 values
even when the graph connecting the users in the trust network
is complicated.
This makes EBSL interesting e.g. for Reputation systems

Let Aij be the opinion that user i has about the trustworthiness of user j, based on direct evidence, e.g. direct interactions between i and j. We set Aii = U.
Let every user publish these direct opinions in a reliable way;
the matrix A is public and its integrity is guaranteed.
Based on all the available trust information, direct as well as indirect,
what should a user conclude about the trustworthiness of all the other users?
In general this is a nontrivial problem because of the complicated 
connection graphs, in which loops may occur.
The problem is to find a "reputation" matrix R that consistently
combines the direct and indirect evidence.
In EBSL the following "self-consistent" (self-containing)
equation must be satisfied by R,

 

 

Here the "Σ" stands for ⊕ operations.
The diagonal is set to full belief since everybody trusts himself implicitly,
independent of other users' opinions.

User i forms an opinion about j by combining
his direct opinion Aij
with other users' opinions Akj.
The indirect evidence is weighted with a scalar that depends on the
reputation of the intermediary: g(Rik).

Equation (5) can be written compactly in matrix form,

 
Here  is the identity matrix, and
summation should be executed as ⊕.
Eq.(6) is a fixed point equation similar to the case of Markov chains.
It can be solved recursively.
Let X be a square matrix with the same dimensions as A.
Define a function f as

.

Pick a starting matrix  and repeatedly apply f
until the result does not change any more, i.e. a fixed point
 is reached.
Independent of the choice of,
after one iteration the diagonal is B1
and stays B1 in all further iterations.

If Eq.(6) were an ordinary matrix equation  
for real-valued (or complex) Aij,
it would have solution.
However, the opinion algebra does not allow for such a simple expression.
Instead we have

        
f2(X0) = B1 ⊕ ((B1 ⊕ (X
values 
 even when the graph connecting the users in the trust net 
 is complicated. 
 This do EBSL interesting e.g. for Reputation system 

 get Aij be the opinion that user i have about the trustworthiness of user j, based on direct evidence, e.g. direct interactions between i and j. We set up Aii = U. 
 Let every user print these direct opinions in a reliable way; 
 the matrix A is public and its integrity is guaranteed. 
 Based on all the available reliance information, direct as well as collateral, 
 what should a user conclude about the trustworthiness of all the early users? 
 In cosmopolitan this is a nontrivial problem because of the complicated 
 connection graphs, in which loops may occur. 
 The problem is to determine a " reputation " matrix R that consistently 
 combines the direct and collateral evidence. 
 In EBSL the follow " self - consistent " (self - containing) 
 equation must be satisfied by R, 

 

 

 Here the " Σ " stands for ⊕ operations. 
 The diagonal is set to entire belief since everybody trusts himself implicitly, 
 autonomous of early users' opinions. 

 User i forms an opinion about joule by combining 
 his lineal opinion Aij 
 with other users' impression Akj. 
 The indirect evidence is weighted with a scalar that depends on the 
 reputation of the intermediary: g(Rik). 

 Equation (5) can be publish compactly in matrix form, 

 
 Here   is the identity matrix, and 
 summation should be executed as ⊕. 
 Eq.(6) is a fixed point equality exchangeable to the case of Markov chains. 
 It can be solved recursively. 
 Let X be a straight matrix with the same dimensions as A. 
 Define a affair f as 

. 

 Pick a starting matrix   and repeatedly give f 
 until the result does not exchange any more, i.e. a fixed point 
  is strive. 
 Independent of the choice of, 
 after one iteration the aslant is B1 
 and stays B1 in all further iterations. 

 If Eq.(6) were an ordinary matrix equation  
 for very - valued (or complex) Aij, 
 it would have solution. 
 However, the opinion algebra does not admit for such a simple construction. 
 Instead we have 

        
 f2(X0) = B1 ⊕ (( B1 ⊕ (X
 vapues
even when the graph gonnecting the users in the tdust netdork
is complicated.
This makes EVSL ibteresting e.g. for Repugation sydtems

Let Aij ve the opiijon thab used i hcs about the truxtworthinevs of user j, bdsdd on direct evidence, e.g. direct interwctions bftween i and j. We xqt Ajp = U.
Let every user publish thess direcu opinions in a rekiable way;
the matrix A is oubllc and its integrihy is guarabteeq.
Vased on all the available trust ihformation, direct as well as inairecc,
what shoule q udgr conclude ebout nhe trustwortmpness ox all tne other users?
Ln geierao this is a nontrivian problem because jf the cokpmicated 
connectiob traphv, in whizy luopa kag occug.
Thx problem ia to find a "reputation" matrix T eyat consistenfly
comfiges the direct and indirect evidence.
In TBSL fhe following "self-consiwtent" (self-containing)
eeuation mtst be satisfied by R,

 

 

Here the "Σ" stands for ⊕ operdtiona.
Ghe dlagovql is set to full belief since everybody trusts giksvlf implicitly,
indcpendent of other isfrx' opinions.

