching a common methodology with [Kaplan et al., 2013]


Acknowledgement

Research was sponsored by US Army Research laboratoryand the UK Ministry of Defence and was accomplished underAgreement Number W911NF-06-3-0001. The views andconclusions contained in this document are those of theauthors and should not be interpreted as representing theofficial policies, either expressed or implied, of the US ArmyResearch Laboratory, the U.S. Government, the UK Ministryof Defense, or the UK Government. The US and UKGovernments are authorized to reproduce and distributereprints for Government purposes notwithstanding anycopyright notation hereon.


References I

[Castelfranchi and Falcone, 2010] Castelfranchi, C. and Falcone, R. (2010).Trust theory: A socio-cognitive and computational model.Wiley Series in Agent Technology.

[Ismail and Jøsang, 2002] Ismail, R. and Jøsang, A. (2002).The beta reputation system.In Prooceedings of BLED 2002.

[Jøsang et al., 2006] Jøsang, A., Pope, S., and Marsh, S. (2006).Exploring different types of trust propagation.In Proceedings of the 4th International Conference on Trust Management (iTrust’06).

[Kaplan et al., 2013] Kaplan, L. M., Sensoy, M., Tang, Y., Chakraborty, S., Bisdikian, C., and de Mel,G. (2013).Reasoning under uncertainty: Variations of subjective logic deduction.In Proceedings of the Sixteenth International Conference on Information Fusion.


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