W. Yuan et al Knowledge-Based Systems 23(2010)232-238 Recommender Coverage MTPD Fig. 6. The MAE of TARS: the MAE of the conventional TARS model can be any of the Fig. 7. The rating coverage and the recommender coverage of TARS: the coverages nine dots, while the rectangular dot represents the MAE of our proposed modeL f the conventional TArS model can be any of the nine dots, while the rectangular dots represent the coverages of our proposed model 4. 2. Experimental results dots shown in Fig. 7 represent the rating coverage and the recom- three aspects to verify its effectiveness. These three aspects inclugx mender coverage of our proposed model, in which [L is selected as examine he performance of our proposed TARS model the value of MTPD. The experimental results show that:(1)If the rating prediction accuracy, the rating prediction coverage and MTPD is set to be smaller than our suggested value, both the rating the computational complexity The data used for simulations are coverage and the recommender coverage of TARS decrease, in those shown in Section 4.1 which the recommender coverage decreases significantly. (2)If The rating prediction accuracy is measured by the error of the MTPD is set to be greater than our suggested value, the rating co predicted ratings of TARS. Specifically, we calculate the mean abso- erage and the recommender coverage of TARS do not significantly lute error(MAe)since it is very appropriate and useful for evaluat ing prediction accuracy in offline tests [1]. To calculate MAE, the coverage of Tars are both very high, more than 99%, by using our predicted rating is compared with the real rating and the difference ggested value of MTPD in absolute value)is the prediction error, this error is then aver- The computational complexity of constructing the trust net ged over all predictions to obtain the overall MAe. By predicting work for TARS is o(k ma ), as mentioned in Section 2. Therefore, if the rating on each rated item of our explored experimental data, MTPD (represented by dmax )is set to be smaller than our suggested value, the computational complexity of constructing trust net we report the MAE of TARS with different values of MiPd in works for TARS is exponentially less expensive. On the other hand to choose the value of MTPD, its MAE can be any of the nine dots if MTPD is set to be greater than our suggested value, the compu- shown in Fig. 6. The rectangular dot shown in Fig. 6 represens er tional complexity of constructing trust networks for TARS is the MAE of our proposed model, in which [L l is selected as the lue of MTPD. The experimental results show that: (1)If MTPD is set To sum up, though setting MTPD smaller than our suggested va- to be smaller than our suggested value, the rating prediction accu- lue is computational less expensive, the rating prediction accuracy racy of TARS is getting worse. 2)If MTPD is set to be greater than and the rating prediction coverage of TARS are worse; while setting our suggested value, the rating prediction accuracy of TArs dose MTPD greater than our suggested value leads to similar rating pre- not significantly change. diction accuracy and similar rating prediction coverage of tar The coverage of TARS is measured by both the rating coverage but it is computational exponentially more expensive. We there- fore draw the conclusion that [LI is a good estimation of MTPD and the recommender coverage. The rating coverage is the portion which provides the maximum rating prediction coverage and the of items that TARS is able to predict, i.e., the portion of items that the active user can get at least one recommendation. However this tional complexity. This verifies the effectiveness of our proposed naximum rating prediction accuracy with the minimum computa- quantity is not always informative about the quality of TARS TARS is sometimes good on the rating coverage, but only involve small model portion of recommenders. This is because an item usually has a number of recommendations, so a good rating coverage does not 5 Conclusions and future work ecessaril a good coverage on the recommenders. Since it facilities the rating prediction by involving as many recommenda- Analyzing five trust networks obtained from the real online ions as possible in TARS, we introduce the term recommender sites, we verify that the trust network is the small-world network. coverage. The recommender coverage is the portion of recom- This means that it is able to build up the trust relationship between menders that could be involved in TARS. By using different value two randomly selected users of the trust network within limited of MTPD, the rating coverage and the recommender coverage of number of hops, and the average path length of the trust network ur explored experimental data are given in Fig. 7. Since the con is similar to that of the random network that has the same number entional TARS model did not mention how to choose the value of users and same number of edges per user as the trust network. of MTPD, its rating coverage and the recommender coverage can This verified small-world nature of the trust network can facilitate be any of the nine dots shown in the lines of Fig. 7. The rectangular its usage in various applications. In this paper, we use tarS as an4.2. Experimental results We examine the performance of our proposed TARS model in three aspects to verify its effectiveness. These three aspects include the rating