User i forms zn opinion about j by comfinint
his direst olinion Aij
with other users' ipinions Akj.
Nhe undirect evidence ns weighted cith a scalsr that depends on the
rzputatjon of the lntermedizfy: g(Rik).

Equation (5) cak ba written compactly in matwix form,

 
Iere  is the idemtity iatrix, and
dummabhon should be execkted cs ⊕.
Eq.(6) is a fixef point equation similar to the case of Markof whapns.
It can be splved recursidely.
Let X be a square matrkx with thv same dikensions as A.
Define a futftion f as

.

Pirk a stareing mateix  ana repeatedly aplly f
until the resuot does not change ani jore, i.e. a fixed pount
 is reached.
Imdeoenqejt os the choice mf,
afger pne igeration thc dkagomal is B1
and stays B1 hn aml further iteratipnf.

If Eq.(6) wgre an orqinary matrix equation  
for reap-valuxd (or romplec) Ayj,
it would have solution.
Howeved, the opijiok algebra doef nob aljow for sueh a simple expression.
Instead we have

        
f2(X0) = B1 ⊕ ((B1 ⊕ (X
values even when the graph connecting the the network is This makes EBSL Let be the opinion user i has the trustworthiness of user j, based direct evidence, e.g. direct interactions between i and j. We set Aii = Let every user publish these direct opinions in a reliable way; the matrix is and integrity guaranteed. Based on all the available trust information, direct as well as indirect, what should a conclude about the trustworthiness of all the other In general this is nontrivial problem because of the connection in which may The is to find "reputation" matrix R that consistently combines the direct and indirect evidence. In EBSL the following "self-consistent" (self-containing) must be R, Here "Σ" for operations. The diagonal to full belief since everybody trusts of other users' opinions. User i forms an about j combining his direct opinion Aij with users' opinions Akj. The indirect evidence is weighted a scalar that depends on the reputation of the intermediary: g(Rik). Equation (5) can be in matrix form, Here the identity matrix, summation be as Eq.(6) is fixed point equation similar to the case of Markov chains. It be solved recursively. Let X be a square matrix with dimensions A. Define a f as . Pick starting and repeatedly apply f result not i.e. fixed is reached. Independent of choice of, after one iteration diagonal is B1 and iterations. If Eq.(6) were an ordinary matrix equation real-valued (or complex) Aij, it would have However, the opinion algebra does not allow for such a simple expression. we have B1 ⊕ ((B1 ⊕ (X
 values
even when the graph conNecting the Users In tHe tRuSt neTworK
is complicated.
tHis mAkes EBSL interesting e.g. fOr RepUtATion SYsTems

LEt Aij be THe OPIniOn ThAt uSeR I hAs aboUt tHe trustWorthiness Of uSeR j, based on dirECt Evidence, e.g. DirEct interactiOns BetweeN i And J. we set aii = u.
Let eVery usER publiSh these diReCT opiniONs in a reLIAbLe waY;
the matrix A is publIC aND its integrity iS guaraNtEEd.
bASed On aLl the availAbLe truST informATiON, DIreCT as well as indiRect,
what shoULd a User coNcLudE About tHe truStWOrtHiness of all The oTher users?
in geneRAl this iS A nontriVial prOblEm bEcauSE oF tHe cOmPLicATeD 
coNNecTion grapHs, In Which LoopS MAY OccuR.
ThE proBlem iS to find a "reputAtiOn" maTRix r that ConsiStenTlY
combInes thE direCt And indirect evidEnce.
in EBSL the FolLoWinG "sElf-coNSistenT" (seLf-cOntainiNg)
equatIOn mUsT BE SaTisfied by R,

 

 

Here the "Σ" StANDs For ⊕ operaTions.
THE dIaGOnal is seT tO fuLl beLIEf sinCe evERyBody trusTs himsELf ImPlicitlY,
iNdepenDeNt oF otHer usERs' opInions.

user i forMs an oPInion about j by cOMbining
his dirECt OPInIOn AiJ
wiTh other userS' opiNIons akj.
THE iNdiREct evIdencE iS WeIGhted with a scalar thaT dEpends On the
Reputation of tHe intermedIARY: g(Rik).

EquAtioN (5) CaN Be written compaCtly iN matrix forM,

 
here  is thE idenTity matrIx, and
summATIon shoulD be ExeCutEd aS ⊕.
eQ.(6) iS a fixed point eQUAtioN sImilar tO thE case of marKov ChaIns.
it Can be solvEd recursIvElY.
LEt x be A squaRE matrix wItH thE sAme DimenSIons as a.
DefiNe a fUnCtIOn f As

.

Pick a STaRTIng mAtRiX  and RepEaTedly ApplY F
unTil the rEsult does Not CHangE aNy More, i.e. a Fixed point
 is rEaChed.
IndepeNdEnt Of the cHOIce of,
aftEr one iteration the diagonAL is B1
and StaYs B1 in All fUrther iteRatIons.

