112 G. Pitsilis et al To measure the accuracy we calculated the Mean Absolute Error between the di- ectly calculated similarity s and the one derived from the transitive trust S.We use the following formula in which Cmax and Cmin are the maximum/minimum val ues of the Correlation Coefficient(I and -1 respectively) MAE (5.1) The evaluation algorithm can be described in pseudo-code as in fig. 3. Let us call dt j the direct trust between entities i and j and iti j the indirect one. Assuming that j is within 2 hops of i in the constructed trust graph, the indirect trust of i for j can be calculated using subjective logic in two steps: First, the derived trust of every alter- native path that begins from i and ends to j is calculated separately as a transitive lationship using the suggestion operator & Then all the values of the alternative paths along with dTi,j are combined together using the consensus operator e which gives the value of iTij. In general the consensus is expressed in the following for mula where A and B are two different agents which hold about the statement p re- spectively the opinions @n ando Let be the set of all users Let r be the set of all ratings over items Let R,cR be the set of the ratings of some user u k/m/会D s Cardinality of set of ratings of user i Let ecR The set of ratings of user i Let Mcki: pEM, EpCR and E, nE2 10 For j in M do p has 10 common ratings S←CC(,) Pearson' s similarity李 T←iast(,j Derived Indirect trust s S←f(T) Derived Similarity from our formula s s Absolute mean error value s MAE← End For j End for i Fig. 3. The evaluation algorithm The consensus opinion held by an imaginary agent A, B is o=o2={b2,d2,u (5.2) More about this can be found in[20]. In our particular case the statement p is th trustworthiness of the target j. A, B represent the alternative paths from i to The algorithm for calculating the indirect trust between the origin i and the target j is shown in figure 4To measure the accuracy we calculated the Mean Absolute Error between the di￾rectly calculated similarity S and the one derived from the transitive trust S’. We use the following formula in which Cmax and Cmin are the maximum/minimum val￾ues of the Correlation Coefficient (1 and -1 respectively): max min ' CC SS MAE   (5.1) The evaluation algorithm can be described in pseudo-code as in fig. 3. Let us call dti,,j the direct trust between entities i and j and iti,,j the indirect one. Assuming that j is within 2 hops of i in the constructed trust graph, the indirect trust of i for j can be calculated using subjective logic in two steps: First, the derived trust of every alter￾native path that begins from i and ends to j is calculated separately as a transitive re￾lationship using the suggestion operator … . Then all the values of the alternative paths along with dTi,,j are combined together using the consensus operator † which gives the value of iTi,,j. In general the consensus is expressed in the following for￾mula where A and B are two different agents which hold about the statement p re￾spectively the opinions A Z p and B Z p . Fig. 3. The evaluation algorithm The consensus opinion held by an imaginary agent A,B is: },,{ , ,,, BA p BA p BA p B p A p BA p ZZZ † udb (5.2) More about this can be found in [20]. In our particular case the statement p is the trustworthiness of the target j. A, B represent the alternative paths from i to j. The algorithm for calculating the indirect trust between the origin i and the target j is shown in figure 4. Let K be the set of all users Let R be the set of all ratings over items Let RuR be the set of the ratings of some user u Let KiK : Ru t10 * Cardinality of set of ratings of user i * For i in Ki Let Ei R * The set of ratings of user i * Let MKi : pM , EpR and ˆ EE pi t 10 For j in M do * p has 10 common ratings with i m CCS ji ),( * Pearson’s similarity * m iTrustT ji ),( * Derived Indirect trust * )( ' m TfS * Derived Similarity from our formula f * max min ' CC SS MAE   m * Absolute Mean Error value * End For j End For i Average(MAE) 112 G. Pitsilis et al