prediction accuracy, the rating prediction coverage and the computational complexity. The data used for simulations are those shown in Section 4.1. The rating prediction accuracy is measured by the error of the predicted ratings of TARS. Specifically, we calculate the mean abso￾lute error (MAE) since it is very appropriate and useful for evaluat￾ing prediction accuracy in offline tests [1]. To calculate MAE, the predicted rating is compared with the real rating and the difference (in absolute value) is the prediction error, this error is then aver￾aged over all predictions to obtain the overall MAE. By predicting the rating on each rated item of our explored experimental data, we report the MAE of TARS with different values of MTPD in Fig. 6. Since the conventional TARS model did not mention how to choose the value of MTPD, its MAE can be any of the nine dots shown in Fig. 6. The rectangular dot shown in Fig. 6 represents the MAE of our proposed model, in which dL e is selected as the va￾lue of MTPD. The experimental results show that: (1) If MTPD is set to be smaller than our suggested value, the rating prediction accu￾racy of TARS is getting worse. (2) If MTPD is set to be greater than our suggested value, the rating prediction accuracy of TARS dose not significantly change. The coverage of TARS is measured by both the rating coverage and the recommender coverage. The rating coverage is the portion of items that TARS is able to predict, i.e., the portion of items that the active user can get at least one recommendation. However, this quantity is not always informative about the quality of TARS. TARS is sometimes good on the rating coverage, but only involve small portion of recommenders. This is because an item usually has a number of recommendations, so a good rating coverage does not necessarily imply a good coverage on the recommenders. Since it facilities the rating prediction by involving as many recommenda￾tions as possible in TARS, we introduce the term recommender coverage. The recommender coverage is the portion of recom￾menders that could be involved in TARS. By using different values of MTPD, the rating coverage and the recommender coverage of our explored experimental data are given in Fig. 7. Since the con￾ventional TARS model did not mention how to choose the value of MTPD, its rating coverage and the recommender coverage can be any of the nine dots shown in the lines of Fig. 7. The rectangular dots shown in Fig. 7 represent the rating coverage and the recom￾mender coverage of our proposed model, in which dLe is selected as the value of MTPD. The experimental results show that: (1) If MTPD is set to be smaller than our suggested value, both the rating coverage and the recommender coverage of TARS decrease, in which the recommender coverage decreases significantly. (2) If MTPD is set to be greater than our suggested value, the rating cov￾erage and the recommender coverage of TARS do not significantly change. This is because the rating coverage and the recommender coverage of TARS are both very high, more than 99%, by using our suggested value of MTPD. The computational complexity of constructing the trust net￾work for TARS is Oðk dmax Þ, as mentioned in Section 2. Therefore, if MTPD (represented by dmax) is set to be smaller than our suggested value, the computational complexity of constructing trust net￾works for TARS is exponentially less expensive. On the other hand, if MTPD is set to be greater than our suggested value, the compu￾tational complexity of constructing trust networks for TARS is exponentially more expensive. To sum up, though setting MTPD smaller than our suggested va￾lue is computational less expensive, the rating prediction accuracy and the rating prediction coverage of TARS are worse; while setting MTPD greater than our suggested value leads to similar rating pre￾diction accuracy and similar rating prediction coverage of TARS, but it is computational exponentially more expensive. We there￾fore draw the conclusion that dLe is a good estimation of MTPD which provides the maximum rating prediction coverage and the maximum rating prediction accuracy with the minimum computa￾tional complexity. This verifies the effectiveness of our proposed model. 5. Conclusions and future work Analyzing five trust networks obtained from the real online sites, we verify that the trust network is the small-world network. This means that it is able to build up the trust relationship between two randomly selected users of the trust network within limited number of hops, and the average path length of the trust network is similar to that of the random network that has the same number of users and same number of edges per user as the trust network. This verified small-world nature of the trust network can facilitate its usage in various applications. In this paper, we use TARS as an 1 2 3 4 5 6 7 8 9 0.7 0.75 0.8 0.85 MTPD MAE Fig. 6. The MAE of TARS: the MAE of the conventional TARS model can be any of the nine dots, while the rectangular dot represents the MAE of our proposed model. 1 2 3 4 5 6 7 8 9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 MTPD Coverage Rating Coverage Recommender Coverage Fig. 7. The rating coverage and the recommender coverage of TARS: the coverages of the conventional TARS model can be any of the nine dots, while the rectangular dots represent the coverages of our proposed model. W. 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