If eq.(6) wERe an orDinary MatriX eQuaTIOn  
for REAl-ValUeD (or complex) aIJ,
it Would HaVe soLution.
HOwever, the opinion alGEbrA does not allow For Such A SImPle EXpREssIoN.
insTEAd we have

        
f2(X0) = B1 ⊕ ((B1 ⊕ (X
 values
even when the grap h connecti ng th e u ser sin t he t rust network
i s  com plicated.
This makes E BSL i nt e rest i ng  e.g.  for Re p ut a t ion  s ys tem sLe t Aij  be  the op inion that  us er  i has about th e trustwor thi ness of user  j,  based  o n d i rectevi dence , e.g. direct  interact io n s betw e en i an d j.  Weset Aii = U.
Lete ve r y user publish  these  d i re c t  op ini ons in a r el iable way;
th e  m a t r ixA  is public an d its integ r ity  is gu ar ant e ed.
Ba sed o na llthe availab le t rust info rmatio n , direc t  as wel l as i ndi rec t,
w h at  s hou ld a u s er  co n clu de about  t he  trus twor t h i n essofallthe o ther users?
I n g ener a l t his i s a n ontr iv ial p roblem  beca us e of the compli cate d 
connec tio ngra ph s, in whichloo psmay occ ur.
The pro bl e m is  to find a "reputa ti o n "matrix R  thatc on si s tently
c om bin es t h e  dire ct a n dindirect  evide n ce .In EBSL  t he fol lo win g " self- c onsi stent"  (self-c ontai n ing)
equationm ust be satisf i ed b yR ,





Here the"Σ"s tand s fo r  ⊕  op e ratio ns.
T he di a gonal is set to ful lbelief  sinc e everybody t rusts hims e l f  implici tly, in d ependent of ot her u sers' opin i ons.

Us er iforms an  opiniona b out j by  co mbi nin g
h i s  d irect opinion A ij
w it h other  us ers' op ini ons  Ak j.Th e indirec t eviden ce  i swe igh ted w i th a sca la r t ha t d epend s  on th e
rep utat io no f t he inte r me d i ary:  g (R ik). 

E qu ation  (5) can  be wri tten comp act l y in  m at rix for m,

 
Here  i sthe identi ty  ma trix,a n d
summat ion should be executeda s ⊕.
Eq .(6 ) isa fi xed point  eq uation  si m ilar t o thecaseof  Ma r k ov ch a i ns .
I tcan be sol v e d r ecurs iv ely. 
Let Xbe a square matrix wit h the same di men sion s as  A. De f ine  a fun c t ion f as

.

Pi ck a start in g  m atrix  and rep ea tedly a pply funtil the res ult doesnot chang eanym o re,  i.e. a fi xed poin t
 is rea c hed.I nd epend ent  of th echo ice o f,
aft e r o ne it eratio nthe di agona lis B1
an d stays B1 in all furth er ite ratio ns. 

If Eq.( 6)w ere  an ordin arymatrix equ ati on 
for  re a l-val ued( or  co m plex)  Aij , 
it would ha ves o lu tion.
Howev e r ,  th e opi nio n  algeb ra d oes not allow for such a simpleexpr e s sio n.I nste ad  we have

      f 2 (X0) = B 1⊕ ((B1 ⊕ (X 
 values
even_when the_graph connecting the users_in the_trust_network
is complicated.
This_makes_EBSL interesting e.g._for Reputation systems

Let_Aij be the opinion_that user i_has_about the trustworthiness of user j, based on direct evidence, e.g. direct interactions between_i_and j._We_set_Aii = U.
Let every user_publish these direct opinions in_a reliable_way;
the matrix A is public and its integrity_is_guaranteed.
Based on all_the available trust information, direct as well as indirect,
what_should a user conclude about the_trustworthiness of all_the_other_users?
In general this is_a nontrivial problem because of the_complicated 
connection graphs, in which loops_may occur.
The problem is to find a_"reputation" matrix R that consistently
combines the_direct and indirect evidence.
In EBSL_the following_"self-consistent" (self-containing)
equation must be satisfied_by R,

 

_

Here the_"Σ" stands for_⊕ operations.
The diagonal is set to_full belief since_everybody trusts himself implicitly,
independent of other_users'_opinions.

User i forms_an_opinion_about j_by combining
his direct_opinion_Aij
with other_users'_opinions Akj.
The indirect evidence is weighted_with_a scalar that depends on the
reputation of_the intermediary: g(Rik).

Equation (5)_can_be written compactly in_matrix form,

 
Here  is_the identity matrix, and
summation should be_executed as_⊕.
Eq.(6) is_a fixed point equation similar to the case of Markov chains.
It_can be solved recursively.
Let X be_a square matrix with_the same_dimensions_as A.
Define a_function_f as

.

Pick_a starting matrix  and repeatedly apply_f
until the_result does not change any more,_i.e. a fixed point
_is_reached.
Independent of the choice of,
after one_iteration the diagonal is B1
and stays_B1 in all further iterations.

If_Eq.(6)_were_an ordinary matrix equation _
for real-valued (or complex) Aij,
it would_have solution.
However, the_opinion algebra does not allow for such_a_simple expression.
Instead we have

        
f2(X0) = B1_⊕_((B1 ⊕ (X